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API Reference Overview

The Monet Stats API provides a comprehensive collection of statistical metrics and utilities for atmospheric sciences applications. This reference covers all available functions, their parameters, return values, and use cases.

API Structure

Monet Stats is organized into several functional modules:

Core Modules

Import Conventions

Standard Imports

# Import entire library
import monet_stats as ms

# Import specific modules
from monet_stats import contingency_metrics, correlation_metrics

# Import specific functions
from monet_stats import R2, RMSE, POD, FAR

import monet_stats as ms
import numpy as np
import xarray as xr

Data Format Support

NumPy Arrays

import numpy as np

obs = np.array([1, 2, 3, 4, 5])
mod = np.array([1.1, 2.1, 2.9, 4.1, 4.8])

r2 = ms.R2(obs, mod)  # Works with 1D arrays
rmse = ms.RMSE(obs, mod)

Multi-dimensional Arrays

# 2D arrays (e.g., spatial fields)
obs_2d = np.random.normal(20, 2, (50, 50))
mod_2d = obs_2d + np.random.normal(0, 1, (50, 50))

fss = ms.FSS(obs_2d, mod_2d, window=5)

Pandas DataFrames

import pandas as pd

df = pd.DataFrame({
    'observed': np.random.normal(20, 2, 100),
    'modeled': np.random.normal(20.5, 2.5, 100),
    'station': ['A'] * 50 + ['B'] * 50
})

# Apply metrics by group
results = df.groupby('station').apply(
    lambda x: pd.Series({
        'RMSE': ms.RMSE(x['observed'], x['modeled']),
        'R2': ms.R2(x['observed'], x['modeled'])
    })
)

XArray DataArrays

import xarray as xr

obs_da = xr.DataArray(
    np.random.normal(20, 2, (10, 10, 365)),
    dims=['lat', 'lon', 'time'],
    coords={
        'lat': range(10),
        'lon': range(10),
        'time': pd.date_range('2020-01-01', periods=365, freq='D')
    }
)

mod_da = obs_da + xr.DataArray(
    np.random.normal(0, 1, (10, 10, 365)),
    dims=['lat', 'lon', 'time'],
    coords=obs_da.coords
)

# Metrics preserve coordinates and dimensions
skill = ms.R2(obs_da, mod_da)  # Returns DataArray with same coordinates

Common Parameters

Core Parameters

Most metrics accept these common parameters:

  • obs: Observed values (array-like)
  • mod: Modeled/predicted values (array-like)
  • axis: Axis along which to compute metrics (int, optional)
  • nan_policy: How to handle NaN values ('omit', 'propagate', 'raise')

Threshold Parameters

Many metrics use threshold parameters for categorical analysis:

  • minval: Minimum threshold for event definition
  • maxval: Maximum threshold for event definition (optional)

Spatial Parameters

Spatial metrics often include:

  • window: Size of spatial window (int)
  • threshold: Event threshold for spatial analysis

Return Value Types

Scalar Values

Most metrics return single scalar values:

r2 = ms.R2(obs, mod)  # float
rmse = ms.RMSE(obs, mod)  # float

Arrays

Some metrics return arrays for multi-dimensional input:

# For 2D spatial data
fss = ms.FSS(obs_2d, mod_2d)  # float

DataArrays (xarray)

When using xarray inputs, metrics return DataArrays:

skill = ms.R2(obs_da, mod_da)  # DataArray with coordinates

Error Handling

Data Shape Validation

try:
    result = ms.R2(obs_1d, mod_2d)  # Will raise ValueError
except ValueError as e:
    print(f"Shape mismatch: {e}")

NaN Handling

# Data with NaN values
obs_with_nan = np.array([1, 2, np.nan, 4])
mod_with_nan = np.array([1.1, 2.1, 3.1, 4.1])

# Functions automatically handle NaN by default
rmse = ms.RMSE(obs_with_nan, mod_with_nan)  # Uses valid pairs only

Type Validation

# Invalid types will raise TypeError
try:
    result = ms.R2("invalid", "data")  # TypeError
except TypeError as e:
    print(f"Invalid data type: {e}")

Performance Considerations

Vectorized Operations

All metrics use NumPy and Xarray vectorized operations for optimal performance. Loop-free implementations ensure maximum speed on modern hardware.

Out-of-Core Processing with Dask

For datasets larger than RAM, monet-stats is fully compatible with Dask. Most metrics are "lazy-aware" and will preserve the Dask computation graph.

# Open large dataset with chunks (Aero Protocol recommended)
ds = xr.open_dataset("large_data.nc", chunks={"time": "auto", "lat": 100, "lon": 100})
obs = xr.open_dataset("obs_data.nc", chunks={"time": "auto", "lat": 100, "lon": 100})

# Metrics stay lazy and don't trigger loading
skill = ms.RMSE(obs.var, ds.var, axis="time")

# Execution only happens on compute() or plotting
result = skill.compute()

Scientific Provenance

When using Xarray DataArrays, monet-stats automatically updates the attrs['history'] to track which statistical operations were applied to the data, ensuring scientific reproducibility.

Example Usage Patterns

Basic Error Analysis

import monet_stats as ms
import numpy as np

# Sample data
obs = np.array([1.0, 2.5, 3.2, 4.8, 5.0])
mod = np.array([1.2, 2.3, 3.5, 4.6, 5.2])

# Error metrics
error_analysis = {
    'RMSE': ms.RMSE(obs, mod),
    'MAE': ms.MAE(obs, mod),
    'MB': ms.MB(obs, mod),
    'NMB': ms.NMB(obs, mod),
    'NME': ms.NME(obs, mod)
}

Comprehensive Model Evaluation

def evaluate_model(observed, modeled):
    """Comprehensive model evaluation suite"""

    metrics = {
        # Error measures
        'RMSE': ms.RMSE(observed, modeled),
        'MAE': ms.MAE(observed, modeled),
        'MB': ms.MB(observed, modeled),
        'NMB': ms.NMB(observed, modeled),

        # Skill scores
        'R2': ms.R2(observed, modeled),
        'NSE': ms.NSE(observed, modeled),
        'KGE': ms.KGE(observed, modeled),
        'IOA': ms.IOA(observed, modeled),

        # Relative measures
        'MPE': ms.MPE(observed, modeled),
        'NME': ms.NME(observed, modeled)
    }

    return metrics

# Usage
results = evaluate_model(obs, mod)
for metric, value in results.items():
    print(f"{metric}: {value:.4f}")

Categorical Event Analysis

# Binary event analysis
obs_events = np.array([0, 1, 1, 0, 1, 0, 1, 1, 0, 0])
mod_events = np.array([0, 1, 0, 0, 1, 1, 1, 0, 0, 1])

# Contingency table metrics
contingency_metrics = {
    'POD': ms.POD(obs_events, mod_events, threshold=0.5),
    'FAR': ms.FAR(obs_events, mod_events, threshold=0.5),
    'CSI': ms.CSI(obs_events, mod_events, threshold=0.5),
    'HSS': ms.HSS(obs_events, mod_events, threshold=0.5),
    'ETS': ms.ETS(obs_events, mod_events, threshold=0.5)
}

API Reference

The following sections provide auto-generated documentation for each core module based on docstrings.

Contingency Metrics

BSS_binary(obs, mod, threshold, axis=None)

Binary Brier Skill Score for deterministic forecasts.

Typical Use Cases

  • Evaluating the accuracy of deterministic binary forecasts (e.g., precipitation yes/no).
  • Used in meteorology and environmental modeling to assess forecast skill relative to a reference.

Typical Values and Range

  • Range: -∞ to 1
  • 1: Perfect forecast
  • 0: Same skill as reference forecast
  • Negative: Worse than reference forecast

Parameters

obs : numpy.ndarray or xarray.DataArray Observed binary outcomes or continuous values. mod : numpy.ndarray or xarray.DataArray Forecast binary outcomes or continuous values. threshold : float Threshold value to convert continuous forecasts to binary. axis : int, str, or iterable of such, optional Axis along which to compute the score.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Binary Brier Skill Score.

Examples

import numpy as np from monet_stats.contingency_metrics import BSS_binary obs = np.array([0, 1, 1, 0]) mod = np.array([0, 1, 0, 0]) BSS_binary(obs, mod, threshold=0.5) 0.5

Source code in src/monet_stats/contingency_metrics.py 
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def BSS_binary(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    threshold: float,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Binary Brier Skill Score for deterministic forecasts.

    Typical Use Cases
    -----------------
    - Evaluating the accuracy of deterministic binary forecasts (e.g.,
      precipitation yes/no).
    - Used in meteorology and environmental modeling to assess forecast skill
      relative to a reference.

    Typical Values and Range
    ------------------------
    - Range: -∞ to 1
    - 1: Perfect forecast
    - 0: Same skill as reference forecast
    - Negative: Worse than reference forecast

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed binary outcomes or continuous values.
    mod : numpy.ndarray or xarray.DataArray
        Forecast binary outcomes or continuous values.
    threshold : float
        Threshold value to convert continuous forecasts to binary.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the score.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Binary Brier Skill Score.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import BSS_binary
    >>> obs = np.array([0, 1, 1, 0])
    >>> mod = np.array([0, 1, 0, 0])
    >>> BSS_binary(obs, mod, threshold=0.5)
    0.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_binary = (obs >= threshold).astype(float)
        mod_binary = (mod >= threshold).astype(float)

        bs = ((mod_binary - obs_binary) ** 2).mean(dim=dim)
        obs_clim = obs_binary.mean(dim=dim)
        bs_ref = ((obs_clim - obs_binary) ** 2).mean(dim=dim)

        result = xr.where(bs_ref > 0, 1.0 - (bs / bs_ref), 0.0)
        history = f"BSS_binary computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_binary = (np.asarray(obs) >= threshold).astype(float)
        mod_binary = (np.asarray(mod) >= threshold).astype(float)

        bs = np.nanmean((mod_binary - obs_binary) ** 2, axis=axis)
        obs_clim = np.nanmean(obs_binary, axis=axis)
        if axis is not None:
            # Need to keep dims for subtraction
            obs_clim_kd = np.nanmean(obs_binary, axis=axis, keepdims=True)
        else:
            obs_clim_kd = obs_clim
        bs_ref = np.nanmean((obs_clim_kd - obs_binary) ** 2, axis=axis)

        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(bs_ref > 0, 1.0 - (bs / bs_ref), 0.0)
            return result.item() if np.ndim(result) == 0 else result

CSI(obs, mod, minval, maxval=None, axis=None)

Critical Success Index (CSI).

Typical Use Cases

  • Evaluating forecast skill for rare or binary events (e.g., precipitation, air quality exceedances).
  • Used in meteorology and environmental modeling to assess event prediction accuracy.

Typical Values and Range

  • Range: 0 to 1
  • 1: Perfect forecast
  • 0: No skill (no correct predictions)

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Modeled values. minval : float Minimum threshold value for event detection. maxval : float, optional Maximum threshold value for event detection. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray CSI value for the given threshold.

Examples

import numpy as np from monet_stats.contingency_metrics import CSI obs = np.array([1, 0, 1, 0]) mod = np.array([1, 1, 0, 0]) CSI(obs, mod, minval=0.5) 0.3333333333333333

Source code in src/monet_stats/contingency_metrics.py 
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def CSI(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Critical Success Index (CSI).

    Typical Use Cases
    -----------------
    - Evaluating forecast skill for rare or binary events (e.g., precipitation,
      air quality exceedances).
    - Used in meteorology and environmental modeling to assess event prediction
      accuracy.

    Typical Values and Range
    ------------------------
    - Range: 0 to 1
    - 1: Perfect forecast
    - 0: No skill (no correct predictions)

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Modeled values.
    minval : float
        Minimum threshold value for event detection.
    maxval : float, optional
        Maximum threshold value for event detection.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        CSI value for the given threshold.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import CSI
    >>> obs = np.array([1, 0, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> CSI(obs, mod, minval=0.5)
    0.3333333333333333
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    denom = a + b + c
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, a / denom, np.nan)
        history = f"CSI computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, a / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

ETS(obs, mod, minval, maxval=None, axis=None)

Equitable Threat Score (ETS).

Typical Use Cases

  • Evaluating forecast skill for rare events (e.g., precipitation, air quality exceedances).
  • Used in meteorology and environmental modeling to assess binary event prediction accuracy.

Typical Values and Range

  • Range: -1/3 to 1
  • 1: Perfect forecast
  • 0: No skill (random forecast)
  • Negative values: Worse than random

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Modeled values. minval : float Minimum threshold value for event detection. maxval : float, optional Maximum threshold value for event detection. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray ETS value for the given threshold.

Examples

import numpy as np from monet_stats.contingency_metrics import ETS obs = np.array([1, 0, 1, 0]) mod = np.array([1, 1, 0, 0]) ETS(obs, mod, minval=0.5) -0.2

Source code in src/monet_stats/contingency_metrics.py 
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def ETS(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Equitable Threat Score (ETS).

    Typical Use Cases
    -----------------
    - Evaluating forecast skill for rare events (e.g., precipitation, air quality
      exceedances).
    - Used in meteorology and environmental modeling to assess binary event
      prediction accuracy.

    Typical Values and Range
    ------------------------
    - Range: -1/3 to 1
    - 1: Perfect forecast
    - 0: No skill (random forecast)
    - Negative values: Worse than random

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Modeled values.
    minval : float
        Minimum threshold value for event detection.
    maxval : float, optional
        Maximum threshold value for event detection.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        ETS value for the given threshold.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import ETS
    >>> obs = np.array([1, 0, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> ETS(obs, mod, minval=0.5)
    -0.2
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    total = a + b + c + d
    random_hits = ((a + b) * (a + c)) / total
    denom = a + b + c - random_hits
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, (a - random_hits) / denom, np.nan)
        history = f"ETS computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, (a - random_hits) / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

ETS_max_threshold(obs, mod, minval_range, maxval_range, step_size=1.0)

Find the threshold that maximizes the Equitable Threat Score (ETS) over a range.

Typical Use Cases

  • Finding the optimal threshold for binary classification in meteorological or environmental modeling.
  • Used to optimize event detection thresholds in forecast systems.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval_range : float Minimum value of threshold range to test. maxval_range : float Maximum value of threshold range to test. step_size : float, optional Step size for testing thresholds. Default is 1.0.

Returns

optimal_threshold : float Threshold value that maximizes ETS. max_ets : float Maximum ETS value achieved.

Examples

import numpy as np obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5]) ETS_max_threshold(obs, mod, 1, 5, 0.5) (2.5, 1.0)

Source code in src/monet_stats/contingency_metrics.py 
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def ETS_max_threshold(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval_range: float,
    maxval_range: float,
    step_size: float = 1.0,
) -> Tuple[float, float]:
    """
    Find the threshold that maximizes the Equitable Threat Score (ETS) over a range.

    Typical Use Cases
    -----------------
    - Finding the optimal threshold for binary classification in meteorological
      or environmental modeling.
    - Used to optimize event detection thresholds in forecast systems.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval_range : float
        Minimum value of threshold range to test.
    maxval_range : float
        Maximum value of threshold range to test.
    step_size : float, optional
        Step size for testing thresholds. Default is 1.0.

    Returns
    -------
    optimal_threshold : float
        Threshold value that maximizes ETS.
    max_ets : float
        Maximum ETS value achieved.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5])
    >>> ETS_max_threshold(obs, mod, 1, 5, 0.5)
    (2.5, 1.0)
    """
    thresholds = np.arange(minval_range, maxval_range, step_size)
    ets_values = []

    for threshold in thresholds:
        ets_val = ETS(obs, mod, threshold)
        if isinstance(ets_val, xr.DataArray):
            ets_val = ets_val.values.item()
        ets_values.append(ets_val)

    # Find the threshold that gives the maximum ETS
    max_idx = np.nanargmax(ets_values)
    optimal_threshold = thresholds[max_idx]
    max_ets = ets_values[max_idx]

    return float(optimal_threshold), float(max_ets)

FAR(obs, mod, minval, maxval=None, axis=None)

False Alarm Rate (FAR) for a given event threshold.

Typical Use Cases

  • Evaluating the frequency of false alarms in categorical forecasts (e.g., precipitation, air quality events).
  • Used in meteorology and environmental modeling to assess forecast reliability.

Typical Values and Range

  • Range: 0 to 1
  • 0: No false alarms (perfect reliability)
  • 1: All alarms are false (no reliability)

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval : float Minimum event threshold. maxval : float, optional Maximum event threshold. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray False alarm rate.

Examples

import numpy as np from monet_stats.contingency_metrics import FAR obs = np.array([0, 1, 1, 0]) mod = np.array([1, 1, 0, 0]) FAR(obs, mod, minval=0.5) 0.5

Source code in src/monet_stats/contingency_metrics.py 
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def FAR(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    False Alarm Rate (FAR) for a given event threshold.

    Typical Use Cases
    -----------------
    - Evaluating the frequency of false alarms in categorical forecasts (e.g.,
      precipitation, air quality events).
    - Used in meteorology and environmental modeling to assess forecast
      reliability.

    Typical Values and Range
    ------------------------
    - Range: 0 to 1
    - 0: No false alarms (perfect reliability)
    - 1: All alarms are false (no reliability)

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval : float
        Minimum event threshold.
    maxval : float, optional
        Maximum event threshold.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        False alarm rate.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import FAR
    >>> obs = np.array([0, 1, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> FAR(obs, mod, minval=0.5)
    0.5
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    denom = a + c
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, c / denom, np.nan)
        history = f"FAR computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, c / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

FAR_min_threshold(obs, mod, minval_range, maxval_range, step_size=1.0)

Find the threshold that minimizes the False Alarm Rate (FAR) over a range.

Typical Use Cases

  • Finding the optimal threshold for minimizing false alarms in meteorological or environmental modeling.
  • Used to optimize event detection thresholds in forecast systems.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval_range : float Minimum value of threshold range to test. maxval_range : float Maximum value of threshold range to test. step_size : float, optional Step size for testing thresholds. Default is 1.0.

Returns

optimal_threshold : float Threshold value that minimizes FAR. min_far : float Minimum FAR value achieved.

Examples

import numpy as np obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5]) FAR_min_threshold(obs, mod, 1, 5, 0.5) (2.5, 0.0)

Source code in src/monet_stats/contingency_metrics.py 
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def FAR_min_threshold(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval_range: float,
    maxval_range: float,
    step_size: float = 1.0,
) -> Tuple[float, float]:
    """
    Find the threshold that minimizes the False Alarm Rate (FAR) over a range.

    Typical Use Cases
    -----------------
    - Finding the optimal threshold for minimizing false alarms in meteorological
      or environmental modeling.
    - Used to optimize event detection thresholds in forecast systems.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval_range : float
        Minimum value of threshold range to test.
    maxval_range : float
        Maximum value of threshold range to test.
    step_size : float, optional
        Step size for testing thresholds. Default is 1.0.

    Returns
    -------
    optimal_threshold : float
        Threshold value that minimizes FAR.
    min_far : float
        Minimum FAR value achieved.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5])
    >>> FAR_min_threshold(obs, mod, 1, 5, 0.5)
    (2.5, 0.0)
    """
    thresholds = np.arange(minval_range, maxval_range, step_size)
    far_values = []

    for threshold in thresholds:
        far_val = FAR(obs, mod, threshold)
        if isinstance(far_val, xr.DataArray):
            far_val = far_val.values.item()
        far_values.append(far_val)

    # Find the threshold that gives the minimum FAR
    min_idx = np.nanargmin(far_values)
    optimal_threshold = thresholds[min_idx]
    min_far = far_values[min_idx]

    return float(optimal_threshold), float(min_far)

FBI(obs, mod, minval, maxval=None, axis=None)

Frequency Bias Index (FBI) for a given event threshold.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval : float Minimum event threshold. maxval : float, optional Maximum event threshold. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

fbi : numpy.number, numpy.ndarray, or xarray.DataArray Frequency bias index.

Examples

import numpy as np from monet_stats.contingency_metrics import FBI obs = np.array([0, 1, 1, 0]) mod = np.array([1, 1, 0, 0]) FBI(obs, mod, minval=0.5) 1.0

Source code in src/monet_stats/contingency_metrics.py 
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def FBI(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Frequency Bias Index (FBI) for a given event threshold.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval : float
        Minimum event threshold.
    maxval : float, optional
        Maximum event threshold.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    fbi : numpy.number, numpy.ndarray, or xarray.DataArray
        Frequency bias index.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import FBI
    >>> obs = np.array([0, 1, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> FBI(obs, mod, minval=0.5)
    1.0
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    denom = a + b
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, (a + c) / denom, np.nan)
        history = f"FBI computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, (a + c) / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

HSS(obs, mod, minval, maxval=None, axis=None)

Heidke Skill Score (HSS).

Typical Use Cases

  • Evaluating categorical forecast skill (e.g., precipitation, air quality events).
  • Used in meteorology and environmental modeling to assess binary event prediction accuracy.

Typical Values and Range

  • Range: -∞ to 1
  • 1: Perfect forecast
  • 0: No skill (random forecast)
  • Negative values: Worse than random

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Modeled values. minval : float Minimum threshold value for event detection. maxval : float, optional Maximum threshold value for event detection. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray HSS value for the given threshold.

Examples

import numpy as np from monet_stats.contingency_metrics import HSS obs = np.array([1, 0, 1, 0]) mod = np.array([1, 1, 0, 0]) HSS(obs, mod, minval=0.5) -0.3333333333333333

Source code in src/monet_stats/contingency_metrics.py 
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def HSS(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Heidke Skill Score (HSS).

    Typical Use Cases
    -----------------
    - Evaluating categorical forecast skill (e.g., precipitation, air quality
      events).
    - Used in meteorology and environmental modeling to assess binary event
      prediction accuracy.

    Typical Values and Range
    ------------------------
    - Range: -∞ to 1
    - 1: Perfect forecast
    - 0: No skill (random forecast)
    - Negative values: Worse than random

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Modeled values.
    minval : float
        Minimum threshold value for event detection.
    maxval : float, optional
        Maximum threshold value for event detection.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        HSS value for the given threshold.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import HSS
    >>> obs = np.array([1, 0, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> HSS(obs, mod, minval=0.5)
    -0.3333333333333333
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    denom = (a + c) * (c + d) + (a + b) * (b + d)
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, 2 * (a * d - b * c) / denom, np.nan)
        history = f"HSS computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, 2 * (a * d - b * c) / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

HSS_max_threshold(obs, mod, minval_range, maxval_range, step_size=1.0)

Find the threshold that maximizes the Heidke Skill Score (HSS) over a range.

Typical Use Cases

  • Finding the optimal threshold for binary classification in meteorological or environmental modeling.
  • Used to optimize event detection thresholds in forecast systems.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval_range : float Minimum value of threshold range to test. maxval_range : float Maximum value of threshold range to test. step_size : float, optional Step size for testing thresholds. Default is 1.0.

Returns

optimal_threshold : float Threshold value that maximizes HSS. max_hss : float Maximum HSS value achieved.

Examples

import numpy as np obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5]) HSS_max_threshold(obs, mod, 1, 5, 0.5) (2.5, 1.0)

Source code in src/monet_stats/contingency_metrics.py 
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def HSS_max_threshold(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval_range: float,
    maxval_range: float,
    step_size: float = 1.0,
) -> Tuple[float, float]:
    """
    Find the threshold that maximizes the Heidke Skill Score (HSS) over a range.

    Typical Use Cases
    -----------------
    - Finding the optimal threshold for binary classification in meteorological
      or environmental modeling.
    - Used to optimize event detection thresholds in forecast systems.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval_range : float
        Minimum value of threshold range to test.
    maxval_range : float
        Maximum value of threshold range to test.
    step_size : float, optional
        Step size for testing thresholds. Default is 1.0.

    Returns
    -------
    optimal_threshold : float
        Threshold value that maximizes HSS.
    max_hss : float
        Maximum HSS value achieved.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5])
    >>> HSS_max_threshold(obs, mod, 1, 5, 0.5)
    (2.5, 1.0)
    """
    thresholds = np.arange(minval_range, maxval_range, step_size)
    hss_values = []

    for threshold in thresholds:
        hss_val = HSS(obs, mod, threshold)
        if isinstance(hss_val, xr.DataArray):
            hss_val = hss_val.values.item()
        hss_values.append(hss_val)

    # Find the threshold that gives the maximum HSS
    max_idx = np.nanargmax(hss_values)
    optimal_threshold = thresholds[max_idx]
    max_hss = hss_values[max_idx]

    return float(optimal_threshold), float(max_hss)

POD(obs, mod, minval, maxval=None, axis=None)

Probability of Detection (POD) for a given event threshold.

Typical Use Cases

  • Evaluating how well a model detects events above a critical threshold (e.g., pollution exceedances, precipitation events).
  • Used in contingency table analysis for categorical forecast verification.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval : float Minimum event threshold. maxval : float, optional Maximum event threshold. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

pod : numpy.number, numpy.ndarray, or xarray.DataArray Probability of detection.

Examples

import numpy as np from monet_stats.contingency_metrics import POD obs = np.array([0, 1, 1, 0]) mod = np.array([1, 1, 0, 0]) POD(obs, mod, minval=0.5) 0.5

Source code in src/monet_stats/contingency_metrics.py 
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def POD(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Probability of Detection (POD) for a given event threshold.

    Typical Use Cases
    -----------------
    - Evaluating how well a model detects events above a critical threshold
      (e.g., pollution exceedances, precipitation events).
    - Used in contingency table analysis for categorical forecast verification.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval : float
        Minimum event threshold.
    maxval : float, optional
        Maximum event threshold.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    pod : numpy.number, numpy.ndarray, or xarray.DataArray
        Probability of detection.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import POD
    >>> obs = np.array([0, 1, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> POD(obs, mod, minval=0.5)
    0.5
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    denom = a + b
    if isinstance(denom, xr.DataArray):
        result = xr.where(denom > 0, a / denom, np.nan)
        history = f"POD computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(denom > 0, a / denom, np.nan)
            return result.item() if np.ndim(result) == 0 else result

POD_max_threshold(obs, mod, minval_range, maxval_range, step_size=1.0)

Find the threshold that maximizes the Probability of Detection (POD) over a range.

Typical Use Cases

  • Finding the optimal threshold for maximizing detection rates in meteorological or environmental modeling.
  • Used to optimize event detection thresholds in forecast systems.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval_range : float Minimum value of threshold range to test. maxval_range : float Maximum value of threshold range to test. step_size : float, optional Step size for testing thresholds. Default is 1.0.

Returns

optimal_threshold : float Threshold value that maximizes POD. max_pod : float Maximum POD value achieved.

Examples

import numpy as np obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5]) POD_max_threshold(obs, mod, 1, 5, 0.5) (2.5, 1.0)

Source code in src/monet_stats/contingency_metrics.py 
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def POD_max_threshold(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval_range: float,
    maxval_range: float,
    step_size: float = 1.0,
) -> Tuple[float, float]:
    """
    Find the threshold that maximizes the Probability of Detection (POD) over a range.

    Typical Use Cases
    -----------------
    - Finding the optimal threshold for maximizing detection rates in
      meteorological or environmental modeling.
    - Used to optimize event detection thresholds in forecast systems.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval_range : float
        Minimum value of threshold range to test.
    maxval_range : float
        Maximum value of threshold range to test.
    step_size : float, optional
        Step size for testing thresholds. Default is 1.0.

    Returns
    -------
    optimal_threshold : float
        Threshold value that maximizes POD.
    max_pod : float
        Maximum POD value achieved.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.5, 2.5, 3.5, 4.5, 5.5])
    >>> POD_max_threshold(obs, mod, 1, 5, 0.5)
    (2.5, 1.0)
    """
    thresholds = np.arange(minval_range, maxval_range, step_size)
    pod_values = []

    for threshold in thresholds:
        pod_val = POD(obs, mod, threshold)
        if isinstance(pod_val, xr.DataArray):
            pod_val = pod_val.values.item()
        pod_values.append(pod_val)

    # Find the threshold that gives the maximum POD
    max_idx = np.nanargmax(pod_values)
    optimal_threshold = thresholds[max_idx]
    max_pod = pod_values[max_idx]

    return float(optimal_threshold), float(max_pod)

TSS(obs, mod, minval, maxval=None, axis=None)

Hanssen-Kuipers Discriminant (True Skill Statistic, TSS).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. minval : float Minimum event threshold. maxval : float, optional Maximum event threshold. axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

tss : numpy.number, numpy.ndarray, or xarray.DataArray True skill statistic.

Examples

import numpy as np from monet_stats.contingency_metrics import TSS obs = np.array([0, 1, 1, 0]) mod = np.array([1, 1, 0, 0]) TSS(obs, mod, minval=0.5) 0.0

Source code in src/monet_stats/contingency_metrics.py 
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def TSS(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Hanssen-Kuipers Discriminant (True Skill Statistic, TSS).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    minval : float
        Minimum event threshold.
    maxval : float, optional
        Maximum event threshold.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    tss : numpy.number, numpy.ndarray, or xarray.DataArray
        True skill statistic.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.contingency_metrics import TSS
    >>> obs = np.array([0, 1, 1, 0])
    >>> mod = np.array([1, 1, 0, 0])
    >>> TSS(obs, mod, minval=0.5)
    0.0
    """
    a, b, c, d = _contingency_table(obs, mod, minval, maxval, axis=axis)
    pod_denom = a + b
    pofd_denom = c + d

    if isinstance(pod_denom, xr.DataArray):
        pod = xr.where(pod_denom > 0, a / pod_denom, np.nan)
        pofd = xr.where(pofd_denom > 0, c / pofd_denom, np.nan)
        result = pod - pofd
        history = f"TSS computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        with np.errstate(divide="ignore", invalid="ignore"):
            pod = np.where(pod_denom > 0, a / pod_denom, np.nan)
            pofd = np.where(pofd_denom > 0, c / pofd_denom, np.nan)
            result = pod - pofd
            return result.item() if np.ndim(result) == 0 else result

scores(obs, mod, minval, maxval=None, axis=None)

Calculate scores using the _contingency_table.

Parameters

obs : numpy.ndarray or xarray.DataArray Observation values ("truth"). mod : numpy.ndarray or xarray.DataArray Model values ("prediction"). minval : float Minimum threshold for event. maxval : float, optional Maximum threshold for event. axis : int, str, or iterable of such, optional Axis along which to compute the scores.

Returns

a, b, c, d Counts of hits, misses, false alarms, and correct negatives.

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([1.5, 1.8, 3.2, 3.8]) a, b, c, d = scores(obs, mod, minval=2.5)

Source code in src/monet_stats/contingency_metrics.py 
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def scores(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    minval: float,
    maxval: Optional[float] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Tuple[
    Union[np.number, np.ndarray, xr.DataArray],
    Union[np.number, np.ndarray, xr.DataArray],
    Union[np.number, np.ndarray, xr.DataArray],
    Union[np.number, np.ndarray, xr.DataArray],
]:
    """
    Calculate scores using the _contingency_table.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observation values ("truth").
    mod : numpy.ndarray or xarray.DataArray
        Model values ("prediction").
    minval : float
        Minimum threshold for event.
    maxval : float, optional
        Maximum threshold for event.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the scores.

    Returns
    -------
    a, b, c, d
        Counts of hits, misses, false alarms, and correct negatives.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.5, 1.8, 3.2, 3.8])
    >>> a, b, c, d = scores(obs, mod, minval=2.5)
    """
    return _contingency_table(obs, mod, minval, maxval, axis=axis)

Correlation Metrics

Correlation and Agreement Metrics for Model Evaluation

AC(obs, mod, axis=None)

Anomaly Correlation (AC).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Anomaly correlation coefficient (unitless, -1 to 1).

Examples

import numpy as np from monet_stats.correlation_metrics import AC obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 2.9, 4.1]) AC(obs, mod) 0.9922778767136677

Source code in src/monet_stats/correlation_metrics.py 
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def AC(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Anomaly Correlation (AC).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Anomaly correlation coefficient (unitless, -1 to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import AC
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 2.9, 4.1])
    >>> AC(obs, mod)
    0.9922778767136677
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_bar = obs.mean(dim=dim)
        mod_bar = mod.mean(dim=dim)
        obs_anom = obs - obs_bar
        mod_anom = mod - mod_bar
        p1 = (mod_anom * obs_anom).sum(dim=dim)
        p2 = ((mod_anom**2).sum(dim=dim) * (obs_anom**2).sum(dim=dim)) ** 0.5
        result = p1 / p2
        # Update history
        history = f"AC computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_bar = np.ma.mean(obs, axis=axis)
        mod_bar = np.ma.mean(mod, axis=axis)
        if axis is not None:
            # Need to keep dims for subtraction if axis is not None
            obs_bar_kd = np.ma.mean(obs, axis=axis, keepdims=True)
            mod_bar_kd = np.ma.mean(mod, axis=axis, keepdims=True)
        else:
            obs_bar_kd = obs_bar
            mod_bar_kd = mod_bar
        obs_anom = np.subtract(obs, obs_bar_kd)
        mod_anom = np.subtract(mod, mod_bar_kd)
        p1 = np.ma.sum(np.ma.multiply(mod_anom, obs_anom), axis=axis)
        p2 = np.ma.sqrt(np.ma.multiply(np.ma.sum(obs_anom**2, axis=axis), np.ma.sum(mod_anom**2, axis=axis)))
        return p1 / p2

CCC(obs, mod, axis=None)

Concordance Correlation Coefficient (CCC).

Typical Use Cases

  • Quantifying the agreement between model and observations, accounting for precision and accuracy.
  • Used in model evaluation to assess how well model predictions agree with observations.
  • Measures how far the values deviate from the line of perfect concordance (slope=1, intercept=0).

Typical Values and Range

  • Range: -1 to 1
  • 1: Perfect agreement between model and observations
  • 0: No agreement
  • -1: Perfect negative agreement

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the coefficient.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Concordance correlation coefficient (unitless, -1 to 1).

Examples

import numpy as np from monet_stats.correlation_metrics import CCC obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 2.9, 4.1]) CCC(obs, mod) 0.9984779299847792

Source code in src/monet_stats/correlation_metrics.py 
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def CCC(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Concordance Correlation Coefficient (CCC).

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, accounting for
      precision and accuracy.
    - Used in model evaluation to assess how well model predictions agree with
      observations.
    - Measures how far the values deviate from the line of perfect concordance
      (slope=1, intercept=0).

    Typical Values and Range
    ------------------------
    - Range: -1 to 1
    - 1: Perfect agreement between model and observations
    - 0: No agreement
    - -1: Perfect negative agreement

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the coefficient.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Concordance correlation coefficient (unitless, -1 to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import CCC
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 2.9, 4.1])
    >>> CCC(obs, mod)
    0.9984779299847792
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        # Calculate means
        obs_mean = obs.mean(dim=dim)
        mod_mean = mod.mean(dim=dim)

        # Calculate variances and covariance
        obs_var = obs.var(dim=dim)
        mod_var = mod.var(dim=dim)
        covar = ((obs - obs_mean) * (mod - mod_mean)).mean(dim=dim)

        # Calculate CCC
        numerator = 2 * covar
        denominator = obs_var + mod_var + (obs_mean - mod_mean) ** 2
        result = numerator / denominator
        # Update history
        history = f"CCC computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Calculate means
        obs_mean = np.nanmean(obs, axis=axis)
        mod_mean = np.nanmean(mod, axis=axis)

        # Calculate variances and covariance
        obs_var = np.nanvar(obs, axis=axis)
        mod_var = np.nanvar(mod, axis=axis)
        if axis is not None:
            obs_mean_kd = np.nanmean(obs, axis=axis, keepdims=True)
            mod_mean_kd = np.nanmean(mod, axis=axis, keepdims=True)
        else:
            obs_mean_kd = obs_mean
            mod_mean_kd = mod_mean
        covar = np.nanmean((obs - obs_mean_kd) * (mod - mod_mean_kd), axis=axis)

        # Calculate CCC
        numerator = 2 * covar
        denominator = obs_var + mod_var + (obs_mean - mod_mean) ** 2
        return numerator / denominator

E1(obs, mod, axis=None)

Modified Coefficient of Efficiency (E1).

Typical Use Cases

  • Quantifying the efficiency of model predictions relative to observed mean, robust to outliers.
  • Used in hydrology, meteorology, and model skill assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Modified coefficient of efficiency (unitless, -inf to 1).

Examples

import numpy as np from monet_stats.correlation_metrics import E1 obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) E1(obs, mod) 0.0

Source code in src/monet_stats/correlation_metrics.py 
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def E1(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Coefficient of Efficiency (E1).

    Typical Use Cases
    -----------------
    - Quantifying the efficiency of model predictions relative to observed mean,
      robust to outliers.
    - Used in hydrology, meteorology, and model skill assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Modified coefficient of efficiency (unitless, -inf to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import E1
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> E1(obs, mod)
    0.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        num = abs(obs - mod).sum(dim=dim)
        denom = abs(obs - obs.mean(dim=dim)).sum(dim=dim)
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"E1 computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        num = np.ma.abs(np.subtract(obs, mod)).sum(axis=axis)
        mean_obs = np.ma.mean(obs, axis=axis, keepdims=True)
        denom = np.ma.abs(np.subtract(obs, mean_obs)).sum(axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            result = np.where((num == 0) & (denom == 0), 1.0, result)
            result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

E1_prime(obs, mod, axis=None)

Modified Coefficient of Efficiency (E1') - Alternative formulation.

Typical Use Cases

  • Quantifying the efficiency of model predictions relative to observed mean, robust to outliers.
  • Used in hydrology, meteorology, and model skill assessment as an alternative to E1.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Modified coefficient of efficiency (unitless, -inf to 1).

Examples

import numpy as np from monet_stats.correlation_metrics import E1_prime obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) E1_prime(obs, mod) 0.0

Source code in src/monet_stats/correlation_metrics.py 
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def E1_prime(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Coefficient of Efficiency (E1') - Alternative formulation.

    Typical Use Cases
    -----------------
    - Quantifying the efficiency of model predictions relative to observed mean,
      robust to outliers.
    - Used in hydrology, meteorology, and model skill assessment as an
      alternative to E1.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Modified coefficient of efficiency (unitless, -inf to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import E1_prime
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> E1_prime(obs, mod)
    0.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_mean = obs.mean(dim=dim)
        num = abs(obs - mod).sum(dim=dim)
        denom = abs(obs - obs_mean).sum(dim=dim)
        # Handle case where denominator is 0
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"E1_prime computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obs_c, mod_c = matchedcompressed(obs, mod)
            obs_mean_kd = np.nanmean(obs_c)
        else:
            obs_c, mod_c = obs, mod
            obs_mean_kd = np.nanmean(obs_c, axis=axis, keepdims=True)

        num = np.nansum(np.abs(obs_c - mod_c), axis=axis)
        denom = np.nansum(np.abs(obs_c - obs_mean_kd), axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            if np.ndim(result) == 0:
                if num == 0 and denom == 0:
                    result = np.array(1.0)
                elif denom == 0:
                    result = np.array(-np.inf)
            else:
                result = np.where((num == 0) & (denom == 0), 1.0, result)
                result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

IOA(obs, mod, axis=None)

Index of Agreement (IOA).

Typical Use Cases

  • Quantifying the agreement between model and observations, normalized by total deviation.
  • Used in model evaluation for skill assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import IOA obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) IOA(obs, mod) 0.8

Source code in src/monet_stats/correlation_metrics.py 
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def IOA(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Index of Agreement (IOA).

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, normalized by
      total deviation.
    - Used in model evaluation for skill assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import IOA
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> IOA(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obsmean = obs.mean(dim=dim)
        num = ((obs - mod) ** 2).sum(dim=dim)
        denom = ((abs(mod - obsmean) + abs(obs - obsmean)) ** 2).sum(dim=dim)
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"IOA computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obsmean = np.ma.mean(obs, axis=axis, keepdims=True)
        num = (np.ma.abs(np.subtract(obs, mod)) ** 2).sum(axis=axis)
        denom = ((np.ma.abs(np.subtract(mod, obsmean)) + np.ma.abs(np.subtract(obs, obsmean))) ** 2).sum(axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            result = np.where((num == 0) & (denom == 0), 1.0, result)
            result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

IOA_m(obs, mod, axis=None)

Index of Agreement (IOA), robust to masked arrays.

Typical Use Cases

  • Quantifying the agreement between model and observations, normalized by total deviation.
  • Used in model evaluation for skill assessment, robust to masked arrays.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import IOA_m obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) IOA_m(obs, mod) 0.8

Source code in src/monet_stats/correlation_metrics.py 
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def IOA_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Index of Agreement (IOA), robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, normalized by
      total deviation.
    - Used in model evaluation for skill assessment, robust to masked arrays.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import IOA_m
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> IOA_m(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obsmean = obs.mean(dim=dim)
        num = ((obs - mod) ** 2).sum(dim=dim)
        denom = ((abs(mod - obsmean) + abs(obs - obsmean)) ** 2).sum(dim=dim)
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"IOA_m computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obsmean = np.ma.mean(obs, axis=axis, keepdims=True)
        num = (np.ma.abs(np.subtract(obs, mod)) ** 2).sum(axis=axis)
        denom = ((np.ma.abs(np.subtract(mod, obsmean)) + np.ma.abs(np.subtract(obs, obsmean))) ** 2).sum(axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            result = np.where((num == 0) & (denom == 0), 1.0, result)
            result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

IOA_prime(obs, mod, axis=None)

Index of Agreement (IOA') - Alternative formulation.

Typical Use Cases

  • Quantifying the agreement between model and observations, normalized by total deviation.
  • Used in model evaluation for skill assessment as an alternative to IOA.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import IOA_prime obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) IOA_prime(obs, mod) 0.8

Source code in src/monet_stats/correlation_metrics.py 
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def IOA_prime(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Index of Agreement (IOA') - Alternative formulation.

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, normalized by
      total deviation.
    - Used in model evaluation for skill assessment as an alternative to IOA.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import IOA_prime
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> IOA_prime(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obsmean = obs.mean(dim=dim)
        num = ((obs - mod) ** 2).sum(dim=dim)
        denom = ((abs(mod - obsmean) + abs(obs - obsmean)) ** 2).sum(dim=dim)
        # Handle case where denominator is 0
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"IOA_prime computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obs_c, mod_c = matchedcompressed(obs, mod)
            obsmean_kd = np.nanmean(obs_c)
        else:
            obs_c, mod_c = obs, mod
            obsmean_kd = np.nanmean(obs_c, axis=axis, keepdims=True)

        num = np.nansum((obs_c - mod_c) ** 2, axis=axis)
        denom = np.nansum((np.abs(mod_c - obsmean_kd) + np.abs(obs_c - obsmean_kd)) ** 2, axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            if np.ndim(result) == 0:
                if num == 0 and denom == 0:
                    result = np.array(1.0)
                elif denom == 0:
                    result = np.array(-np.inf)
            else:
                result = np.where((num == 0) & (denom == 0), 1.0, result)
                result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

KGE(obs, mod, axis=None)

Kling-Gupta Efficiency (KGE).

Typical Use Cases

  • Quantifying the overall agreement between model and observations, combining correlation, bias, and variability.
  • Used in hydrology, meteorology, and environmental model evaluation.

Typical Values and Range

  • Range: -∞ to 1
  • 1: Perfect agreement between model and observations
  • 0: Moderate skill
  • Negative values: Poor skill

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis along which to compute KGE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Kling-Gupta efficiency (unitless, -∞ to 1).

Examples

import numpy as np from monet_stats.correlation_metrics import KGE obs = np.array([1, 2, 3]) mod = np.array([1.1, 1.9, 3.2]) KGE(obs, mod) 0.8988771192996924

Source code in src/monet_stats/correlation_metrics.py 
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def KGE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Kling-Gupta Efficiency (KGE).

    Typical Use Cases
    -----------------
    - Quantifying the overall agreement between model and observations,
      combining correlation, bias, and variability.
    - Used in hydrology, meteorology, and environmental model evaluation.

    Typical Values and Range
    ------------------------
    - Range: -∞ to 1
    - 1: Perfect agreement between model and observations
    - 0: Moderate skill
    - Negative values: Poor skill

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute KGE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Kling-Gupta efficiency (unitless, -∞ to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import KGE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([1.1, 1.9, 3.2])
    >>> KGE(obs, mod)
    0.8988771192996924
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        r = xr.corr(obs, mod, dim=dim)
        alpha = mod.std(dim=dim) / obs.std(dim=dim)
        beta = mod.mean(dim=dim) / obs.mean(dim=dim)
        result = 1.0 - ((r - 1.0) ** 2 + (alpha - 1.0) ** 2 + (beta - 1.0) ** 2) ** 0.5
        # Update history
        history = f"KGE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            from scipy.stats import pearsonr

            obsc, modc = matchedcompressed(obs, mod)
            if len(obsc) < 2:
                r = 0.0
            else:
                r, _ = pearsonr(obsc, modc)
        else:
            # Manual vectorized correlation for numpy with axis
            obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
            mod_mean = np.nanmean(mod, axis=axis, keepdims=True)
            obs_std = obs - obs_mean
            mod_std = mod - mod_mean
            num = np.nansum(obs_std * mod_std, axis=axis)
            den = np.sqrt(np.nansum(obs_std**2, axis=axis) * np.nansum(mod_std**2, axis=axis))
            with np.errstate(divide="ignore", invalid="ignore"):
                r = num / den
                r = np.where(np.isnan(r), 0.0, r)

        alpha = np.ma.std(mod, axis=axis) / np.ma.std(obs, axis=axis)
        beta = np.ma.mean(mod, axis=axis) / np.ma.mean(obs, axis=axis)
        result = 1.0 - ((r - 1.0) ** 2 + (alpha - 1.0) ** 2 + (beta - 1.0) ** 2) ** 0.5
        return result.item() if np.ndim(result) == 0 else result

R2(obs, mod, axis=None)

Coefficient of Determination (R^2, unitless).

Typical Use Cases

  • Quantifying how well model predictions explain the variance in observations.
  • Used in regression analysis, model skill assessment, and forecast verification.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Coefficient of determination (R^2).

Examples

import numpy as np from monet_stats.correlation_metrics import R2 obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 1.9, 3.2, 3.8]) R2(obs, mod) 0.9846153846153847

Source code in src/monet_stats/correlation_metrics.py 
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def R2(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Coefficient of Determination (R^2, unitless).

    Typical Use Cases
    -----------------
    - Quantifying how well model predictions explain the variance in observations.
    - Used in regression analysis, model skill assessment, and forecast
      verification.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Coefficient of determination (R^2).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import R2
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 1.9, 3.2, 3.8])
    >>> R2(obs, mod)
    0.9846153846153847
    """
    from scipy.stats import pearsonr

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            # Default to all dimensions if None
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _pearsonr2(a, b):
            if np.var(a) == 0 or np.var(b) == 0:
                return 0.0
            r_val, _ = pearsonr(a, b)
            if np.isnan(r_val):
                return 0.0
            return r_val**2

        result = xr.apply_ufunc(
            _pearsonr2,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"R2 computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obsc, modc = matchedcompressed(obs, mod)
            if np.var(obsc) == 0 or np.var(modc) == 0:
                return 0.0
            r_val, _ = pearsonr(obsc, modc)
            if np.isnan(r_val):
                return 0.0
            return r_val**2
        else:
            # Manual vectorized R2
            obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
            mod_mean = np.nanmean(mod, axis=axis, keepdims=True)
            obs_std = obs - obs_mean
            mod_std = mod - mod_mean
            num = np.nansum(obs_std * mod_std, axis=axis)
            den = np.sqrt(np.nansum(obs_std**2, axis=axis) * np.nansum(mod_std**2, axis=axis))
            with np.errstate(divide="ignore", invalid="ignore"):
                r = num / den
                result = np.where(np.isnan(r), 0.0, r**2)
                return result.item() if np.ndim(result) == 0 else result

RMSE(obs, mod, axis=None)

Root Mean Square Error (RMSE, model unit).

Typical Use Cases

  • Quantifying the average magnitude of errors between model and observations.
  • Used in model evaluation, forecast verification, and regression analysis.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root mean square error value(s).

Examples

import numpy as np from monet_stats.correlation_metrics import RMSE obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) RMSE(obs, mod) 1.118033988749895

Source code in src/monet_stats/correlation_metrics.py 
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def RMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Mean Square Error (RMSE, model unit).

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of errors between model and observations.
    - Used in model evaluation, forecast verification, and regression analysis.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root mean square error value(s).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import RMSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> RMSE(obs, mod)
    1.118033988749895
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        result = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"RMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.sqrt(np.ma.mean((np.subtract(mod, obs)) ** 2, axis=axis))

RMSEs(obs, mod, axis=None)

Root Mean Squared Error between observations and regression fit.

(RMSEs, model unit)

Typical Use Cases

  • Quantifying the error between observations and a regression fit to the model predictions.
  • Used in model evaluation to assess how well a regression fit to the model matches the observations.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray, optional Root mean squared error value(s), or None if regression fails.

Examples

import numpy as np from monet_stats.correlation_metrics import RMSEs obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) RMSEs(obs, mod) 0.7071067811865476

Source code in src/monet_stats/correlation_metrics.py 
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def RMSEs(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray, None]:
    """
    Root Mean Squared Error between observations and regression fit.

    (RMSEs, model unit)

    Typical Use Cases
    -----------------
    - Quantifying the error between observations and a regression fit to the
      model predictions.
    - Used in model evaluation to assess how well a regression fit to the model
      matches the observations.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray, optional
        Root mean squared error value(s), or None if regression fails.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import RMSEs
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> RMSEs(obs, mod)
    0.7071067811865476
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _rmses(a, b):
            from scipy.stats import linregress

            mask = ~np.isnan(a) & ~np.isnan(b)
            if not np.any(mask):
                return np.nan
            m, c, _, _, _ = linregress(a[mask], b[mask])
            mod_hat = c + m * a
            return np.sqrt(np.mean((mod_hat - a) ** 2))

        result = xr.apply_ufunc(
            _rmses,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"RMSEs computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            try:
                from scipy.stats import linregress

                obsc, modc = matchedcompressed(obs, mod)
                m, b, _, _, _ = linregress(obsc, modc)
                mod_hat = b + m * obs
                return RMSE(obs, mod_hat, axis=axis)
            except (ValueError, ZeroDivisionError):
                return None
        else:
            # Manual vectorized regression for numpy with axis
            obs = np.asarray(obs)
            mod = np.asarray(mod)
            if axis < 0:
                axis = obs.ndim + axis

            obs_moved = np.moveaxis(obs, axis, -1)
            mod_moved = np.moveaxis(mod, axis, -1)
            other_shape = obs_moved.shape[:-1]
            obs_flat = obs_moved.reshape(-1, obs_moved.shape[-1])
            mod_flat = mod_moved.reshape(-1, mod_moved.shape[-1])

            results = []
            from scipy.stats import linregress

            for i in range(len(obs_flat)):
                mask = ~np.isnan(obs_flat[i]) & ~np.isnan(mod_flat[i])
                if np.sum(mask) < 2:
                    results.append(np.nan)
                else:
                    m, b, _, _, _ = linregress(obs_flat[i][mask], mod_flat[i][mask])
                    mod_hat = b + m * obs_flat[i]
                    results.append(np.sqrt(np.nanmean((mod_hat - obs_flat[i]) ** 2)))
            return np.array(results).reshape(other_shape)

RMSEu(obs, mod, axis=None)

Root Mean Squared Error between regression fit and model predictions.

(RMSEu, model unit)

Typical Use Cases

  • Quantifying the error between a linear regression fit to observations and the model predictions.
  • Used in model evaluation to assess how well a regression fit to obs matches the model output.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray, optional Root mean squared error value(s), or None if regression fails.

Examples

import numpy as np from monet_stats.correlation_metrics import RMSEu obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) RMSEu(obs, mod) 0.7071067811865476

Source code in src/monet_stats/correlation_metrics.py 
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def RMSEu(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray, None]:
    """
    Root Mean Squared Error between regression fit and model predictions.

    (RMSEu, model unit)

    Typical Use Cases
    -----------------
    - Quantifying the error between a linear regression fit to observations and
      the model predictions.
    - Used in model evaluation to assess how well a regression fit to obs
      matches the model output.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray, optional
        Root mean squared error value(s), or None if regression fails.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import RMSEu
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> RMSEu(obs, mod)
    0.7071067811865476
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _rmseu(a, b):
            from scipy.stats import linregress

            mask = ~np.isnan(a) & ~np.isnan(b)
            if not np.any(mask):
                return np.nan
            m, c, _, _, _ = linregress(a[mask], b[mask])
            mod_hat = c + m * a
            return np.sqrt(np.mean((mod_hat - b) ** 2))

        result = xr.apply_ufunc(
            _rmseu,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"RMSEu computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            try:
                from scipy.stats import linregress

                obsc, modc = matchedcompressed(obs, mod)
                m, b, _, _, _ = linregress(obsc, modc)
                mod_hat = b + m * obs
                return RMSE(mod_hat, mod, axis=axis)
            except (ValueError, ZeroDivisionError):
                return None
        else:
            obs = np.asarray(obs)
            mod = np.asarray(mod)
            if axis < 0:
                axis = obs.ndim + axis

            obs_moved = np.moveaxis(obs, axis, -1)
            mod_moved = np.moveaxis(mod, axis, -1)
            other_shape = obs_moved.shape[:-1]
            obs_flat = obs_moved.reshape(-1, obs_moved.shape[-1])
            mod_flat = mod_moved.reshape(-1, mod_moved.shape[-1])

            results = []
            from scipy.stats import linregress

            for i in range(len(obs_flat)):
                mask = ~np.isnan(obs_flat[i]) & ~np.isnan(mod_flat[i])
                if np.sum(mask) < 2:
                    results.append(np.nan)
                else:
                    m, b, _, _, _ = linregress(obs_flat[i][mask], mod_flat[i][mask])
                    mod_hat = b + m * obs_flat[i]
                    results.append(np.sqrt(np.nanmean((mod_hat - mod_flat[i]) ** 2)))
            return np.array(results).reshape(other_shape)

WDAC(obs, mod, axis=None)

Wind Direction Anomaly Correlation (WDAC).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Modeled wind direction values (degrees). axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray WDAC value(s).

Examples

import numpy as np from monet_stats.correlation_metrics import WDAC obs = np.array([350, 10, 20]) mod = np.array([345, 15, 25]) WDAC(obs, mod) 0.9992386127814763

Source code in src/monet_stats/correlation_metrics.py 
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def WDAC(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Anomaly Correlation (WDAC).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Modeled wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        WDAC value(s).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import WDAC
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([345, 15, 25])
    >>> WDAC(obs, mod)
    0.9992386127814763
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_rad = obs * np.pi / 180.0
        mod_rad = mod * np.pi / 180.0
        obs_anom = obs_rad - obs_rad.mean(dim=dim)
        mod_anom = mod_rad - mod_rad.mean(dim=dim)
        numerator = (np.sin(obs_anom) * np.sin(mod_anom)).sum(dim=dim)
        denominator = np.sqrt((np.sin(obs_anom) ** 2).sum(dim=dim) * (np.sin(mod_anom) ** 2).sum(dim=dim))
        result = numerator / denominator
        # Update history
        history = f"WDAC computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_rad = np.deg2rad(obs)
        mod_rad = np.deg2rad(mod)
        if axis is not None:
            obs_bar_rad = np.ma.mean(obs_rad, axis=axis, keepdims=True)
            mod_bar_rad = np.ma.mean(mod_rad, axis=axis, keepdims=True)
        else:
            obs_bar_rad = np.ma.mean(obs_rad)
            mod_bar_rad = np.ma.mean(mod_rad)

        obs_anom = obs_rad - obs_bar_rad
        mod_anom = mod_rad - mod_bar_rad
        numerator = np.ma.sum(np.sin(obs_anom) * np.sin(mod_anom), axis=axis)
        denominator = np.ma.sqrt(
            np.ma.sum(np.sin(obs_anom) ** 2, axis=axis) * np.ma.sum(np.sin(mod_anom) ** 2, axis=axis)
        )
        return numerator / denominator

WDIOA(obs, mod, axis=None)

Wind Direction Index of Agreement (WDIOA).

Standard version.

Typical Use Cases

  • Quantifying the agreement between observed and modeled wind directions, accounting for circularity.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Typical Values and Range

  • Range: 0 to 1
  • 1: Perfect agreement between observed and modeled wind directions
  • 0: No agreement (as bad as using the mean of observations)

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Modeled wind direction values (degrees). axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Wind direction index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import WDIOA obs = np.array([350, 10, 20]) mod = np.array([345, 15, 25]) WDIOA(obs, mod) 0.8

Source code in src/monet_stats/correlation_metrics.py 
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def WDIOA(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Index of Agreement (WDIOA).

    Standard version.

    Typical Use Cases
    -----------------
    - Quantifying the agreement between observed and modeled wind directions,
      accounting for circularity.
    - Used in wind energy, meteorology, and air quality studies to assess wind
      direction model performance.

    Typical Values and Range
    ------------------------
    - Range: 0 to 1
    - 1: Perfect agreement between observed and modeled wind directions
    - 0: No agreement (as bad as using the mean of observations)

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Modeled wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Wind direction index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import WDIOA
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([345, 15, 25])
    >>> WDIOA(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        num = abs(circlebias(obs - mod)).sum(dim=dim)
        mean_obs = obs.mean(dim=dim)
        denom = (abs(circlebias(mod - mean_obs)) + abs(circlebias(obs - mean_obs))).sum(dim=dim)

        result = 1.0 - (num / denom)
        result = xr.where(denom == 0, 1.0, result)

        # Update history
        history = f"WDIOA computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        num = np.ma.sum(np.ma.abs(circlebias(np.subtract(obs, mod))), axis=axis)
        mean_obs = np.ma.mean(obs, axis=axis, keepdims=True)
        denom = np.ma.sum(
            np.ma.abs(circlebias(np.subtract(mod, mean_obs))) + np.ma.abs(circlebias(np.subtract(obs, mean_obs))),
            axis=axis,
        )
        result = np.where(denom == 0, 1.0, 1.0 - (num / denom))
        return result.item() if np.ndim(result) == 0 else result

WDIOA_m(obs, mod, axis=None)

Wind Direction Index of Agreement (WDIOA_m).

Robust to masked arrays.

Typical Use Cases

  • Quantifying the agreement between observed and modeled wind directions, accounting for circularity.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Typical Values and Range

  • Range: 0 to 1
  • 1: Perfect agreement between observed and modeled wind directions
  • 0: No agreement (as bad as using the mean of observations)

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Modeled wind direction values (degrees). axis : int, str, or iterable of such, optional Axis along which to compute the metric.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Wind direction index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import WDIOA_m obs = np.array([350, 10, 20]) mod = np.array([345, 15, 25]) WDIOA_m(obs, mod) 0.8

Source code in src/monet_stats/correlation_metrics.py 
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def WDIOA_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Index of Agreement (WDIOA_m).

    Robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the agreement between observed and modeled wind directions,
      accounting for circularity.
    - Used in wind energy, meteorology, and air quality studies to assess wind
      direction model performance.

    Typical Values and Range
    ------------------------
    - Range: 0 to 1
    - 1: Perfect agreement between observed and modeled wind directions
    - 0: No agreement (as bad as using the mean of observations)

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Modeled wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis along which to compute the metric.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Wind direction index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import WDIOA_m
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([345, 15, 25])
    >>> WDIOA_m(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obsmean = obs.mean(dim=dim)
        num = (abs(circlebias_m(obs - mod))).sum(dim=dim)
        denom = (abs(circlebias_m(mod - obsmean)) + abs(circlebias_m(obs - obsmean))).sum(dim=dim)

        result = 1.0 - (num / denom)
        result = xr.where(denom == 0, 1.0, result)

        # Update history
        history = f"WDIOA_m computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obsmean = np.ma.mean(obs, axis=axis, keepdims=True)
        num = np.ma.sum(np.ma.abs(circlebias_m(np.subtract(obs, mod))), axis=axis)
        denom = np.ma.sum(
            np.ma.abs(circlebias_m(np.subtract(mod, obsmean))) + np.ma.abs(circlebias_m(np.subtract(obs, obsmean))),
            axis=axis,
        )
        result = np.where(denom == 0, 1.0, 1.0 - (num / denom))
        return result.item() if np.ndim(result) == 0 else result

WDRMSE(obs, mod, axis=None)

Wind Direction Root Mean Square Error (WDRMSE, model unit).

Standard version.

Typical Use Cases

  • Quantifying the average magnitude of wind direction errors, accounting for circularity.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Model predicted wind direction values (degrees). axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Wind direction root mean square error (degrees).

Examples

import numpy as np from monet_stats.correlation_metrics import WDRMSE obs = np.array([350, 10, 20]) mod = np.array([10, 20, 30]) WDRMSE(obs, mod) 20.0

Source code in src/monet_stats/correlation_metrics.py 
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def WDRMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Root Mean Square Error (WDRMSE, model unit).

    Standard version.

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of wind direction errors, accounting for
      circularity.
    - Used in wind energy, meteorology, and air quality studies to assess wind
      direction model performance.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Wind direction root mean square error (degrees).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import WDRMSE
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([10, 20, 30])
    >>> WDRMSE(obs, mod)
    20.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        result = (circlebias(mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"WDRMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.sqrt(np.ma.mean((circlebias(np.subtract(mod, obs))) ** 2, axis=axis))

WDRMSE_m(obs, mod, axis=None)

Wind Direction Root Mean Square Error (WDRMSE, model unit).

Robust to masked arrays.

Typical Use Cases

  • Quantifying the average magnitude of wind direction errors, accounting for circularity, robust to masked arrays.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Model predicted wind direction values (degrees). axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Wind direction root mean square error (degrees).

Examples

import numpy as np from monet_stats.correlation_metrics import WDRMSE_m obs = np.array([350, 10, 20]) mod = np.array([10, 20, 30]) WDRMSE_m(obs, mod) 20.0

Source code in src/monet_stats/correlation_metrics.py 
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def WDRMSE_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Root Mean Square Error (WDRMSE, model unit).

    Robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of wind direction errors, accounting for
      circularity, robust to masked arrays.
    - Used in wind energy, meteorology, and air quality studies to assess wind
      direction model performance.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Wind direction root mean square error (degrees).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import WDRMSE_m
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([10, 20, 30])
    >>> WDRMSE_m(obs, mod)
    20.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        result = (circlebias_m(mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"WDRMSE_m computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.sqrt(np.ma.mean((circlebias_m(np.subtract(mod, obs))) ** 2, axis=axis))

d1(obs, mod, axis=None)

Modified Index of Agreement (d1).

Typical Use Cases

  • Quantifying the agreement between model and observations, less sensitive to outliers than IOA.
  • Used in model evaluation for robust skill assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Modified index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import d1 obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) d1(obs, mod) 0.5

Source code in src/monet_stats/correlation_metrics.py 
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def d1(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Index of Agreement (d1).

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, less sensitive
      to outliers than IOA.
    - Used in model evaluation for robust skill assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Modified index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import d1
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> d1(obs, mod)
    0.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        num = abs(obs - mod).sum(dim=dim)
        mean_obs = obs.mean(dim=dim)
        denom = (abs(mod - mean_obs) + abs(obs - mean_obs)).sum(dim=dim)
        result = 1.0 - (num / denom)
        result = xr.where((num == 0) & (denom == 0), 1.0, result)
        result = xr.where((num != 0) & (denom == 0), -np.inf, result)

        # Update history
        history = f"d1 computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        num = np.ma.abs(np.subtract(obs, mod)).sum(axis=axis)
        mean_obs = np.ma.mean(obs, axis=axis, keepdims=True)
        denom = (np.ma.abs(np.subtract(mod, mean_obs)) + np.ma.abs(np.subtract(obs, mean_obs))).sum(axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
            result = np.where((num == 0) & (denom == 0), 1.0, result)
            result = np.where((num != 0) & (denom == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

kendalltau(obs, mod, axis=None)

Kendall rank correlation coefficient.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension name along which to compute the coefficient.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Kendall rank correlation coefficient.

Examples

import numpy as np from monet_stats.correlation_metrics import kendalltau obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) kendalltau(obs, mod) 1.0

Source code in src/monet_stats/correlation_metrics.py 
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def kendalltau(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Kendall rank correlation coefficient.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension name along which to compute the coefficient.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Kendall rank correlation coefficient.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import kendalltau
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> kendalltau(obs, mod)
    1.0
    """
    from scipy.stats import kendalltau as _kendalltau

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _kendalltau_onlytau(a, b):
            mask = ~np.isnan(a) & ~np.isnan(b)
            if np.sum(mask) < 2:
                return np.nan
            return _kendalltau(a[mask], b[mask])[0]

        result = xr.apply_ufunc(
            _kendalltau_onlytau,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"kendalltau computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obsc, modc = matchedcompressed(obs, mod)
            if len(obsc) < 2:
                return np.nan
            return _kendalltau(obsc, modc)[0]
        else:
            # Fallback for numpy with axis: manual loop over other axes
            obs = np.asarray(obs)
            mod = np.asarray(mod)
            if axis < 0:
                axis = obs.ndim + axis

            obs_moved = np.moveaxis(obs, axis, -1)
            mod_moved = np.moveaxis(mod, axis, -1)

            other_shape = obs_moved.shape[:-1]
            obs_flat = obs_moved.reshape(-1, obs_moved.shape[-1])
            mod_flat = mod_moved.reshape(-1, mod_moved.shape[-1])

            results = []
            for i in range(len(obs_flat)):
                mask = ~np.isnan(obs_flat[i]) & ~np.isnan(mod_flat[i])
                if np.sum(mask) < 2:
                    results.append(np.nan)
                else:
                    results.append(_kendalltau(obs_flat[i][mask], mod_flat[i][mask])[0])

            return np.array(results).reshape(other_shape)

matchmasks(a1, a2)

Match and combine masks from two masked arrays.

Typical Use Cases

  • Ensuring that two arrays have the same mask for paired statistical calculations.
  • Used in metrics that require both arrays to have valid data at the same locations (e.g., correlation, regression).

Parameters

a1 : array-like or numpy.ma.MaskedArray First input array. a2 : array-like or numpy.ma.MaskedArray Second input array.

Returns

tuple of numpy.ma.MaskedArray Tuple of (a1_masked, a2_masked) with combined mask.

Examples

import numpy as np from monet.util import stats a1 = np.ma.array([1, 2, 3], mask=[0, 1, 0]) a2 = np.ma.array([4, 5, 6], mask=[0, 0, 1]) stats.matchmasks(a1, a2) (masked_array(data=[1, --, 3], mask=[False, True, False]), masked_array(data=[4, --, --], mask=[False, False, True]))

Source code in src/monet_stats/correlation_metrics.py 
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def matchmasks(a1: ArrayLike, a2: ArrayLike) -> Tuple[np.ma.MaskedArray, np.ma.MaskedArray]:
    """
    Match and combine masks from two masked arrays.

    Typical Use Cases
    -----------------
    - Ensuring that two arrays have the same mask for paired statistical calculations.
    - Used in metrics that require both arrays to have valid data at the same locations (e.g., correlation, regression).

    Parameters
    ----------
    a1 : array-like or numpy.ma.MaskedArray
        First input array.
    a2 : array-like or numpy.ma.MaskedArray
        Second input array.

    Returns
    -------
    tuple of numpy.ma.MaskedArray
        Tuple of (a1_masked, a2_masked) with combined mask.

    Examples
    --------
    >>> import numpy as np
    >>> from monet.util import stats
    >>> a1 = np.ma.array([1, 2, 3], mask=[0, 1, 0])
    >>> a2 = np.ma.array([4, 5, 6], mask=[0, 0, 1])
    >>> stats.matchmasks(a1, a2)
    (masked_array(data=[1, --, 3], mask=[False,  True, False]),
     masked_array(data=[4, --, --], mask=[False, False,  True]))
    """
    mask = np.ma.getmaskarray(a1) | np.ma.getmaskarray(a2)
    return np.ma.masked_where(mask, a1), np.ma.masked_where(mask, a2)

pearsonr(obs, mod, axis=None)

Pearson correlation coefficient.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension name along which to compute the coefficient.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Pearson correlation coefficient.

Examples

import numpy as np from monet_stats.correlation_metrics import pearsonr obs = np.array([1, 2, 3]) mod = np.array([2, 4, 6]) pearsonr(obs, mod) 1.0

Source code in src/monet_stats/correlation_metrics.py 
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def pearsonr(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Pearson correlation coefficient.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension name along which to compute the coefficient.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Pearson correlation coefficient.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import pearsonr
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 4, 6])
    >>> pearsonr(obs, mod)
    1.0
    """
    from scipy.stats import pearsonr as _pearsonr

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _pearsonr_onlyr(a, b):
            mask = ~np.isnan(a) & ~np.isnan(b)
            if np.sum(mask) < 2 or np.var(a[mask]) == 0 or np.var(b[mask]) == 0:
                return np.nan
            return _pearsonr(a[mask], b[mask])[0]

        result = xr.apply_ufunc(
            _pearsonr_onlyr,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"pearsonr computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obsc, modc = matchedcompressed(obs, mod)
            if len(obsc) < 2 or np.var(obsc) == 0 or np.var(modc) == 0:
                return 0.0
            r_val, _ = _pearsonr(obsc, modc)
            return r_val if not np.isnan(r_val) else 0.0
        else:
            # For numpy with axis, use manual vectorized correlation
            obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
            mod_mean = np.nanmean(mod, axis=axis, keepdims=True)
            obs_std = obs - obs_mean
            mod_std = mod - mod_mean
            num = np.nansum(obs_std * mod_std, axis=axis)
            den = np.sqrt(np.nansum(obs_std**2, axis=axis) * np.nansum(mod_std**2, axis=axis))
            with np.errstate(divide="ignore", invalid="ignore"):
                result = num / den
                return result.item() if np.ndim(result) == 0 else result

spearmanr(obs, mod, axis=None)

Spearman rank correlation coefficient.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the coefficient.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Spearman rank correlation coefficient.

Examples

import numpy as np from monet_stats.correlation_metrics import spearmanr obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) spearmanr(obs, mod) 0.8660254037844387

Source code in src/monet_stats/correlation_metrics.py 
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def spearmanr(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Spearman rank correlation coefficient.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the coefficient.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Spearman rank correlation coefficient.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import spearmanr
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> spearmanr(obs, mod)
    0.8660254037844387
    """
    from scipy.stats import spearmanr as _spearmanr

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dim = obs.dims
        elif isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        def _spearmanr_onlyrho(a, b):
            mask = ~np.isnan(a) & ~np.isnan(b)
            if np.sum(mask) < 2:
                return np.nan
            return _spearmanr(a[mask], b[mask])[0]

        result = xr.apply_ufunc(
            _spearmanr_onlyrho,
            obs,
            mod,
            input_core_dims=[[dim] if isinstance(dim, str) else list(dim)] * 2,
            output_core_dims=[[]],
            vectorize=True,
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
        )
        # Update history
        history = f"spearmanr computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        if axis is None:
            obsc, modc = matchedcompressed(obs, mod)
            if len(obsc) < 2:
                return np.nan
            return _spearmanr(obsc, modc)[0]
        else:
            # Fallback for numpy with axis: manual loop over other axes
            obs = np.asarray(obs)
            mod = np.asarray(mod)
            if axis < 0:
                axis = obs.ndim + axis

            # Move axis to last position
            obs_moved = np.moveaxis(obs, axis, -1)
            mod_moved = np.moveaxis(mod, axis, -1)

            # Reshape all other axes into one
            other_shape = obs_moved.shape[:-1]
            obs_flat = obs_moved.reshape(-1, obs_moved.shape[-1])
            mod_flat = mod_moved.reshape(-1, mod_moved.shape[-1])

            results = []
            for i in range(len(obs_flat)):
                mask = ~np.isnan(obs_flat[i]) & ~np.isnan(mod_flat[i])
                if np.sum(mask) < 2:
                    results.append(np.nan)
                else:
                    results.append(_spearmanr(obs_flat[i][mask], mod_flat[i][mask])[0])

            return np.array(results).reshape(other_shape)

taylor_skill(obs, mod, axis=None)

Taylor Skill Score (TSS).

Typical Use Cases

  • Summarizing model performance in a single skill score for use in Taylor diagrams.
  • Used in climate, weather, and environmental model evaluation.

Typical Values and Range

  • Range: 0 to 1
  • 1: Perfect agreement between model and observations
  • 0: No skill

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis along which to compute the skill score.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Taylor skill score (unitless, 0-1).

Examples

import numpy as np from monet_stats.correlation_metrics import taylor_skill obs = np.array([1, 2, 3]) mod = np.array([1.1, 1.9, 3.2]) taylor_skill(obs, mod) 0.9995574044955781

Source code in src/monet_stats/correlation_metrics.py 
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def taylor_skill(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Taylor Skill Score (TSS).

    Typical Use Cases
    -----------------
    - Summarizing model performance in a single skill score for use in Taylor
      diagrams.
    - Used in climate, weather, and environmental model evaluation.

    Typical Values and Range
    ------------------------
    - Range: 0 to 1
    - 1: Perfect agreement between model and observations
    - 0: No skill

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis along which to compute the skill score.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Taylor skill score (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.correlation_metrics import taylor_skill
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([1.1, 1.9, 3.2])
    >>> taylor_skill(obs, mod)
    0.9995574044955781
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        std_obs = obs.std(dim=dim)
        std_mod = mod.std(dim=dim)
        corr = xr.corr(obs, mod, dim=dim)

        # Calculate Taylor Skill Score using the common formula
        # S = 4 * (1 + R) / ( (sigma_p/sigma_o + sigma_o/sigma_p)^2 * (1 + R_max) )
        # Assuming R_max = 1.0
        norm_std = std_mod / std_obs
        result = (4.0 * (corr + 1.0)) / ((norm_std + 1.0 / norm_std) ** 2 * 2.0)
        # Update history
        history = f"taylor_skill computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        std_obs = np.ma.std(obs, axis=axis)
        std_mod = np.ma.std(mod, axis=axis)
        from scipy.stats import pearsonr

        if axis is None:
            if np.ma.is_masked(obs):
                corr = pearsonr(obs.compressed(), mod.compressed())[0]
            else:
                corr = pearsonr(obs, mod)[0]
        else:
            # Vectorized correlation over axis for numpy
            obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
            mod_mean = np.nanmean(mod, axis=axis, keepdims=True)
            obs_anom = obs - obs_mean
            mod_anom = mod - mod_mean
            num_corr = np.nansum(obs_anom * mod_anom, axis=axis)
            den_corr = np.sqrt(np.nansum(obs_anom**2, axis=axis) * np.nansum(mod_anom**2, axis=axis))
            with np.errstate(divide="ignore", invalid="ignore"):
                corr = num_corr / den_corr

        norm_std = std_mod / std_obs
        with np.errstate(divide="ignore", invalid="ignore"):
            result = (4.0 * (corr + 1.0)) / ((norm_std + 1.0 / norm_std) ** 2 * 2.0)
            result = np.where(np.isnan(result) | np.isinf(result), 1.0, result)
        return result.item() if np.ndim(result) == 0 else result

Error Metrics

Error Metrics for Model Evaluation

COE(obs, mod, axis=None)

Center of Mass Error (COE).

The COE measures the displacement between the centroids (centers of mass) of two fields. For spatial data, this represents the shift in the center of a feature (e.g., a storm or a pollutant plume).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values (typically 2D spatial field). mod : numpy.ndarray or xarray.DataArray Model or predicted values (typically 2D spatial field). axis : int, str, or iterable of such, optional Axis or dimension(s) over which to compute the centroid. If None, computes over all axes.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Center of mass error (Euclidean distance between centroids).

Examples

import numpy as np from monet_stats.error_metrics import COE obs = np.zeros((5, 5)) obs[2, 2] = 1.0 # Peak at center (2, 2) mod = np.zeros((5, 5)) mod[3, 3] = 1.0 # Peak shifted to (3, 3)

Displacement is sqrt(1^2 + 1^2) = sqrt(2) approx 1.414

np.allclose(COE(obs, mod), np.sqrt(2)) True

Source code in src/monet_stats/error_metrics.py 
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def COE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Center of Mass Error (COE).

    The COE measures the displacement between the centroids (centers of mass)
    of two fields. For spatial data, this represents the shift in the center
    of a feature (e.g., a storm or a pollutant plume).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values (typically 2D spatial field).
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values (typically 2D spatial field).
    axis : int, str, or iterable of such, optional
        Axis or dimension(s) over which to compute the centroid.
        If None, computes over all axes.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Center of mass error (Euclidean distance between centroids).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import COE
    >>> obs = np.zeros((5, 5))
    >>> obs[2, 2] = 1.0  # Peak at center (2, 2)
    >>> mod = np.zeros((5, 5))
    >>> mod[3, 3] = 1.0  # Peak shifted to (3, 3)
    >>> # Displacement is sqrt(1^2 + 1^2) = sqrt(2) approx 1.414
    >>> np.allclose(COE(obs, mod), np.sqrt(2))
    True
    """
    from .utils_stats import _update_history

    def _get_centroid(da: xr.DataArray, dims: Iterable[str]) -> List[xr.DataArray]:
        """Helper to calculate centroid of a DataArray."""
        total = da.sum(dim=dims)
        # Handle zero sum to avoid division by zero
        total_safe = xr.where(total == 0, 1e-10, total)
        coords_list = []
        for d in dims:
            # Check if coord exists and is numeric
            if d in da.coords and np.issubdtype(da.coords[d].dtype, np.number):
                coord = da.coords[d]
            else:
                # Fallback to dimension indices
                coord = xr.DataArray(np.arange(da.sizes[d]), dims=d, name=d)
            # Weighted mean of coordinate
            c = (da * coord).sum(dim=dims) / total_safe
            coords_list.append(c)
        return coords_list

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        if axis is None:
            dims = list(obs.dims)
        elif isinstance(axis, (int, str)):
            dims = [obs.dims[axis] if isinstance(axis, int) else axis]
        else:
            dims = [obs.dims[d] if isinstance(d, int) else d for d in axis]

        c_obs = _get_centroid(obs, dims)
        c_mod = _get_centroid(mod, dims)

        # Euclidean distance
        dist_sq = sum((cm - co) ** 2 for cm, co in zip(c_mod, c_obs))
        result = dist_sq**0.5

        return _update_history(result, "Center of Mass Error (COE)")

    # Fallback to numpy
    obs_arr = np.asanyarray(obs)
    mod_arr = np.asanyarray(mod)

    if axis is None:
        axes = tuple(range(obs_arr.ndim))
    elif isinstance(axis, int):
        axes = (axis,)
    else:
        axes = tuple(axis)

    def _get_numpy_centroid(arr: np.ndarray, axes_tuple: Tuple[int, ...]) -> List[np.ndarray]:
        """Helper to calculate centroid of a NumPy array."""
        total = np.sum(arr, axis=axes_tuple)
        total_safe = np.where(total == 0, 1e-10, total)
        c_list = []
        for ax in axes_tuple:
            # Create coordinate array for this axis
            shape = [1] * arr.ndim
            shape[ax] = arr.shape[ax]
            coord = np.arange(arr.shape[ax]).reshape(shape)
            c = np.sum(arr * coord, axis=axes_tuple) / total_safe
            c_list.append(c)
        return c_list

    c_obs_np = _get_numpy_centroid(obs_arr, axes)
    c_mod_np = _get_numpy_centroid(mod_arr, axes)

    dist_sq_np = sum((cm - co) ** 2 for cm, co in zip(c_mod_np, c_obs_np))
    return dist_sq_np**0.5

CORR_INDEX(obs, mod, axis=None)

Correlation Index (CORR_INDEX).

Typical Use Cases

  • Measuring the linear relationship between observed and modeled values.
  • Used as a component in model evaluation.
  • Quantifies how well model captures observed patterns.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute correlation index.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Correlation index (unitless, -1 to 1).

Examples

import numpy as np from monet_stats.error_metrics import CORR_INDEX obs = np.array([1, 2, 3, 4]) mod = np.array([2, 4, 6, 8]) CORR_INDEX(obs, mod) 1.0

Source code in src/monet_stats/error_metrics.py 
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def CORR_INDEX(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Correlation Index (CORR_INDEX).

    Typical Use Cases
    -----------------
    - Measuring the linear relationship between observed and modeled values.
    - Used as a component in model evaluation.
    - Quantifies how well model captures observed patterns.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute correlation index.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Correlation index (unitless, -1 to 1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import CORR_INDEX
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 4, 6, 8])
    >>> CORR_INDEX(obs, mod)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        # Using xarray's built-in correlation function
        result = xr.corr(obs, mod, dim=dim)
        # Update history
        history = f"CORR_INDEX computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Fallback to numpy-compatible logic
        obs = np.asarray(obs)
        mod = np.asarray(mod)
        if axis is None:
            from scipy.stats import pearsonr

            return pearsonr(obs.flatten(), mod.flatten())[0]
        else:
            # Manual vectorized correlation over axis for robustness across scipy versions
            obs_mean = np.mean(obs, axis=axis, keepdims=True)
            mod_mean = np.mean(mod, axis=axis, keepdims=True)
            obs_std = obs - obs_mean
            mod_std = mod - mod_mean
            num = np.sum(obs_std * mod_std, axis=axis)
            den = np.sqrt(np.sum(obs_std**2, axis=axis) * np.sum(mod_std**2, axis=axis))
            return num / den

CRMSE(obs, mod, axis=None)

Centered Root Mean Square Error (CRMSE).

Typical Use Cases

  • Quantifying the error between anomalies (deviations from mean) of model and observations.
  • Used in Taylor diagrams, model evaluation, and forecast verification.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute CRMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Centered root mean square error.

Examples

import numpy as np from monet_stats.error_metrics import CRMSE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) CRMSE(obs, mod) 0.4714045207910317

Source code in src/monet_stats/error_metrics.py 
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def CRMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Centered Root Mean Square Error (CRMSE).

    Typical Use Cases
    -----------------
    - Quantifying the error between anomalies (deviations from mean) of model
      and observations.
    - Used in Taylor diagrams, model evaluation, and forecast verification.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute CRMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Centered root mean square error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import CRMSE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> CRMSE(obs, mod)
    0.4714045207910317
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        o_ = obs - obs.mean(dim=dim)
        m_ = mod - mod.mean(dim=dim)
        result = ((m_ - o_) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"CRMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        o_ = np.subtract(obs, np.mean(obs, axis=axis, keepdims=True))
        m_ = np.subtract(mod, np.mean(mod, axis=axis, keepdims=True))
        return (np.ma.abs(m_ - o_) ** 2).mean(axis=axis) ** 0.5

IOA(obs, mod, axis=None)

Index of Agreement (IOA).

Typical Use Cases

  • Quantifying the agreement between model and observations, normalized by total deviation.
  • Used in model evaluation for skill assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute IOA.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.error_metrics import IOA obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) IOA(obs, mod) 0.8

Source code in src/monet_stats/error_metrics.py 
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def IOA(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Index of Agreement (IOA).

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, normalized by
      total deviation.
    - Used in model evaluation for skill assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute IOA.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import IOA
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> IOA(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        obs_mean = obs.mean(dim=dim)
        num = ((obs - mod) ** 2).sum(dim=dim)
        denom = ((abs(mod - obs_mean) + abs(obs - obs_mean)) ** 2).sum(dim=dim)
        result = 1.0 - (num / denom)
        # Update history
        history = f"IOA computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_mean = np.mean(obs, axis=axis, keepdims=True)
        num = np.sum((obs - mod) ** 2, axis=axis)
        denom = np.sum((np.abs(mod - obs_mean) + np.abs(obs - obs_mean)) ** 2, axis=axis)
        return 1.0 - (num / denom)

IOA_m(obs, mod, axis=None)

Index of Agreement (IOA) - robust to masked arrays.

Typical Use Cases

  • Quantifying the agreement between model and observations, normalized by total deviation, robust to missing data.
  • Used in model evaluation for skill assessment with incomplete datasets.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute IOA.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Index of agreement (unitless, 0-1).

Examples

import numpy as np from monet_stats.error_metrics import IOA_m obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) IOA_m(obs, mod) 0.8

Source code in src/monet_stats/error_metrics.py 
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def IOA_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Index of Agreement (IOA) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the agreement between model and observations, normalized by
      total deviation, robust to missing data.
    - Used in model evaluation for skill assessment with incomplete datasets.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute IOA.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Index of agreement (unitless, 0-1).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import IOA_m
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> IOA_m(obs, mod)
    0.8
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # IOA implementation for xarray already handles NaNs
        return IOA(obs, mod, axis=axis)
    else:
        obs_mean = np.ma.mean(obs, axis=axis, keepdims=True)
        num = np.ma.sum((obs - mod) ** 2, axis=axis)
        denom = np.ma.sum((np.ma.abs(mod - obs_mean) + np.ma.abs(obs - obs_mean)) ** 2, axis=axis)
        return 1.0 - (num / denom)

LOG_ERROR(obs, mod, axis=None)

Logarithmic Error Metric.

Typical Use Cases

  • Quantifying errors for variables that span several orders of magnitude.
  • Used in atmospheric sciences for concentration data (e.g., pollutants).
  • Helpful when relative rather than absolute errors are important.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values (should be positive). mod : numpy.ndarray or xarray.DataArray Model or predicted values (should be positive). axis : int, str, or iterable of such, optional Axis or dimension along which to compute log error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Logarithmic error metric.

Examples

import numpy as np from monet_stats.error_metrics import LOG_ERROR obs = np.array([1, 100]) mod = np.array([2, 200]) LOG_ERROR(obs, mod) 0.34657359027997264

Source code in src/monet_stats/error_metrics.py 
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def LOG_ERROR(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Logarithmic Error Metric.

    Typical Use Cases
    -----------------
    - Quantifying errors for variables that span several orders of magnitude.
    - Used in atmospheric sciences for concentration data (e.g., pollutants).
    - Helpful when relative rather than absolute errors are important.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values (should be positive).
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values (should be positive).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute log error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Logarithmic error metric.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import LOG_ERROR
    >>> obs = np.array([1, 100])
    >>> mod = np.array([2, 200])
    >>> LOG_ERROR(obs, mod)
    0.34657359027997264
    """
    # Add small epsilon to avoid log(0) and handle negative values
    epsilon = 1e-10

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        # Use abs to handle potential negative values, then add epsilon
        obs_safe = abs(obs) + epsilon
        mod_safe = abs(mod) + epsilon
        obs_log = np.log(obs_safe)
        mod_log = np.log(mod_safe)
        result = ((mod_log - obs_log) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"LOG_ERROR computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Use abs to handle potential negative values, then add epsilon
        obs_safe = np.abs(obs) + epsilon
        mod_safe = np.abs(mod) + epsilon
        obs_log = np.log(obs_safe)
        mod_log = np.log(mod_safe)

        result = np.sqrt(np.mean((mod_log - obs_log) ** 2, axis=axis))
        # Return 0 for perfect agreement
        if np.array_equal(obs, mod):
            return 0.0
        return result

MAE(obs, mod, axis=None)

Mean Absolute Error (MAE).

Typical Use Cases

  • Quantifying the average magnitude of errors between model and observations, regardless of direction.
  • Used in model evaluation, forecast verification, and regression analysis.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MAE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute error.

Examples

import numpy as np from monet_stats.error_metrics import MAE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MAE(obs, mod) 0.6666666666666666

Source code in src/monet_stats/error_metrics.py 
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def MAE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Error (MAE).

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of errors between model and observations,
      regardless of direction.
    - Used in model evaluation, forecast verification, and regression analysis.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MAE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MAE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MAE(obs, mod)
    0.6666666666666666
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = abs(mod - obs).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MAE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.abs(np.subtract(mod, obs)).mean(axis=axis)

MAE_m(obs, mod, axis=None)

Mean Absolute Error (MAE) - robust to masked arrays.

Typical Use Cases

  • Quantifying the average magnitude of errors between model and observations, regardless of direction, robust to missing data.
  • Used in model evaluation, forecast verification, and regression analysis with incomplete datasets.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MAE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute error.

Examples

import numpy as np from monet_stats.error_metrics import MAE_m obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MAE_m(obs, mod) 0.6666666666666666

Source code in src/monet_stats/error_metrics.py 
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def MAE_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Error (MAE) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of errors between model and
      observations, regardless of direction, robust to missing data.
    - Used in model evaluation, forecast verification, and regression analysis
      with incomplete datasets.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MAE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MAE_m
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MAE_m(obs, mod)
    0.6666666666666666
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # MAE implementation for xarray already handles NaNs
        return MAE(obs, mod, axis=axis)
    else:
        return np.ma.mean(np.ma.abs(np.subtract(mod, obs)), axis=axis)

MAE_norm(obs, mod, axis=None)

Normalized Mean Absolute Error (MAE_norm).

Normalizes MAE by the range of observations.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute normalized MAE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Normalized mean absolute error (unitless).

Source code in src/monet_stats/error_metrics.py 
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def MAE_norm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Normalized Mean Absolute Error (MAE_norm).

    Normalizes MAE by the range of observations.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute normalized MAE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Normalized mean absolute error (unitless).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        mae = abs(mod - obs).mean(dim=dim, keep_attrs=True)
        obs_min = obs.min(dim=dim)
        obs_max = obs.max(dim=dim)
        obs_range = obs_max - obs_min
        # Avoid division by zero
        result = xr.where(obs_range == 0, mae, mae / obs_range)
        # Update history
        history = f"MAE_norm computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        mae = np.mean(np.abs(np.subtract(mod, obs)), axis=axis)
        obs_min = np.min(obs, axis=axis)
        obs_max = np.max(obs, axis=axis)
        obs_range = obs_max - obs_min
        # Avoid division by zero
        result = np.where(obs_range == 0, mae, mae / obs_range)
        return result.item() if np.ndim(result) == 0 else result

MAPE(obs, mod, axis=None)

Mean Absolute Percentage Error (MAPE).

Typical Use Cases

  • Quantifying the average relative error between model and observations as a percentage.
  • Used in time series forecasting, regression, and model evaluation for percentage-based error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MAPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute percentage error (in percent).

Examples

import numpy as np from monet_stats.error_metrics import MAPE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MAPE(obs, mod) 50.0

Source code in src/monet_stats/error_metrics.py 
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def MAPE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Percentage Error (MAPE).

    Typical Use Cases
    -----------------
    - Quantifying the average relative error between model and observations
      as a percentage.
    - Used in time series forecasting, regression, and model evaluation for
      percentage-based error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MAPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute percentage error (in percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MAPE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MAPE(obs, mod)
    50.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (100 * abs(mod - obs) / abs(obs)).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MAPE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return (100 * np.ma.abs(np.subtract(mod, obs)) / np.ma.abs(obs)).mean(axis=axis)

MAPE_mod(obs, mod, axis=None)

Modified Mean Absolute Percentage Error (MAPE).

This version handles cases where observations might be zero or near zero by using a small epsilon to avoid division by zero.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MAPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute percentage error (in percent).

Source code in src/monet_stats/error_metrics.py 
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def MAPE_mod(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Mean Absolute Percentage Error (MAPE).

    This version handles cases where observations might be zero or near zero
    by using a small epsilon to avoid division by zero.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MAPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute percentage error (in percent).
    """
    # Small epsilon to avoid division by zero
    epsilon = 1e-8

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        # Add epsilon to avoid division by zero
        obs_safe = xr.where(abs(obs) < epsilon, epsilon, obs)
        result = (100 * abs(mod - obs) / abs(obs_safe)).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MAPE_mod computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Add epsilon to avoid division by zero
        obs_safe = np.where(np.abs(obs) < epsilon, epsilon, obs)
        return (100 * np.abs(np.subtract(mod, obs)) / np.abs(obs_safe)).mean(axis=axis)

MAPEm(obs, mod, axis=None)

Mean Absolute Percentage Error (MAPE) - robust to masked arrays.

Typical Use Cases

  • Quantifying the average relative error between model and observations as a percentage, robust to missing data.
  • Used in time series forecasting, regression, and model evaluation for percentage-based error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MAPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute percentage error (in percent).

Examples

import numpy as np from monet_stats.error_metrics import MAPEm obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MAPEm(obs, mod) 50.0

Source code in src/monet_stats/error_metrics.py 
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def MAPEm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Percentage Error (MAPE) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the average relative error between model and observations as
      a percentage, robust to missing data.
    - Used in time series forecasting, regression, and model evaluation for
      percentage-based error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MAPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute percentage error (in percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MAPEm
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MAPEm(obs, mod)
    50.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # MAPE implementation for xarray already handles NaNs
        return MAPE(obs, mod, axis=axis)
    else:
        return 100 * np.ma.mean(np.ma.abs((mod - obs) / obs), axis=axis)

MASE(obs, mod, axis=None)

Mean Absolute Scaled Error (MASE).

Typical Use Cases

  • Quantifying model error relative to the error of a simple baseline model (e.g., naive forecast).
  • Used in time series forecasting and model evaluation.
  • Provides scale-independent comparison across different datasets.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MASE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute scaled error (unitless).

Examples

import numpy as np from monet_stats.error_metrics import MASE obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 3.1, 4.1]) MASE(obs, mod) 0.1

Source code in src/monet_stats/error_metrics.py 
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def MASE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Scaled Error (MASE).

    Typical Use Cases
    -----------------
    - Quantifying model error relative to the error of a simple baseline model
      (e.g., naive forecast).
    - Used in time series forecasting and model evaluation.
    - Provides scale-independent comparison across different datasets.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MASE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute scaled error (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MASE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 3.1, 4.1])
    >>> MASE(obs, mod)
    0.1
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        # Calculate naive forecast error (using previous observation)
        # Assuming 'time' is a dimension for shift
        # If 'time' is not present, we use the provided dimension or axis
        if "time" in obs.dims:
            naive_error = abs(obs - obs.shift(time=1)).mean(dim=dim, skipna=True)
        else:
            # Fallback if time is not named 'time'
            naive_error = abs(obs - obs.shift({obs.dims[0]: 1})).mean(dim=dim, skipna=True)

        model_error = abs(mod - obs).mean(dim=dim, keep_attrs=True)
        result = model_error / naive_error
        # Update history
        history = f"MASE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Calculate naive forecast error (using previous observation)
        if axis is not None:
            naive_diff = np.diff(obs, axis=axis)
            naive_error = np.mean(np.abs(naive_diff), axis=axis)
        else:
            naive_diff = np.diff(obs)
            naive_error = np.mean(np.abs(naive_diff))
        model_error = np.mean(np.abs(np.subtract(mod, obs)), axis=axis)
        return model_error / naive_error

MASE_mod(obs, mod, axis=None)

Modified Mean Absolute Scaled Error (MASE).

This version handles cases where the naive forecast error is zero by using a small epsilon to avoid division by zero.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MASE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute scaled error (unitless).

Source code in src/monet_stats/error_metrics.py 
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def MASE_mod(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Mean Absolute Scaled Error (MASE).

    This version handles cases where the naive forecast error is zero
    by using a small epsilon to avoid division by zero.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MASE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute scaled error (unitless).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        # Calculate naive forecast error (using previous observation)
        if "time" in obs.dims:
            naive_error = abs(obs - obs.shift(time=1)).mean(dim=dim, skipna=True)
        else:
            naive_error = abs(obs - obs.shift({obs.dims[0]: 1})).mean(dim=dim, skipna=True)

        model_error = abs(mod - obs).mean(dim=dim, keep_attrs=True)
        # Avoid division by zero
        result = xr.where(naive_error == 0, model_error, model_error / naive_error)
        # Update history
        history = f"MASE_mod computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        # Calculate naive forecast error (using previous observation)
        if axis is not None:
            naive_diff = np.diff(obs, axis=axis)
            naive_error = np.mean(np.abs(naive_diff), axis=axis)
        else:
            naive_diff = np.diff(obs)
            naive_error = np.mean(np.abs(naive_diff))
        model_error = np.mean(np.abs(np.subtract(mod, obs)), axis=axis)
        # Avoid division by zero
        result = np.where(naive_error == 0, model_error, model_error / naive_error)
        return result.item() if np.ndim(result) == 0 else result

MASEm(obs, mod, axis=None)

Mean Absolute Scaled Error (MASE) - robust to masked arrays.

Typical Use Cases

  • Quantifying model error relative to the error of a simple baseline model (e.g., naive forecast), robust to masked arrays.
  • Used in time series forecasting and model evaluation with missing data.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MASE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean absolute scaled error (unitless).

Examples

import numpy as np from monet_stats.error_metrics import MASEm obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 3.1, 4.1]) MASEm(obs, mod) 0.1

Source code in src/monet_stats/error_metrics.py 
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def MASEm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Absolute Scaled Error (MASE) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying model error relative to the error of a simple baseline model
      (e.g., naive forecast), robust to masked arrays.
    - Used in time series forecasting and model evaluation with missing data.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MASE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean absolute scaled error (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MASEm
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 3.1, 4.1])
    >>> MASEm(obs, mod)
    0.1
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # MASE implementation for xarray already handles NaNs with skipna=True
        return MASE(obs, mod, axis=axis)
    else:
        # Calculate naive forecast error (using previous observation) with masked arrays
        if axis is not None:
            # Use numpy's gradient-like approach for masked arrays
            naive_diff = np.ma.diff(obs, axis=axis)
            naive_error = np.ma.mean(np.ma.abs(naive_diff), axis=axis)
        else:
            naive_diff = np.ma.diff(obs)
            naive_error = np.ma.mean(np.ma.abs(naive_diff))
        model_error = np.ma.mean(np.ma.abs(np.subtract(mod, obs)), axis=axis)
        return model_error / naive_error

MB(obs, mod, axis=None)

Mean Bias (MB).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the mean bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean bias value(s) = mean(model - observation). Positive values indicate model overestimation.

Source code in src/monet_stats/error_metrics.py 
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def MB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Bias (MB).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the mean bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean bias value(s) = mean(model - observation).
        Positive values indicate model overestimation.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (mod - obs).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.mean(np.subtract(mod, obs), axis=axis)

MNB(obs, mod, axis=None)

Mean Normalized Bias (%).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean normalized bias (percent).

Source code in src/monet_stats/error_metrics.py 
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def MNB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Normalized Bias (%).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean normalized bias (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = ((mod - obs) / obs).mean(dim=dim, keep_attrs=True) * 100.0
        # Update history
        history = f"MNB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.masked_invalid((mod - obs) / obs).mean(axis=axis) * 100.0

MNE(obs, mod, axis=None)

Mean Normalized Gross Error (%).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean normalized gross error (percent).

Source code in src/monet_stats/error_metrics.py 
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def MNE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Normalized Gross Error (%).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean normalized gross error (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (abs(mod - obs) / obs).mean(dim=dim, keep_attrs=True) * 100.0
        # Update history
        history = f"MNE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.masked_invalid(np.ma.abs(mod - obs) / obs).mean(axis=axis) * 100.0

MO(obs, mod, axis=None)

Mean Error (MO) - Mean of (model - observation).

Typical Use Cases

  • Quantifying the average bias between model predictions and observations.
  • Used in model evaluation to assess systematic errors.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the mean error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean error (model - observation) in observation units. Returns 0.0 for perfect agreement.

Examples

import numpy as np from monet_stats.error_metrics import MO obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.1, 2.1, 3.1, 4.1, 5.1]) MO(obs, mod) 0.1

Source code in src/monet_stats/error_metrics.py 
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def MO(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Error (MO) - Mean of (model - observation).

    Typical Use Cases
    -----------------
    - Quantifying the average bias between model predictions and observations.
    - Used in model evaluation to assess systematic errors.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the mean error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean error (model - observation) in observation units.
        Returns 0.0 for perfect agreement.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MO
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.1, 2.1, 3.1, 4.1, 5.1])
    >>> MO(obs, mod)
    0.1
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (mod - obs).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MO computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.mean(np.subtract(mod, obs), axis=axis)

MP(obs=None, mod=None, axis=None)

Mean Predictions (model unit).

Typical Use Cases

  • Calculating the average value of model predictions for baseline or climatological reference.
  • Used in normalization, anomaly calculation, and summary statistics for model output.

Parameters

obs : numpy.ndarray or xarray.DataArray, optional Observed values (not used for MP but included for signature matching). mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the mean.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean of predictions.

Source code in src/monet_stats/error_metrics.py 
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def MP(
    obs: Optional[Union[np.ndarray, xr.DataArray]] = None,
    mod: Union[np.ndarray, xr.DataArray] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Predictions (model unit).

    Typical Use Cases
    -----------------
    - Calculating the average value of model predictions for baseline or
      climatological reference.
    - Used in normalization, anomaly calculation, and summary statistics for
      model output.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray, optional
        Observed values (not used for MP but included for signature matching).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the mean.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean of predictions.
    """
    if isinstance(mod, xr.DataArray):
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = mod.dims[axis]
        else:
            dim = axis
        result = mod.mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"MP computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.mean(mod, axis=axis)

MdnB(obs, mod, axis=None)

Median Bias (MdnB).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the median bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median bias value(s) = median(model - observation). Positive values indicate model overestimation.

Source code in src/monet_stats/error_metrics.py 
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def MdnB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Bias (MdnB).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the median bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median bias value(s) = median(model - observation).
        Positive values indicate model overestimation.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (mod - obs).median(dim=dim, keep_attrs=True)
        # Update history
        history = f"MdnB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.median(np.subtract(mod, obs), axis=axis)

MdnNB(obs, mod, axis=None)

Median Normalized Bias (%).

Typical Use Cases

  • Assessing the central tendency of model bias relative to observations, less sensitive to outliers than mean.
  • Useful for robust model evaluation in the presence of skewed or non-normal error distributions.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median normalized bias (percent).

Source code in src/monet_stats/error_metrics.py 
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def MdnNB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Normalized Bias (%).

    Typical Use Cases
    -----------------
    - Assessing the central tendency of model bias relative to observations,
      less sensitive to outliers than mean.
    - Useful for robust model evaluation in the presence of skewed or non-normal
      error distributions.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median normalized bias (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = ((mod - obs) / obs).median(dim=dim, keep_attrs=True) * 100.0
        # Update history
        history = f"MdnNB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.median(np.ma.masked_invalid((mod - obs) / obs), axis=axis) * 100.0

MdnNE(obs, mod, axis=None)

Median Normalized Gross Error (%).

Typical Use Cases

  • Evaluating the typical magnitude of model errors relative to observations, robust to outliers.
  • Useful for summarizing error magnitude in non-Gaussian or heavy-tailed error distributions.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median normalized gross error (percent).

Source code in src/monet_stats/error_metrics.py 
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def MdnNE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Normalized Gross Error (%).

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of model errors relative to observations,
      robust to outliers.
    - Useful for summarizing error magnitude in non-Gaussian or heavy-tailed
      error distributions.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median normalized gross error (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (abs(mod - obs) / obs).median(dim=dim, keep_attrs=True) * 100.0
        # Update history
        history = f"MdnNE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.median(np.ma.masked_invalid(np.ma.abs(mod - obs) / obs), axis=axis) * 100.0

MdnO(obs, mod, axis=None)

Median Error (MdnO) - Median of (model - observation).

Typical Use Cases

  • Quantifying the typical bias between model predictions and observations, robust to outliers.
  • Used in robust model evaluation for non-parametric error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the median error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median error (model - observation) in observation units. Returns 0.0 for perfect agreement.

Examples

import numpy as np from monet_stats.error_metrics import MdnO obs = np.array([1, 2, 3, 4, 5]) mod = np.array([1.1, 2.1, 3.1, 4.1, 5.1]) MdnO(obs, mod) 0.1

Source code in src/monet_stats/error_metrics.py 
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def MdnO(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Error (MdnO) - Median of (model - observation).

    Typical Use Cases
    -----------------
    - Quantifying the typical bias between model predictions and observations,
      robust to outliers.
    - Used in robust model evaluation for non-parametric error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the median error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median error (model - observation) in observation units.
        Returns 0.0 for perfect agreement.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MdnO
    >>> obs = np.array([1, 2, 3, 4, 5])
    >>> mod = np.array([1.1, 2.1, 3.1, 4.1, 5.1])
    >>> MdnO(obs, mod)
    0.1
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (mod - obs).median(dim=dim, keep_attrs=True)
        # Update history
        history = f"MdnO computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.median(np.subtract(mod, obs), axis=axis)

MdnP(obs, mod, axis=None)

Median Error (MdnP) - Median of (model - observation).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the median error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median error (model - observation) in model units. Returns 0.0 for perfect agreement.

Source code in src/monet_stats/error_metrics.py 
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def MdnP(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Error (MdnP) - Median of (model - observation).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the median error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median error (model - observation) in model units.
        Returns 0.0 for perfect agreement.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (mod - obs).median(dim=dim, keep_attrs=True)
        # Update history
        history = f"MdnP computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.median(np.subtract(mod, obs), axis=axis)

MedAE(obs, mod, axis=None)

Median Absolute Error (MedAE).

Typical Use Cases

  • Evaluating the typical magnitude of errors, robust to outliers and non-normal error distributions.
  • Used in robust regression, model evaluation, and forecast verification.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MedAE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median absolute error.

Examples

import numpy as np from monet_stats.error_metrics import MedAE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MedAE(obs, mod) 1.0

Source code in src/monet_stats/error_metrics.py 
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def MedAE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Absolute Error (MedAE).

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of errors, robust to outliers and
      non-normal error distributions.
    - Used in robust regression, model evaluation, and forecast verification.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MedAE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median absolute error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MedAE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MedAE(obs, mod)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = abs(mod - obs).median(dim=dim, keep_attrs=True)
        # Update history
        history = f"MedAE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.median(np.ma.abs(np.subtract(mod, obs)), axis=axis)

MedAE_m(obs, mod, axis=None)

Median Absolute Error (MedAE) - robust to masked arrays and outliers.

Typical Use Cases

  • Evaluating the typical magnitude of errors, robust to outliers and non-normal error distributions with missing data.
  • Used in robust regression, model evaluation, and forecast verification with incomplete datasets.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute MedAE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median absolute error.

Examples

import numpy as np from monet_stats.error_metrics import MedAE_m obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MedAE_m(obs, mod) 1.0

Source code in src/monet_stats/error_metrics.py 
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def MedAE_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Median Absolute Error (MedAE) - robust to masked arrays and outliers.

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of errors, robust to outliers and
      non-normal error distributions with missing data.
    - Used in robust regression, model evaluation, and forecast verification
      with incomplete datasets.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute MedAE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median absolute error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import MedAE_m
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MedAE_m(obs, mod)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # MedAE implementation for xarray already handles NaNs
        return MedAE(obs, mod, axis=axis)
    else:
        return np.ma.median(np.ma.abs(np.subtract(mod, obs)), axis=axis)

NMSE(obs, mod, axis=None)

Normalized Mean Square Error (NMSE).

Typical Use Cases

  • Quantifying the normalized squared error between model and observations.
  • Used in model evaluation to compare performance across different variables or sites with different scales.
  • Provides dimensionless error metric for cross-comparison.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute NMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Normalized mean square error (unitless).

Examples

import numpy as np from monet_stats.error_metrics import NMSE obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NMSE(obs, mod) 0.25

Source code in src/monet_stats/error_metrics.py 
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def NMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Normalized Mean Square Error (NMSE).

    Typical Use Cases
    -----------------
    - Quantifying the normalized squared error between model and observations.
    - Used in model evaluation to compare performance across different variables
      or sites with different scales.
    - Provides dimensionless error metric for cross-comparison.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute NMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Normalized mean square error (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NMSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NMSE(obs, mod)
    0.25
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        mse = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True)
        obs_var = obs.var(dim=dim)
        # Handle case where variance is 0 (perfect agreement)
        result = xr.where(obs_var == 0, 0, mse / obs_var)
        # Update history
        history = f"NMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        mse = np.mean((np.subtract(mod, obs)) ** 2, axis=axis)
        obs_var = np.var(obs, axis=axis)
        # Handle case where variance is 0 (perfect agreement)
        result = np.where(obs_var == 0, 0, mse / obs_var)
        return result.item() if np.ndim(result) == 0 else result

NMdnGE(obs, mod, axis=None)

Normalized Median Gross Error (%).

Typical Use Cases

  • Comparing the typical (median) error magnitude, normalized by the mean observation, for robust model evaluation.
  • Useful for inter-comparison of model performance across sites or variables with different scales.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Normalized median gross error (percent).

Examples

import numpy as np from monet_stats.error_metrics import NMdnGE obs = np.array([1, 2, 3, 4, 100]) mod = np.array([1.1, 2.1, 3.1, 4.1, 105]) NMdnGE(obs, mod) 0.45454545454545453

Source code in src/monet_stats/error_metrics.py 
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def NMdnGE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Normalized Median Gross Error (%).

    Typical Use Cases
    -----------------
    - Comparing the typical (median) error magnitude, normalized by the mean
      observation, for robust model evaluation.
    - Useful for inter-comparison of model performance across sites or variables
      with different scales.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Normalized median gross error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NMdnGE
    >>> obs = np.array([1, 2, 3, 4, 100])
    >>> mod = np.array([1.1, 2.1, 3.1, 4.1, 105])
    >>> NMdnGE(obs, mod)
    0.45454545454545453
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (abs(mod - obs).median(dim=dim) / obs.mean(dim=dim)) * 100.0
        # Update history
        history = f"NMdnGE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.masked_invalid(np.ma.median(np.ma.abs(mod - obs), axis=axis) / np.ma.mean(obs, axis=axis)) * 100.0

NO(obs, mod=None, axis=None)

N Observations (#).

Typical Use Cases

  • Counting the number of valid (non-masked) observations in a dataset.
  • Used to report sample size for statistical summaries and model evaluation.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray, optional Model predicted values (not used for NO but included for signature matching). axis : int, str, or iterable of such, optional Axis or dimension along which to count.

Returns

int, numpy.ndarray, or xarray.DataArray Number of valid observations.

Source code in src/monet_stats/error_metrics.py 
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def NO(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Optional[Union[np.ndarray, xr.DataArray]] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[int, np.ndarray, xr.DataArray]:
    """
    N Observations (#).

    Typical Use Cases
    -----------------
    - Counting the number of valid (non-masked) observations in a dataset.
    - Used to report sample size for statistical summaries and model evaluation.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray, optional
        Model predicted values (not used for NO but included for signature matching).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to count.

    Returns
    -------
    int, numpy.ndarray, or xarray.DataArray
        Number of valid observations.
    """
    if isinstance(obs, xr.DataArray):
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        return obs.count(dim=dim)
    else:
        return (~np.ma.getmaskarray(obs)).sum(axis=axis)

NOP(obs, mod, axis=None)

N Observations/Prediction Pairs (#).

Typical Use Cases

  • Counting the number of valid observation-prediction pairs for paired statistical analysis.
  • Used to ensure sample size consistency in paired model evaluation metrics.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to count.

Returns

int, numpy.ndarray, or xarray.DataArray Number of valid pairs.

Source code in src/monet_stats/error_metrics.py 
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def NOP(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[int, np.ndarray, xr.DataArray]:
    """
    N Observations/Prediction Pairs (#).

    Typical Use Cases
    -----------------
    - Counting the number of valid observation-prediction pairs for paired
      statistical analysis.
    - Used to ensure sample size consistency in paired model evaluation metrics.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to count.

    Returns
    -------
    int, numpy.ndarray, or xarray.DataArray
        Number of valid pairs.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        # count() on aligned xarray handles NaNs in both by alignment if NaNs are coords,
        # but if NaNs are in data, we need to mask.
        # However, count() counts non-NaN values.
        # To get pairs where BOTH are not NaN:
        mask = obs.notnull() & mod.notnull()
        return mask.sum(dim=dim)
    else:
        obsc, modc = matchmasks(obs, mod)
        return (~np.ma.getmaskarray(obsc)).sum(axis=axis)

NP(obs=None, mod=None, axis=None)

N Predictions (#).

Typical Use Cases

  • Counting the number of valid (non-masked) model predictions in a dataset.
  • Used to report sample size for model output and for filtering invalid predictions.

Parameters

obs : numpy.ndarray or xarray.DataArray, optional Observed values (not used for NP but included for signature matching). mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to count.

Returns

int, numpy.ndarray, or xarray.DataArray Number of valid predictions.

Source code in src/monet_stats/error_metrics.py 
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def NP(
    obs: Optional[Union[np.ndarray, xr.DataArray]] = None,
    mod: Union[np.ndarray, xr.DataArray] = None,
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[int, np.ndarray, xr.DataArray]:
    """
    N Predictions (#).

    Typical Use Cases
    -----------------
    - Counting the number of valid (non-masked) model predictions in a dataset.
    - Used to report sample size for model output and for filtering invalid
      predictions.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray, optional
        Observed values (not used for NP but included for signature matching).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to count.

    Returns
    -------
    int, numpy.ndarray, or xarray.DataArray
        Number of valid predictions.
    """
    if isinstance(mod, xr.DataArray):
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = mod.dims[axis]
        else:
            dim = axis
        return mod.count(dim=dim)
    else:
        return (~np.ma.getmaskarray(mod)).sum(axis=axis)

NRMSE(obs, mod, axis=None)

Normalized Root Mean Square Error (NRMSE).

Typical Use Cases

  • Quantifying the relative error between model and observations, normalized by the range of observations.
  • Used in model evaluation to compare performance across different variables or sites with different scales.
  • Provides dimensionless error metric for cross-comparison.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute NRMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Normalized root mean square error (unitless).

Examples

import numpy as np from monet_stats.error_metrics import NRMSE obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NRMSE(obs, mod) 0.4714045207910317

Source code in src/monet_stats/error_metrics.py 
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def NRMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Normalized Root Mean Square Error (NRMSE).

    Typical Use Cases
    -----------------
    - Quantifying the relative error between model and observations, normalized
      by the range of observations.
    - Used in model evaluation to compare performance across different variables
      or sites with different scales.
    - Provides dimensionless error metric for cross-comparison.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute NRMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Normalized root mean square error (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NRMSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NRMSE(obs, mod)
    0.4714045207910317
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        rmse = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        obs_range = obs.max(dim=dim) - obs.min(dim=dim)
        result = xr.where(obs_range == 0, 0, rmse / obs_range)
        # Update history
        history = f"NRMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        rmse = np.ma.sqrt(np.ma.mean((np.subtract(mod, obs)) ** 2, axis=axis))
        obs_range = np.ma.max(obs, axis=axis) - np.ma.min(obs, axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = np.where(obs_range == 0, 0, rmse / obs_range)
            return result.item() if np.ndim(result) == 0 else result

NSC(obs, mod, axis=None)

Nash-Sutcliffe Coefficient (NSC) - Alternative to NSE.

Typical Use Cases

  • Quantifying the predictive power of hydrological models relative to the mean of observations.
  • Used in hydrology, meteorology, and environmental model evaluation.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute NSC.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Nash-Sutcliffe coefficient (unitless).

Examples

import numpy as np from monet_stats.error_metrics import NSC obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NSC(obs, mod) -0.33333333333333326

Source code in src/monet_stats/error_metrics.py 
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def NSC(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Nash-Sutcliffe Coefficient (NSC) - Alternative to NSE.

    Typical Use Cases
    -----------------
    - Quantifying the predictive power of hydrological models relative to
      the mean of observations.
    - Used in hydrology, meteorology, and environmental model evaluation.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute NSC.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Nash-Sutcliffe coefficient (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NSC
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NSC(obs, mod)
    -0.33333333333333326
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        obs_mean = obs.mean(dim=dim)
        numerator = ((obs - mod) ** 2).sum(dim=dim)
        denominator = ((obs - obs_mean) ** 2).sum(dim=dim)
        result = 1.0 - (numerator / denominator)
        # Update history
        history = f"NSC computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_mean = np.mean(obs, axis=axis, keepdims=True)
        numerator = np.sum((obs - mod) ** 2, axis=axis)
        denominator = np.sum((obs - obs_mean) ** 2, axis=axis)
        return 1.0 - (numerator / denominator)

NSE_alpha(obs, mod, axis=None)

NSE Alpha - Decomposed NSE component measuring ratio of standard deviations.

Typical Use Cases

  • Quantifying the model's ability to capture the variability of observations.
  • Used in model evaluation to assess how well model represents observed variability.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute NSE_alpha.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray NSE alpha component (unitless).

Examples

import numpy as np from monet_stats.error_metrics import NSE_alpha obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NSE_alpha(obs, mod) 0.0

Source code in src/monet_stats/error_metrics.py 
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def NSE_alpha(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    NSE Alpha - Decomposed NSE component measuring ratio of standard deviations.

    Typical Use Cases
    -----------------
    - Quantifying the model's ability to capture the variability of observations.
    - Used in model evaluation to assess how well model represents observed
      variability.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute NSE_alpha.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        NSE alpha component (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NSE_alpha
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NSE_alpha(obs, mod)
    0.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = mod.std(dim=dim) / obs.std(dim=dim)
        # Update history
        history = f"NSE_alpha computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.std(mod, axis=axis) / np.std(obs, axis=axis)

NSE_beta(obs, mod, axis=None)

NSE Beta - Decomposed NSE component measuring bias.

Typical Use Cases

  • Quantifying the systematic bias between model and observations.
  • Used in model evaluation to assess mean differences between model and observations.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute NSE_beta.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray NSE beta component (unitless).

Examples

import numpy as np from monet_stats.error_metrics import NSE_beta obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NSE_beta(obs, mod) 0.5

Source code in src/monet_stats/error_metrics.py 
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def NSE_beta(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    NSE Beta - Decomposed NSE component measuring bias.

    Typical Use Cases
    -----------------
    - Quantifying the systematic bias between model and observations.
    - Used in model evaluation to assess mean differences between model and
      observations.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute NSE_beta.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        NSE beta component (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import NSE_beta
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NSE_beta(obs, mod)
    0.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = mod.mean(dim=dim) / obs.mean(dim=dim)
        # Update history
        history = f"NSE_beta computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.mean(mod, axis=axis) / np.mean(obs, axis=axis)

RM(obs, mod, axis=None)

Root Mean Error (RM) - Root of mean squared error.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root of mean squared error (observation units). Returns 0.0 for perfect agreement.

Source code in src/monet_stats/error_metrics.py 
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def RM(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Mean Error (RM) - Root of mean squared error.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root of mean squared error (observation units).
        Returns 0.0 for perfect agreement.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = np.sqrt(((obs - mod) ** 2).mean(dim=dim, keep_attrs=True))
        # Update history
        history = f"RM computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.sqrt(np.mean((np.subtract(obs, mod)) ** 2, axis=axis))

RMSE(obs, mod, axis=None)

Root Mean Square Error (RMSE).

Typical Use Cases

  • Quantifying the average magnitude of errors between model and observations, accounting for large errors more heavily than MAE.
  • Used in model evaluation, forecast verification, and regression analysis.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute RMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root mean square error.

Examples

import numpy as np from monet_stats.error_metrics import RMSE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) RMSE(obs, mod) 0.816496580927726

Source code in src/monet_stats/error_metrics.py 
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def RMSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Mean Square Error (RMSE).

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of errors between model and observations,
      accounting for large errors more heavily than MAE.
    - Used in model evaluation, forecast verification, and regression analysis.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute RMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root mean square error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import RMSE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> RMSE(obs, mod)
    0.816496580927726
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"RMSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.sqrt(np.mean((np.subtract(mod, obs)) ** 2, axis=axis))

RMSE_m(obs, mod, axis=None)

Root Mean Square Error (RMSE) - robust to masked arrays.

Typical Use Cases

  • Quantifying the average magnitude of errors between model and observations, accounting for large errors more heavily than MAE, robust to missing data.
  • Used in model evaluation, forecast verification, and regression analysis with incomplete datasets.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute RMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root mean square error.

Examples

import numpy as np from monet_stats.error_metrics import RMSE_m obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) RMSE_m(obs, mod) 0.816496580927726

Source code in src/monet_stats/error_metrics.py 
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def RMSE_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Mean Square Error (RMSE) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of errors between model and
      observations, accounting for large errors more heavily than MAE,
      robust to missing data.
    - Used in model evaluation, forecast verification, and regression analysis
      with incomplete datasets.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute RMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root mean square error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import RMSE_m
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> RMSE_m(obs, mod)
    0.816496580927726
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # RMSE implementation for xarray already handles NaNs
        return RMSE(obs, mod, axis=axis)
    else:
        return np.ma.sqrt(np.ma.mean((np.subtract(mod, obs)) ** 2, axis=axis))

RMSE_norm(obs, mod, axis=None)

Normalized Root Mean Square Error (RMSE_norm).

Normalizes RMSE by the range of observations.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute normalized RMSE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Normalized root mean square error (unitless).

Source code in src/monet_stats/error_metrics.py 
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def RMSE_norm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Normalized Root Mean Square Error (RMSE_norm).

    Normalizes RMSE by the range of observations.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute normalized RMSE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Normalized root mean square error (unitless).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        rmse = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        obs_min = obs.min(dim=dim)
        obs_max = obs.max(dim=dim)
        obs_range = obs_max - obs_min
        # Avoid division by zero
        result = xr.where(obs_range == 0, rmse, rmse / obs_range)
        # Update history
        history = f"RMSE_norm computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        rmse = np.sqrt(np.mean((np.subtract(mod, obs)) ** 2, axis=axis))
        obs_min = np.min(obs, axis=axis)
        obs_max = np.max(obs, axis=axis)
        obs_range = obs_max - obs_min
        # Avoid division by zero
        result = np.where(obs_range == 0, rmse, rmse / obs_range)
        return result.item() if np.ndim(result) == 0 else result

RMSPE(obs, mod, axis=None)

Root Mean Square Percentage Error (RMSPE).

Typical Use Cases

  • Quantifying the average relative error between model and observations as a percentage, emphasizing larger errors.
  • Used in time series forecasting, regression, and model evaluation for percentage-based error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute RMSPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root mean square percentage error (in percent).

Examples

import numpy as np from monet_stats.error_metrics import RMSPE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) RMSPE(obs, mod) 50.0

Source code in src/monet_stats/error_metrics.py 
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def RMSPE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Mean Square Percentage Error (RMSPE).

    Typical Use Cases
    -----------------
    - Quantifying the average relative error between model and observations as
      a percentage, emphasizing larger errors.
    - Used in time series forecasting, regression, and model evaluation for
      percentage-based error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute RMSPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root mean square percentage error (in percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import RMSPE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> RMSPE(obs, mod)
    50.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (100 * ((mod - obs) / obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        # Update history
        history = f"RMSPE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return 100 * np.ma.sqrt(np.ma.mean(((mod - obs) / obs) ** 2, axis=axis))

RMdn(obs, mod, axis=None)

Root Median Error (RMdn) - Root of median squared error.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Root of median squared error (observation units). Returns 0.0 for perfect agreement.

Source code in src/monet_stats/error_metrics.py 
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def RMdn(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Root Median Error (RMdn) - Root of median squared error.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Root of median squared error (observation units).
        Returns 0.0 for perfect agreement.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = np.sqrt(((obs - mod) ** 2).median(dim=dim, keep_attrs=True))
        # Update history
        history = f"RMdn computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        squared_errors = (np.subtract(obs, mod)) ** 2
        return np.sqrt(np.median(squared_errors, axis=axis))

STDO(obs, mod, axis=None)

Standard deviation of Observation Errors (obs - mod).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the standard deviation. If None, computes over all axes.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Standard deviation of (observation - model) errors. Returns 0.0 for perfect agreement.

Examples

import numpy as np from monet_stats.error_metrics import STDO obs = np.array([1.0, 2.0, 3.0]) mod = np.array([1.1, 1.9, 3.2]) STDO(obs, mod) 0.1247219128924647

Source code in src/monet_stats/error_metrics.py 
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def STDO(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Standard deviation of Observation Errors (obs - mod).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the standard deviation.
        If None, computes over all axes.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Standard deviation of (observation - model) errors.
        Returns 0.0 for perfect agreement.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import STDO
    >>> obs = np.array([1.0, 2.0, 3.0])
    >>> mod = np.array([1.1, 1.9, 3.2])
    >>> STDO(obs, mod)
    0.1247219128924647
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        errors = obs - mod
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = errors.std(dim=dim, keep_attrs=True)
        # Update history
        history = f"STDO computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result

    # Fallback to numpy-compatible logic
    errors = np.subtract(obs, mod)
    return np.std(errors, axis=axis)

STDP(obs, mod, axis=None)

Standard deviation of Prediction Errors (mod - obs).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the standard deviation. If None, computes over all axes.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Standard deviation of (model - observation) errors. Returns 0.0 for perfect agreement.

Examples

import numpy as np from monet_stats.error_metrics import STDP obs = np.array([1.0, 2.0, 3.0]) mod = np.array([1.1, 1.9, 3.2]) STDP(obs, mod) 0.1247219128924647

Source code in src/monet_stats/error_metrics.py 
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def STDP(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Standard deviation of Prediction Errors (mod - obs).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the standard deviation.
        If None, computes over all axes.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Standard deviation of (model - observation) errors.
        Returns 0.0 for perfect agreement.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import STDP
    >>> obs = np.array([1.0, 2.0, 3.0])
    >>> mod = np.array([1.1, 1.9, 3.2])
    >>> STDP(obs, mod)
    0.1247219128924647
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        errors = mod - obs
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = errors.std(dim=dim, keep_attrs=True)
        # Update history
        history = f"STDP computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result

    # Fallback to numpy-compatible logic
    errors = np.subtract(mod, obs)
    return np.std(errors, axis=axis)

VOLUMETRIC_ERROR(obs, mod, axis=None)

Volumetric Error Metric.

Typical Use Cases

  • Quantifying the volume difference between observed and modeled features.
  • Used in hydrology for flood extent verification.
  • Applied in meteorology for precipitation volume verification.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute volumetric error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Volumetric error metric.

Examples

import numpy as np from monet_stats.error_metrics import VOLUMETRIC_ERROR obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) VOLUMETRIC_ERROR(obs, mod) 0.2

Source code in src/monet_stats/error_metrics.py 
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def VOLUMETRIC_ERROR(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Volumetric Error Metric.

    Typical Use Cases
    -----------------
    - Quantifying the volume difference between observed and modeled features.
    - Used in hydrology for flood extent verification.
    - Applied in meteorology for precipitation volume verification.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute volumetric error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Volumetric error metric.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import VOLUMETRIC_ERROR
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> VOLUMETRIC_ERROR(obs, mod)
    0.2
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        obs_sum = obs.sum(dim=dim)
        mod_sum = mod.sum(dim=dim)
        result = abs(mod_sum - obs_sum) / abs(obs_sum)
        # Update history
        history = f"VOLUMETRIC_ERROR computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_sum = np.sum(obs, axis=axis)
        mod_sum = np.sum(mod, axis=axis)
        return np.abs(mod_sum - obs_sum) / np.abs(obs_sum)

WDMB(obs, mod, axis=None)

Wind Direction Mean Bias (WDMB, standard version).

This version uses circlebias, which is not robust to masked arrays. Use this if your data are dense and do not contain missing values.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Model predicted wind direction values (degrees). axis : int, str, or iterable of such, optional Axis or dimension along which to compute the mean bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean wind direction bias (degrees).

Source code in src/monet_stats/error_metrics.py 
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def WDMB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Mean Bias (WDMB, standard version).

    This version uses circlebias, which is not robust to masked arrays.
    Use this if your data are dense and do not contain missing values.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the mean bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean wind direction bias (degrees).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = circlebias(mod - obs).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"WDMB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.mean(circlebias(np.subtract(mod, obs)), axis=axis)

WDMB_m(obs, mod, axis=None)

Wind Direction Mean Bias (WDMB, robust version for masked arrays).

This version uses circlebias_m, which is robust to masked arrays and missing data. Use this if your data may contain NaNs or masked values.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Model predicted wind direction values (degrees). axis : int, str, or iterable of such, optional Axis or dimension along which to compute the mean bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean wind direction bias (degrees).

Source code in src/monet_stats/error_metrics.py 
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def WDMB_m(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Mean Bias (WDMB, robust version for masked arrays).

    This version uses circlebias_m, which is robust to masked arrays and
    missing data. Use this if your data may contain NaNs or masked values.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the mean bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean wind direction bias (degrees).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = circlebias_m(mod - obs).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"WDMB_m computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.mean(circlebias_m(np.subtract(mod, obs)), axis=axis)

WDMdnB(obs, mod, axis=None)

Wind Direction Median Bias (WDMdnB).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed wind direction values (degrees). mod : numpy.ndarray or xarray.DataArray Model predicted wind direction values (degrees). axis : int, str, or iterable of such, optional Axis or dimension along which to compute the median bias.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Median wind direction bias (degrees).

Source code in src/monet_stats/error_metrics.py 
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def WDMdnB(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Wind Direction Median Bias (WDMdnB).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed wind direction values (degrees).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted wind direction values (degrees).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the median bias.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Median wind direction bias (degrees).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = circlebias(mod - obs).median(dim=dim, keep_attrs=True)
        # Update history
        history = f"WDMdnB computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.ma.median(circlebias(np.subtract(mod, obs)), axis=axis)

bias_fraction(obs, mod, axis=None)

Bias Fraction (BF).

Quantifies the fraction of total error that is due to systematic bias.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute bias fraction.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Bias fraction (unitless, 0-1).

Source code in src/monet_stats/error_metrics.py 
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def bias_fraction(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Bias Fraction (BF).

    Quantifies the fraction of total error that is due to systematic bias.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute bias fraction.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Bias fraction (unitless, 0-1).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        bias = (mod - obs).mean(dim=dim)
        total_error = np.sqrt(((mod - obs) ** 2).mean(dim=dim, keep_attrs=True))
        # Avoid division by zero
        result = xr.where(total_error == 0, 0, (bias**2) / (total_error**2))
        # Update history
        history = f"bias_fraction computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        bias = np.mean(np.subtract(mod, obs), axis=axis)
        total_error = np.sqrt(np.mean((np.subtract(mod, obs)) ** 2, axis=axis))
        # Avoid division by zero
        result = np.where(total_error == 0, 0, (bias**2) / (total_error**2))
        return result.item() if np.ndim(result) == 0 else result

sMAPE(obs, mod, axis=None)

Symmetric Mean Absolute Percentage Error (sMAPE).

Typical Use Cases

  • Quantifying the average relative error between model and observations, normalized by their mean.
  • Used in time series forecasting, regression, and model evaluation for percentage-based error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute sMAPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Symmetric mean absolute percentage error (in percent).

Examples

import numpy as np from monet_stats.error_metrics import sMAPE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) sMAPE(obs, mod) 28.57142857142857

Source code in src/monet_stats/error_metrics.py 
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def sMAPE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Symmetric Mean Absolute Percentage Error (sMAPE).

    Typical Use Cases
    -----------------
    - Quantifying the average relative error between model and observations,
      normalized by their mean.
    - Used in time series forecasting, regression, and model evaluation for
      percentage-based error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute sMAPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Symmetric mean absolute percentage error (in percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import sMAPE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> sMAPE(obs, mod)
    28.57142857142857
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis
        result = (200 * abs(mod - obs) / (abs(mod) + abs(obs))).mean(dim=dim, keep_attrs=True)
        # Update history
        history = f"sMAPE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return (200 * np.ma.abs(np.subtract(mod, obs)) / (np.ma.abs(mod) + np.ma.abs(obs))).mean(axis=axis)

sMAPEm(obs, mod, axis=None)

Symmetric Mean Absolute Percentage Error (sMAPE) - robust to masked arrays.

Typical Use Cases

  • Quantifying the average relative error between model and observations, normalized by their mean, robust to missing data.
  • Used in time series forecasting, regression, and model evaluation for percentage-based error assessment.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute sMAPE.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Symmetric mean absolute percentage error (in percent).

Examples

import numpy as np from monet_stats.error_metrics import sMAPEm obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) sMAPEm(obs, mod) 28.57142857142857

Source code in src/monet_stats/error_metrics.py 
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def sMAPEm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Symmetric Mean Absolute Percentage Error (sMAPE) - robust to masked arrays.

    Typical Use Cases
    -----------------
    - Quantifying the average relative error between model and observations,
      normalized by their mean, robust to missing data.
    - Used in time series forecasting, regression, and model evaluation for
      percentage-based error assessment.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute sMAPE.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Symmetric mean absolute percentage error (in percent).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.error_metrics import sMAPEm
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> sMAPEm(obs, mod)
    28.57142857142857
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # sMAPE implementation for xarray already handles NaNs
        return sMAPE(obs, mod, axis=axis)
    else:
        return 200 * np.ma.mean(np.ma.abs(mod - obs) / (np.ma.abs(mod) + np.ma.abs(obs)), axis=axis)

Efficiency Metrics

Efficiency Metrics for Model Evaluation

MSE(obs, mod, axis=None)

Mean Squared Error (MSE).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the error.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Mean squared error.

Examples

import numpy as np from monet_stats.efficiency_metrics import MSE obs = np.array([1, 2, 3]) mod = np.array([2, 2, 4]) MSE(obs, mod) 0.6666666666666666

Source code in src/monet_stats/efficiency_metrics.py 
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def MSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Mean Squared Error (MSE).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the error.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Mean squared error.

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import MSE
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([2, 2, 4])
    >>> MSE(obs, mod)
    0.6666666666666666
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        result = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True)

        # Update history
        history = f"MSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        return np.nanmean((mod - obs) ** 2, axis=axis)

NSE(obs, mod, axis=None)

Nash-Sutcliffe Efficiency (NSE).

Typical Use Cases

  • Quantifying the predictive power of hydrological models relative to the mean of observations.
  • Used in hydrology, meteorology, and environmental model evaluation.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Nash-Sutcliffe efficiency (unitless).

Examples

import numpy as np from monet_stats.efficiency_metrics import NSE obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 2.9, 4.1]) NSE(obs, mod) 0.98

Source code in src/monet_stats/efficiency_metrics.py 
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def NSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Nash-Sutcliffe Efficiency (NSE).

    Typical Use Cases
    -----------------
    - Quantifying the predictive power of hydrological models relative to the
      mean of observations.
    - Used in hydrology, meteorology, and environmental model evaluation.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Nash-Sutcliffe efficiency (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import NSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 2.9, 4.1])
    >>> NSE(obs, mod)
    0.98
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_mean = obs.mean(dim=dim)
        numerator = ((obs - mod) ** 2).sum(dim=dim)
        denominator = ((obs - obs_mean) ** 2).sum(dim=dim)

        # Handle division by zero
        result = 1.0 - (numerator / denominator)
        result = xr.where((numerator == 0) & (denominator == 0), 1.0, result)
        result = xr.where((numerator != 0) & (denominator == 0), -np.inf, result)

        # Update history
        history = f"NSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
        numerator = np.nansum((obs - mod) ** 2, axis=axis)
        denominator = np.nansum((obs - obs_mean) ** 2, axis=axis)

        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (numerator / denominator)
            result = np.where((numerator == 0) & (denominator == 0), 1.0, result)
            result = np.where((numerator != 0) & (denominator == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

NSElog(obs, mod, axis=None)

Log Nash-Sutcliffe Efficiency (NSElog).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values (positive values only). mod : numpy.ndarray or xarray.DataArray Model predicted values (positive values only). axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Log Nash-Sutcliffe efficiency (unitless).

Examples

import numpy as np from monet_stats.efficiency_metrics import NSElog obs = np.array([1, 10, 100]) mod = np.array([1.1, 9.0, 110]) NSElog(obs, mod) 0.988

Source code in src/monet_stats/efficiency_metrics.py 
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def NSElog(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Log Nash-Sutcliffe Efficiency (NSElog).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values (positive values only).
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values (positive values only).
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Log Nash-Sutcliffe efficiency (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import NSElog
    >>> obs = np.array([1, 10, 100])
    >>> mod = np.array([1.1, 9.0, 110])
    >>> NSElog(obs, mod)
    0.988
    """
    epsilon = 1e-6
    obs_log = np.log(obs + epsilon)
    mod_log = np.log(mod + epsilon)
    return NSE(obs_log, mod_log, axis=axis)

NSEm(obs, mod, axis=None)

Nash-Sutcliffe Efficiency (NSE) - robust to masked arrays.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Nash-Sutcliffe efficiency (unitless).

Examples

import numpy as np from monet_stats.efficiency_metrics import NSEm obs = np.array([1, 2, np.nan, 4]) mod = np.array([1.1, 2.1, 3.0, 4.1]) NSEm(obs, mod) 0.985

Source code in src/monet_stats/efficiency_metrics.py 
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def NSEm(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Nash-Sutcliffe Efficiency (NSE) - robust to masked arrays.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Nash-Sutcliffe efficiency (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import NSEm
    >>> obs = np.array([1, 2, np.nan, 4])
    >>> mod = np.array([1.1, 2.1, 3.0, 4.1])
    >>> NSEm(obs, mod)
    0.985
    """
    # Standard NSE implementation already handles NaNs if using nan-aware functions
    return NSE(obs, mod, axis=axis)

PC(obs, mod, axis=None, tolerance=0.1)

Percent of Correct (PC).

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic. tolerance : float, optional Fraction of observed value used as tolerance (default 0.1).

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Percent of correct predictions (0-100%).

Examples

import numpy as np from monet_stats.efficiency_metrics import PC obs = np.array([1, 2, 3, 4]) mod = np.array([1.05, 2.5, 2.95, 4.05]) PC(obs, mod) 75.0

Source code in src/monet_stats/efficiency_metrics.py 
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def PC(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
    tolerance: float = 0.1,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Percent of Correct (PC).

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.
    tolerance : float, optional
        Fraction of observed value used as tolerance (default 0.1).

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Percent of correct predictions (0-100%).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import PC
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.05, 2.5, 2.95, 4.05])
    >>> PC(obs, mod)
    75.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        tol = tolerance * np.abs(obs)
        correct = np.abs(obs - mod) <= tol
        result = (correct.sum(dim=dim) / correct.count(dim=dim)) * 100.0

        # Update history
        history = f"PC computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        tol = tolerance * np.abs(obs)
        correct = np.abs(obs - mod) <= tol
        total = np.sum(~np.isnan(correct), axis=axis)
        correct_sum = np.nansum(correct, axis=axis)
        return (correct_sum / total) * 100.0

mNSE(obs, mod, axis=None)

Modified Nash-Sutcliffe Efficiency (mNSE).

Uses absolute differences instead of squared differences.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Modified Nash-Sutcliffe efficiency (unitless).

Examples

import numpy as np from monet_stats.efficiency_metrics import mNSE obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 2.9, 4.1]) mNSE(obs, mod) 0.92

Source code in src/monet_stats/efficiency_metrics.py 
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def mNSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Modified Nash-Sutcliffe Efficiency (mNSE).

    Uses absolute differences instead of squared differences.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Modified Nash-Sutcliffe efficiency (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import mNSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 2.9, 4.1])
    >>> mNSE(obs, mod)
    0.92
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_mean = obs.mean(dim=dim)
        numerator = np.abs(obs - mod).sum(dim=dim)
        denominator = np.abs(obs - obs_mean).sum(dim=dim)

        result = 1.0 - (numerator / denominator)
        result = xr.where((numerator == 0) & (denominator == 0), 1.0, result)
        result = xr.where((numerator != 0) & (denominator == 0), -np.inf, result)

        # Update history
        history = f"mNSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
        numerator = np.nansum(np.abs(obs - mod), axis=axis)
        denominator = np.nansum(np.abs(obs - obs_mean), axis=axis)

        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (numerator / denominator)
            result = np.where((numerator == 0) & (denominator == 0), 1.0, result)
            result = np.where((numerator != 0) & (denominator == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

rNSE(obs, mod, axis=None)

Relative Nash-Sutcliffe Efficiency (rNSE).

Normalizes errors by the range of observed values.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model predicted values. axis : int, str, or iterable of such, optional Axis or dimension along which to compute the statistic.

Returns

numpy.number, numpy.ndarray, or xarray.DataArray Relative Nash-Sutcliffe efficiency (unitless).

Examples

import numpy as np from monet_stats.efficiency_metrics import rNSE obs = np.array([1, 2, 3, 4]) mod = np.array([1.1, 2.1, 2.9, 4.1]) rNSE(obs, mod) 0.992

Source code in src/monet_stats/efficiency_metrics.py 
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def rNSE(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Relative Nash-Sutcliffe Efficiency (rNSE).

    Normalizes errors by the range of observed values.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model predicted values.
    axis : int, str, or iterable of such, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    numpy.number, numpy.ndarray, or xarray.DataArray
        Relative Nash-Sutcliffe efficiency (unitless).

    Examples
    --------
    >>> import numpy as np
    >>> from monet_stats.efficiency_metrics import rNSE
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([1.1, 2.1, 2.9, 4.1])
    >>> rNSE(obs, mod)
    0.992
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Handle axis vs dim
        if axis is not None and isinstance(axis, int):
            dim = obs.dims[axis]
        else:
            dim = axis

        obs_mean = obs.mean(dim=dim)
        obs_range = obs.max(dim=dim) - obs.min(dim=dim)
        # Avoid division by zero in normalization
        obs_range_safe = xr.where(obs_range == 0, 1.0, obs_range)

        numerator = (((obs - mod) / obs_range_safe) ** 2).sum(dim=dim)
        denominator = (((obs - obs_mean) / obs_range_safe) ** 2).sum(dim=dim)

        result = 1.0 - (numerator / denominator)
        result = xr.where((numerator == 0) & (denominator == 0), 1.0, result)
        result = xr.where((numerator != 0) & (denominator == 0), -np.inf, result)

        # Update history
        history = f"rNSE computed at {pd.Timestamp.now().isoformat()}"
        result.attrs["history"] = f"{result.attrs.get('history', '')}\n{history}".strip()
        return result
    else:
        obs_mean = np.nanmean(obs, axis=axis, keepdims=True)
        obs_range = np.nanmax(obs, axis=axis, keepdims=True) - np.nanmin(obs, axis=axis, keepdims=True)
        obs_range_safe = np.where(obs_range == 0, 1.0, obs_range)

        with np.errstate(divide="ignore", invalid="ignore"):
            numerator = np.nansum(((obs - mod) / obs_range_safe) ** 2, axis=axis)
            denominator = np.nansum(((obs - obs_mean) / obs_range_safe) ** 2, axis=axis)
            result = 1.0 - (numerator / denominator)
            result = np.where((numerator == 0) & (denominator == 0), 1.0, result)
            result = np.where((numerator != 0) & (denominator == 0), -np.inf, result)
        return result.item() if np.ndim(result) == 0 else result

Relative Metrics

Relative/Percentage Metrics for Model Evaluation (Aero Protocol Compliant)

FB(obs, mod, axis=None)

Fractional Bias (%)

Typical Use Cases

  • Quantifying the average bias as a fraction of the sum of model and observed values.
  • Used in air quality and meteorological model evaluation for normalized bias assessment.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Fractional bias (percent).

Source code in src/monet_stats/relative_metrics.py 
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def FB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Fractional Bias (%)

    Typical Use Cases
    -----------------
    - Quantifying the average bias as a fraction of the sum of model and observed values.
    - Used in air quality and meteorological model evaluation for normalized bias assessment.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Fractional bias (percent).

    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (((mod - obs) / (mod + obs)).mean(dim=dim) * 2.0) * 100.0
        return _update_history(res, "Fractional Bias (FB)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return (np.ma.masked_invalid((mod_arr - obs_arr) / (mod_arr + obs_arr)).mean(axis=axis) * 2.0) * 100.0

FE(obs, mod, axis=None)

Fractional Error (%)

Typical Use Cases

  • Quantifying the average magnitude of model errors as a fraction of the sum of model and observed values.
  • Used in air quality and meteorological model evaluation for normalized error assessment.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Fractional error (percent).

Source code in src/monet_stats/relative_metrics.py 
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def FE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Fractional Error (%)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of model errors as a fraction of the sum of model and observed values.
    - Used in air quality and meteorological model evaluation for normalized error assessment.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Fractional error (percent).

    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod - obs) / (mod + obs)).mean(dim=dim) * 2.0 * 100.0
        return _update_history(res, "Fractional Error (FE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (np.ma.mean(np.ma.abs(mod_arr - obs_arr) / (mod_arr + obs_arr), axis=axis)) * 2.0 * 100.0

ME(obs, mod, axis=None)

Mean Gross Error (model and obs unit)

Typical Use Cases

  • Quantifying the average magnitude of model errors, regardless of direction.
  • Used in model evaluation to summarize overall error size.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Mean gross error value(s).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) ME(obs, mod) 1.0

Source code in src/monet_stats/relative_metrics.py 
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def ME(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Mean Gross Error (model and obs unit)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of model errors, regardless of direction.
    - Used in model evaluation to summarize overall error size.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean gross error value(s).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> ME(obs, mod)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = abs(mod - obs).mean(dim=dim)
        return _update_history(res, "Mean Gross Error (ME)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return np.mean(np.abs(mod_arr - obs_arr), axis=axis)

MNPB(obs, mod, paxis, axis=None)

Mean Normalized Peak Bias (%)

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the mean of normalized peak bias.

Returns

xarray.DataArray or numpy.ndarray or float Mean normalized peak bias (percent).

Source code in src/monet_stats/relative_metrics.py 
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def MNPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Mean Normalized Peak Bias (%)

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the mean of normalized peak bias.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean normalized peak bias (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (((mod.max(dim=pdim) - obs.max(dim=pdim)) / obs.max(dim=pdim)).mean(dim=mdim)) * 100.0
        return _update_history(res, "Mean Normalized Peak Bias (MNPB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            (np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)) / np.ma.max(obs_arr, axis=paxis)
        ).mean(axis=axis) * 100.0

MNPE(obs, mod, paxis, axis=None)

Mean Normalized Peak Error (MNPE, %)

Typical Use Cases

  • Quantifying the average error in peak values between model and observations, normalized by observed peaks.
  • Used in model evaluation for extreme events, such as air quality exceedances or meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the mean of normalized peak error.

Returns

xarray.DataArray or numpy.ndarray or float Mean normalized peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) MNPE(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def MNPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Mean Normalized Peak Error (MNPE, %)

    Typical Use Cases
    -----------------
    - Quantifying the average error in peak values between model and observations, normalized by observed peaks.
    - Used in model evaluation for extreme events, such as air quality exceedances or meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the mean of normalized peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean normalized peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> MNPE(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (abs(mod.max(dim=pdim) - obs.max(dim=pdim)) / obs.max(dim=pdim)).mean(dim=mdim) * 100.0
        return _update_history(res, "Mean Normalized Peak Error (MNPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.abs(np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)) / np.ma.max(obs_arr, axis=paxis)
        ).mean(axis=axis) * 100.0

MPE(obs, mod, axis=None)

Mean Peak Error (%)

Typical Use Cases

  • Quantifying the average error in peak values between model and observations.
  • Used in model evaluation for extreme events, such as air quality exceedances or meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the mean of peak error.

Returns

xarray.DataArray or numpy.ndarray or float Mean peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) MPE(obs, mod) 33.33333333

Source code in src/monet_stats/relative_metrics.py 
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def MPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Mean Peak Error (%)

    Typical Use Cases
    -----------------
    - Quantifying the average error in peak values between model and observations.
    - Used in model evaluation for extreme events, such as air quality exceedances
      or meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the mean of peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> MPE(obs, mod)
    33.33333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod.max(dim=dim) - obs.max(dim=dim)) / obs.max(dim=dim)).mean() * 100.0
        return _update_history(res, "Mean Peak Error (MPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.abs(np.ma.max(mod_arr, axis=axis) - np.ma.max(obs_arr, axis=axis)) / np.ma.max(obs_arr, axis=axis)
        ).mean() * 100.0

MdnE(obs, mod, axis=None)

Median Gross Error (model and obs unit)

Typical Use Cases

  • Evaluating the typical magnitude of model errors, robust to outliers.
  • Used in model evaluation when error distributions are skewed or non-normal.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Median gross error value(s).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) MdnE(obs, mod) 1.0

Source code in src/monet_stats/relative_metrics.py 
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def MdnE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Median Gross Error (model and obs unit)

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of model errors, robust to outliers.
    - Used in model evaluation when error distributions are skewed or non-normal.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Median gross error value(s).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> MdnE(obs, mod)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = abs(mod - obs).median(dim=dim)
        return _update_history(res, "Median Gross Error (MdnE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return np.ma.median(np.ma.abs(mod_arr - obs_arr), axis=axis)

MdnNPB(obs, mod, paxis, axis=None)

Median Normalized Peak Bias (%)

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak bias.

Returns

xarray.DataArray or numpy.ndarray or float Median normalized peak bias (percent).

Source code in src/monet_stats/relative_metrics.py 
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def MdnNPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Median Normalized Peak Bias (%)

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak bias.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Median normalized peak bias (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = ((mod.max(dim=pdim) - obs.max(dim=pdim)) / obs.max(dim=pdim)).median(dim=mdim) * 100.0
        return _update_history(res, "Median Normalized Peak Bias (MdnNPB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.median(
                ((np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)) / np.ma.max(obs_arr, axis=paxis)),
                axis=axis,
            )
            * 100.0
        )

MdnNPE(obs, mod, paxis, axis=None)

Median Normalized Peak Error (MdnNPE, %)

Typical Use Cases

  • Evaluating the typical error in peak values between model and observations, normalized by observed peaks, robust to outliers.
  • Used in robust model evaluation for extreme events, such as air quality exceedances or meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak error.

Returns

xarray.DataArray or numpy.ndarray or float Median normalized peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) MdnNPE(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def MdnNPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Median Normalized Peak Error (MdnNPE, %)

    Typical Use Cases
    -----------------
    - Evaluating the typical error in peak values between model and observations,
      normalized by observed peaks, robust to outliers.
    - Used in robust model evaluation for extreme events, such as air quality exceedances
      or meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Median normalized peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> MdnNPE(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (abs(mod.max(dim=pdim) - obs.max(dim=pdim)) / obs.max(dim=pdim)).median(dim=mdim) * 100.0
        return _update_history(res, "Median Normalized Peak Error (MdnNPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.median(
                (
                    np.ma.abs(np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis))
                    / np.ma.max(obs_arr, axis=paxis)
                ),
                axis=axis,
            )
            * 100.0
        )

MdnPE(obs, mod, axis=None)

Median Peak Error (%)

Typical Use Cases

  • Evaluating the typical error in peak values between model and observations, robust to outliers.
  • Used in robust model evaluation for extreme events, such as air quality exceedances or meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the median of peak error.

Returns

xarray.DataArray or numpy.ndarray or float Median peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) MdnPE(obs, mod) 33.333333333

Source code in src/monet_stats/relative_metrics.py 
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def MdnPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Median Peak Error (%)

    Typical Use Cases
    -----------------
    - Evaluating the typical error in peak values between model and observations,
      robust to outliers.
    - Used in robust model evaluation for extreme events, such as air quality
      exceedances or meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Median peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> MdnPE(obs, mod)
    33.333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod.max(dim=dim) - obs.max(dim=dim)) / obs.max(dim=dim)).median() * 100.0
        return _update_history(res, "Median Peak Error (MdnPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.median(
                (
                    np.ma.abs(np.ma.max(mod_arr, axis=axis) - np.ma.max(obs_arr, axis=axis))
                    / np.ma.max(obs_arr, axis=axis)
                ),
                axis=axis,
            )
            * 100.0
        )

NMB(obs, mod, axis=None)

Normalized Mean Bias (%)

Typical Use Cases

  • Comparing model bias across variables or datasets with different units or scales.
  • Common in regulatory and operational air quality model performance reports.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean bias (percent).

Examples

import numpy as np obs = np.array([1, 2, 3]) mod = np.array([1.1, 2.2, 3.3]) NMB(obs, mod) 10.0

Source code in src/monet_stats/relative_metrics.py 
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def NMB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Bias (%)

    Typical Use Cases
    -----------------
    - Comparing model bias across variables or datasets with different units or scales.
    - Common in regulatory and operational air quality model performance reports.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3])
    >>> mod = np.array([1.1, 2.2, 3.3])
    >>> NMB(obs, mod)
    10.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Ensure we use dimension name if axis is int
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (mod - obs).sum(dim=dim) / obs.sum(dim=dim) * 100.0
        return _update_history(res, "Normalized Mean Bias (NMB)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return (mod_arr - obs_arr).sum(axis=axis) / obs_arr.sum(axis=axis) * 100.0

NMB_ABS(obs, mod, axis=None)

Normalized Mean Bias - Absolute of the denominator (%)

Typical Use Cases

  • Quantifying normalized mean bias when the denominator (sum of observations) may be negative or zero.
  • Used for robust model evaluation in cases with possible sign changes in the observed data sum.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean bias with absolute denominator (percent).

Source code in src/monet_stats/relative_metrics.py 
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def NMB_ABS(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Bias - Absolute of the denominator (%)

    Typical Use Cases
    -----------------
    - Quantifying normalized mean bias when the denominator (sum of observations) may be negative or zero.
    - Used for robust model evaluation in cases with possible sign changes in the observed data sum.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean bias with absolute denominator (percent).
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (mod - obs).sum(dim=dim) / abs(obs.sum(dim=dim)) * 100.0
        return _update_history(res, "Normalized Mean Bias Absolute (NMB_ABS)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return (mod_arr - obs_arr).sum(axis=axis) / np.abs(obs_arr.sum(axis=axis)) * 100.0

NME(obs, mod, axis=None)

Normalized Mean Error (%)

Typical Use Cases

  • Quantifying the average magnitude of model errors relative to observations.
  • Used for model evaluation and comparison across variables or datasets with different scales.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean error (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NME(obs, mod) 37.5

Source code in src/monet_stats/relative_metrics.py 
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def NME(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Error (%)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of model errors relative to observations.
    - Used for model evaluation and comparison across variables or datasets with different scales.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NME(obs, mod)
    37.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod - obs).sum(dim=dim) / obs.sum(dim=dim)) * 100
        return _update_history(res, "Normalized Mean Error (NME)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (np.ma.abs(mod_arr - obs_arr).sum(axis=axis) / obs_arr.sum(axis=axis)) * 100

NME_m(obs, mod, axis=None)

Normalized Mean Error (%) (avoid single block error in np.ma)

Typical Use Cases

  • Quantifying the average magnitude of model errors relative to observations, robust to masked arrays.
  • Used for model evaluation when data may contain masked or missing values.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean error (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NME_m(obs, mod) 37.5

Source code in src/monet_stats/relative_metrics.py 
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def NME_m(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Error (%) (avoid single block error in np.ma)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of model errors relative to observations, robust to masked arrays.
    - Used for model evaluation when data may contain masked or missing values.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NME_m(obs, mod)
    37.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod - obs).sum(dim=dim) / obs.sum(dim=dim)) * 100
        return _update_history(res, "Normalized Mean Error (NME_m)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return (np.abs(mod_arr - obs_arr).sum(axis=axis) / obs_arr.sum(axis=axis)) * 100

NME_m_ABS(obs, mod, axis=None)

Normalized Mean Error (%) - Absolute of the denominator (avoid single block error in np.ma)

Typical Use Cases

  • Quantifying normalized mean error when the denominator (sum of observations) may be negative or zero, robust to masked arrays.
  • Used for model evaluation with possible sign changes or missing values in observed data.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean error with absolute denominator (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NME_m_ABS(obs, mod) 37.5

Source code in src/monet_stats/relative_metrics.py 
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def NME_m_ABS(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Error (%) - Absolute of the denominator
    (avoid single block error in np.ma)

    Typical Use Cases
    -----------------
    - Quantifying normalized mean error when the denominator (sum of observations)
      may be negative or zero, robust to masked arrays.
    - Used for model evaluation with possible sign changes or missing values in observed data.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean error with absolute denominator (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NME_m_ABS(obs, mod)
    37.5
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod - obs).sum(dim=dim) / abs(obs.sum(dim=dim))) * 100
        return _update_history(res, "Normalized Mean Error Absolute (NME_m_ABS)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return (np.abs(mod_arr - obs_arr).sum(axis=axis) / np.abs(obs_arr.sum(axis=axis))) * 100

NMPB(obs, mod, paxis, axis=None)

Normalized Mean Peak Bias (NMPB, %)

Typical Use Cases

  • Quantifying the average bias in peak values, normalized by the mean of observed peaks.
  • Used in model evaluation for extreme events, especially when comparing across sites or time periods.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the mean of normalized peak bias.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean peak bias (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) NMPB(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def NMPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Peak Bias (NMPB, %)

    Typical Use Cases
    -----------------
    - Quantifying the average bias in peak values, normalized by the mean of observed peaks.
    - Used in model evaluation for extreme events, especially when comparing across sites or time periods.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the mean of normalized peak bias.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean peak bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> NMPB(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = ((mod.max(dim=pdim) - obs.max(dim=pdim)).mean(dim=mdim) / obs.max(dim=pdim).mean(dim=mdim)) * 100.0
        return _update_history(res, "Normalized Mean Peak Bias (NMPB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            (np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)).mean(axis=axis)
            / np.ma.max(obs_arr, axis=paxis).mean(axis=axis)
        ) * 100.0

NMPE(obs, mod, paxis, axis=None)

Normalized Mean Peak Error (NMPE, %)

Typical Use Cases

  • Quantifying the average error in peak values, normalized by the mean of observed peaks.
  • Used in model evaluation for extreme events, especially when comparing across sites or time periods.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the mean of normalized peak error.

Returns

xarray.DataArray or numpy.ndarray or float Normalized mean peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) NMPE(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def NMPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Mean Peak Error (NMPE, %)

    Typical Use Cases
    -----------------
    - Quantifying the average error in peak values, normalized by the mean of observed peaks.
    - Used in model evaluation for extreme events, especially when comparing across sites or time periods.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the mean of normalized peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized mean peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> NMPE(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (abs(mod.max(dim=pdim) - obs.max(dim=pdim)).mean(dim=mdim) / obs.max(dim=pdim).mean(dim=mdim)) * 100.0
        return _update_history(res, "Normalized Mean Peak Error (NMPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.abs(np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)).mean(axis=axis)
            / np.ma.max(obs_arr, axis=paxis).mean(axis=axis)
        ) * 100.0

NMdnB(obs, mod, axis=None)

Normalized Median Bias (%)

Typical Use Cases

  • Assessing the central tendency of normalized bias, robust to outliers and non-normal distributions.
  • Used for robust model evaluation across variables or sites with different scales.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median bias (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4, 100]) # 100 is an outlier mod = np.array([1.1, 2.2, 3.3, 4.4, 105]) NMdnB(obs, mod) 10.0

Source code in src/monet_stats/relative_metrics.py 
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def NMdnB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Median Bias (%)

    Typical Use Cases
    -----------------
    - Assessing the central tendency of normalized bias, robust to outliers and non-normal distributions.
    - Used for robust model evaluation across variables or sites with different scales.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4, 100])  # 100 is an outlier
    >>> mod = np.array([1.1, 2.2, 3.3, 4.4, 105])
    >>> NMdnB(obs, mod)
    10.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (mod - obs).median(dim=dim) / obs.median(dim=dim) * 100.0
        return _update_history(res, "Normalized Median Bias (NMdnB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return np.ma.median(mod_arr - obs_arr, axis=axis) / np.ma.median(obs_arr, axis=axis) * 100.0

NMdnE(obs, mod, axis=None)

Normalized Median Error (%)

Typical Use Cases

  • Evaluating the typical magnitude of model errors relative to observations, robust to outliers.
  • Used for robust model evaluation and comparison across variables or datasets with different scales.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median error (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 2]) NMdnE(obs, mod) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def NMdnE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Median Error (%)

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of model errors relative to observations, robust to outliers.
    - Used for robust model evaluation and comparison across variables or datasets with different scales.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 2])
    >>> NMdnE(obs, mod)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = abs(mod - obs).median(dim=dim) / obs.median(dim=dim) * 100
        return _update_history(res, "Normalized Median Error (NMdnE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return np.ma.median(np.ma.abs(mod_arr - obs_arr), axis=axis) / np.ma.median(obs_arr, axis=axis) * 100

NMdnPB(obs, mod, paxis, axis=None)

Normalized Median Peak Bias (NMdnPB, %)

Typical Use Cases

  • Evaluating the typical bias in peak values, normalized by the median of observed peaks, robust to outliers.
  • Used in robust model evaluation for extreme events, especially when comparing across sites or time periods.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak bias.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median peak bias (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) NMdnPB(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def NMdnPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Median Peak Bias (NMdnPB, %)

    Typical Use Cases
    -----------------
    - Evaluating the typical bias in peak values, normalized by the median of observed peaks, robust to outliers.
    - Used in robust model evaluation for extreme events, especially when comparing across sites or time periods.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak bias.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median peak bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> NMdnPB(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (mod.max(dim=pdim) - obs.max(dim=pdim)).median(dim=mdim) / obs.max(dim=pdim).median(dim=mdim) * 100.0
        return _update_history(res, "Normalized Median Peak Bias (NMdnPB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.median(np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis), axis=axis)
            / np.ma.median(np.ma.max(obs_arr, axis=paxis), axis=axis)
        ) * 100.0

NMdnPE(obs, mod, paxis, axis=None)

Normalized Median Peak Error (NMdnPE, %)

Typical Use Cases

  • Evaluating the typical error in peak values, normalized by the median of observed peaks, robust to outliers.
  • Used in robust model evaluation for extreme events, especially when comparing across sites or time periods.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. paxis : int or str Axis or dimension along which to compute the peak (e.g., time or space). axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak error.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) NMdnPE(obs, mod, paxis=1) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def NMdnPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    paxis: Union[int, str],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Normalized Median Peak Error (NMdnPE, %)

    Typical Use Cases
    -----------------
    - Evaluating the typical error in peak values, normalized by the median of observed peaks, robust to outliers.
    - Used in robust model evaluation for extreme events, especially when comparing across sites or time periods.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    paxis : int or str
        Axis or dimension along which to compute the peak (e.g., time or space).
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> NMdnPE(obs, mod, paxis=1)
    33.33333333333333
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        pdim = paxis
        if isinstance(paxis, int):
            pdim = obs.dims[paxis]
        mdim = axis
        if isinstance(axis, int):
            mdim = obs.dims[axis]
        res = (abs(mod.max(dim=pdim) - obs.max(dim=pdim))).median(dim=mdim) / obs.max(dim=pdim).median(dim=mdim) * 100.0
        return _update_history(res, "Normalized Median Peak Error (NMdnPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.median(
                np.ma.abs(np.ma.max(mod_arr, axis=paxis) - np.ma.max(obs_arr, axis=paxis)),
                axis=axis,
            )
            / np.ma.median(np.ma.max(obs_arr, axis=paxis), axis=axis)
        ) * 100.0

PSUTMNPB(obs, mod, axis=None)

Paired Space/Unpaired Time Mean Normalized Peak Bias (PSUTMNPB, %)

Wrapper for MNPB with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTMNPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Mean Normalized Peak Bias (PSUTMNPB, %)

    Wrapper for MNPB with paxis=0, axis=None.
    """
    return MNPB(obs, mod, paxis=0, axis=None)

PSUTMNPE(obs, mod, axis=None)

Paired Space/Unpaired Time Mean Normalized Peak Error (PSUTMNPE, %)

Wrapper for MNPE with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTMNPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Mean Normalized Peak Error (PSUTMNPE, %)

    Wrapper for MNPE with paxis=0, axis=None.
    """
    return MNPE(obs, mod, paxis=0, axis=None)

PSUTMdnNPB(obs, mod, axis=None)

Paired Space/Unpaired Time Median Normalized Peak Bias (PSUTMdnNPB, %)

Wrapper for MdnNPB with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTMdnNPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Median Normalized Peak Bias (PSUTMdnNPB, %)

    Wrapper for MdnNPB with paxis=0, axis=None.
    """
    return MdnNPB(obs, mod, paxis=0, axis=None)

PSUTMdnNPE(obs, mod, axis=None)

Paired Space/Unpaired Time Median Normalized Peak Error (PSUTMdnNPE, %)

Wrapper for MdnNPE with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTMdnNPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Median Normalized Peak Error (PSUTMdnNPE, %)

    Wrapper for MdnNPE with paxis=0, axis=None.
    """
    return MdnNPE(obs, mod, paxis=0, axis=None)

PSUTNMPB(obs, mod, axis=None)

Paired Space/Unpaired Time Normalized Mean Peak Bias (PSUTNMPB, %)

Wrapper for NMPB with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTNMPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Normalized Mean Peak Bias (PSUTNMPB, %)

    Wrapper for NMPB with paxis=0, axis=None.
    """
    return NMPB(obs, mod, paxis=0, axis=None)

PSUTNMPE(obs, mod, axis=None)

Paired Space/Unpaired Time Normalized Mean Peak Error (PSUTNMPE, %)

Wrapper for NMPE with paxis=0, axis=None.

Source code in src/monet_stats/relative_metrics.py 
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def PSUTNMPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Normalized Mean Peak Error (PSUTNMPE, %)

    Wrapper for NMPE with paxis=0, axis=None.
    """
    return NMPE(obs, mod, paxis=0, axis=None)

PSUTNMdnPB(obs, mod, axis=None)

Paired Space/Unpaired Time Normalized Median Peak Bias (PSUTNMdnPB, %)

Typical Use Cases

  • Evaluating the normalized median peak bias for spatially paired, temporally unpaired datasets, robust to outliers.
  • Used in robust model evaluation for spatial ensemble or multi-time analysis.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak bias.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median peak bias (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) PSUTNMdnPB(obs, mod) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def PSUTNMdnPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Normalized Median Peak Bias (PSUTNMdnPB, %)

    Typical Use Cases
    -----------------
    - Evaluating the normalized median peak bias for spatially paired, temporally unpaired datasets, robust to outliers.
    - Used in robust model evaluation for spatial ensemble or multi-time analysis.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak bias.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median peak bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> PSUTNMdnPB(obs, mod)
    33.33333333333333
    """
    return NMdnPB(obs, mod, paxis=0, axis=None)

PSUTNMdnPE(obs, mod, axis=None)

Paired Space/Unpaired Time Normalized Median Peak Error (PSUTNMdnPE, %)

Typical Use Cases

  • Evaluating the normalized median peak error for spatially paired, temporally unpaired datasets, robust to outliers.
  • Used in robust model evaluation for spatial ensemble or multi-time analysis.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the median of normalized peak error.

Returns

xarray.DataArray or numpy.ndarray or float Normalized median peak error (percent).

Examples

import numpy as np obs = np.array([[1, 2, 3], [2, 3, 4]]) mod = np.array([[2, 2, 2], [2, 2, 5]]) PSUTNMdnPE(obs, mod) 33.33333333333333

Source code in src/monet_stats/relative_metrics.py 
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def PSUTNMdnPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Paired Space/Unpaired Time Normalized Median Peak Error (PSUTNMdnPE, %)

    Typical Use Cases
    -----------------
    - Evaluating the normalized median peak error for spatially paired, temporally unpaired
      datasets, robust to outliers.
    - Used in robust model evaluation for spatial ensemble or multi-time analysis.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the median of normalized peak error.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Normalized median peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([[1, 2, 3], [2, 3, 4]])
    >>> mod = np.array([[2, 2, 2], [2, 2, 5]])
    >>> PSUTNMdnPE(obs, mod)
    33.33333333333333
    """
    return NMdnPE(obs, mod, paxis=0, axis=None)

USUTPB(obs, mod, axis=None)

Unpaired Space/Unpaired Time Peak Bias (%)

Typical Use Cases

  • Assessing the bias in peak values between model and observations, regardless of spatial or temporal pairing.
  • Used in event-based or extreme value model evaluation, especially for air quality and meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Peak bias (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 5]) USUTPB(obs, mod) 25.0

Source code in src/monet_stats/relative_metrics.py 
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def USUTPB(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Unpaired Space/Unpaired Time Peak Bias (%)

    Typical Use Cases
    -----------------
    - Assessing the bias in peak values between model and observations, regardless of spatial or temporal pairing.
    - Used in event-based or extreme value model evaluation, especially for air quality and meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Peak bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 5])
    >>> USUTPB(obs, mod)
    25.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = ((mod.max(dim=dim) - obs.max(dim=dim)) / obs.max(dim=dim)) * 100.0
        return _update_history(res, "Unpaired Space/Unpaired Time Peak Bias (USUTPB)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return ((np.ma.max(mod_arr, axis=axis) - np.ma.max(obs_arr, axis=axis)) / np.ma.max(obs_arr, axis=axis)) * 100.0

USUTPE(obs, mod, axis=None)

Unpaired Space/Unpaired Time Peak Error (%)

Typical Use Cases

  • Quantifying the error in peak values between model and observations, regardless of spatial or temporal pairing.
  • Used in event-based or extreme value model evaluation, especially for air quality and meteorological extremes.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed values. mod : xarray.DataArray or numpy.ndarray Model predicted values. axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Peak error (percent).

Examples

import numpy as np obs = np.array([1, 2, 3, 4]) mod = np.array([2, 2, 2, 5]) USUTPE(obs, mod) 25.0

Source code in src/monet_stats/relative_metrics.py 
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def USUTPE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Unpaired Space/Unpaired Time Peak Error (%)

    Typical Use Cases
    -----------------
    - Quantifying the error in peak values between model and observations, regardless of spatial or temporal pairing.
    - Used in event-based or extreme value model evaluation, especially for air quality and meteorological extremes.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    mod : xarray.DataArray or numpy.ndarray
        Model predicted values.
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Peak error (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([1, 2, 3, 4])
    >>> mod = np.array([2, 2, 2, 5])
    >>> USUTPE(obs, mod)
    25.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = (abs(mod.max(dim=dim) - obs.max(dim=dim)) / obs.max(dim=dim)) * 100.0
        return _update_history(res, "Unpaired Space/Unpaired Time Peak Error (USUTPE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return (
            np.ma.abs(np.ma.max(mod_arr, axis=axis) - np.ma.max(obs_arr, axis=axis)) / np.ma.max(obs_arr, axis=axis)
        ) * 100.0

WDME(obs, mod, axis=None)

Wind Direction Mean Gross Error (model and obs unit)

Typical Use Cases

  • Quantifying the average magnitude of wind direction errors, regardless of direction.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed wind direction values (degrees). mod : xarray.DataArray or numpy.ndarray Model predicted wind direction values (degrees). axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Mean gross error in wind direction (degrees).

Examples

import numpy as np obs = np.array([350, 10, 20]) mod = np.array([10, 20, 30]) WDME(obs, mod) 20.0

Source code in src/monet_stats/relative_metrics.py 
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def WDME(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Wind Direction Mean Gross Error (model and obs unit)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of wind direction errors, regardless of direction.
    - Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed wind direction values (degrees).
    mod : xarray.DataArray or numpy.ndarray
        Model predicted wind direction values (degrees).
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean gross error in wind direction (degrees).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([10, 20, 30])
    >>> WDME(obs, mod)
    20.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = abs(circlebias(mod - obs)).mean(dim=dim)
        return _update_history(res, "Wind Direction Mean Gross Error (WDME)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        return np.ma.mean(np.ma.abs(circlebias(mod_arr - obs_arr)), axis=axis)

WDME_m(obs, mod, axis=None)

Wind Direction Mean Gross Error (model and obs unit) (avoid single block error in np.ma)

Typical Use Cases

  • Quantifying the average magnitude of wind direction errors, regardless of direction.
  • Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed wind direction values (degrees). mod : xarray.DataArray or numpy.ndarray Model predicted wind direction values (degrees). axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Mean gross error in wind direction (degrees).

Examples

import numpy as np obs = np.array([350, 10, 20]) mod = np.array([10, 20, 30]) WDME_m(obs, mod) 20.0

Source code in src/monet_stats/relative_metrics.py 
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def WDME_m(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Wind Direction Mean Gross Error (model and obs unit)
    (avoid single block error in np.ma)

    Typical Use Cases
    -----------------
    - Quantifying the average magnitude of wind direction errors, regardless of direction.
    - Used in wind energy, meteorology, and air quality studies to assess wind direction model performance.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed wind direction values (degrees).
    mod : xarray.DataArray or numpy.ndarray
        Model predicted wind direction values (degrees).
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Mean gross error in wind direction (degrees).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([10, 20, 30])
    >>> WDME_m(obs, mod)
    20.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        res = abs(circlebias_m(mod - obs)).mean(dim=dim)
        return _update_history(res, "Wind Direction Mean Gross Error (WDME_m)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        return np.abs(circlebias_m(mod_arr - obs_arr)).mean(axis=axis)

WDMdnE(obs, mod, axis=None)

Wind Direction Median Gross Error (model and obs unit)

Typical Use Cases

  • Evaluating the typical magnitude of wind direction errors, robust to outliers.
  • Used in wind energy and meteorological applications for robust wind direction model evaluation.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed wind direction values (degrees). mod : xarray.DataArray or numpy.ndarray Model predicted wind direction values (degrees). axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Median gross error in wind direction (degrees).

Examples

import numpy as np obs = np.array([350, 10, 20]) mod = np.array([10, 20, 30]) WDMdnE(obs, mod) 10.0

Source code in src/monet_stats/relative_metrics.py 
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def WDMdnE(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Wind Direction Median Gross Error (model and obs unit)

    Typical Use Cases
    -----------------
    - Evaluating the typical magnitude of wind direction errors, robust to outliers.
    - Used in wind energy and meteorological applications for robust wind direction model evaluation.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed wind direction values (degrees).
    mod : xarray.DataArray or numpy.ndarray
        Model predicted wind direction values (degrees).
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Median gross error in wind direction (degrees).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([10, 20, 30])
    >>> WDMdnE(obs, mod)
    10.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        cb = circlebias(mod - obs)
        res = abs(cb).median(dim=dim)
        return _update_history(res, "Wind Direction Median Gross Error (WDMdnE)")
    else:
        obs_arr = np.ma.asanyarray(obs)
        mod_arr = np.ma.asanyarray(mod)
        cb = circlebias(mod_arr - obs_arr)
        return np.ma.median(np.ma.abs(cb), axis=axis)

WDNMB_m(obs, mod, axis=None)

Wind Direction Normalized Mean Bias (%) (avoid single block error in np.ma)

Typical Use Cases

  • Comparing the average wind direction bias, normalized by observed wind direction, across sites or time periods.
  • Used in wind energy and meteorological model evaluation for directionally normalized performance.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed wind direction values (degrees). mod : xarray.DataArray or numpy.ndarray Model predicted wind direction values (degrees). axis : int or str or None, optional Axis or dimension along which to compute the statistic.

Returns

xarray.DataArray or numpy.ndarray or float Wind direction normalized mean bias (percent).

Examples

import numpy as np obs = np.array([350, 10, 20]) mod = np.array([345, 15, 25]) WDNMB_m(obs, mod) -5.0

Source code in src/monet_stats/relative_metrics.py 
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def WDNMB_m(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    axis: Optional[Union[int, str]] = None,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Wind Direction Normalized Mean Bias (%) (avoid single block error in np.ma)

    Typical Use Cases
    -----------------
    - Comparing the average wind direction bias, normalized by observed wind direction, across sites or time periods.
    - Used in wind energy and meteorological model evaluation for directionally normalized performance.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed wind direction values (degrees).
    mod : xarray.DataArray or numpy.ndarray
        Model predicted wind direction values (degrees).
    axis : int or str or None, optional
        Axis or dimension along which to compute the statistic.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Wind direction normalized mean bias (percent).

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.array([350, 10, 20])
    >>> mod = np.array([345, 15, 25])
    >>> WDNMB_m(obs, mod)
    -5.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        dim = axis
        if isinstance(axis, int):
            dim = obs.dims[axis]
        diff = mod - obs
        cb = circlebias_m(diff)
        res = cb.sum(dim=dim) / obs.sum(dim=dim) * 100.0
        return _update_history(res, "Wind Direction Normalized Mean Bias (WDNMB_m)")
    else:
        obs_arr = np.asanyarray(obs)
        mod_arr = np.asanyarray(mod)
        diff = mod_arr - obs_arr
        return circlebias_m(diff).sum(axis=axis) / obs_arr.sum(axis=axis) * 100.0

Spatial & Ensemble Metrics

Spatial and Ensemble Metrics for Atmospheric Sciences (Aero Protocol Compliant)

BSS(obs, mod, threshold)

Brier Skill Score (BSS) for probabilistic forecasts.

Typical Use Cases

  • Evaluating the accuracy of probabilistic binary forecasts relative to climatology.
  • Common in meteorological verification for event occurrence.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed binary outcomes (0 or 1) or continuous values (will be binarized). mod : xarray.DataArray or numpy.ndarray Forecast probabilities (0 to 1) or continuous values (will be binarized). threshold : float Threshold for converting values to binary events.

Returns

xarray.DataArray or numpy.ndarray or float Brier Skill Score.

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def BSS(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    threshold: float,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Brier Skill Score (BSS) for probabilistic forecasts.

    Typical Use Cases
    -----------------
    - Evaluating the accuracy of probabilistic binary forecasts relative to climatology.
    - Common in meteorological verification for event occurrence.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed binary outcomes (0 or 1) or continuous values (will be binarized).
    mod : xarray.DataArray or numpy.ndarray
        Forecast probabilities (0 to 1) or continuous values (will be binarized).
    threshold : float
        Threshold for converting values to binary events.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Brier Skill Score.
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        # Binarize if not already
        o_bin = (obs >= threshold).astype(float)
        m_prob = (mod >= threshold).astype(float)

        bs = ((m_prob - o_bin) ** 2).mean()
        obs_clim = o_bin.mean()
        bs_ref = ((obs_clim - o_bin) ** 2).mean()

        res = xr.where(bs_ref != 0, 1 - (bs / bs_ref), 0.0)
        return _update_history(res, "Brier Skill Score (BSS)")

    o = np.asarray(obs)
    m = np.asarray(mod)
    o_bin = (o >= threshold).astype(float)
    m_prob = (m >= threshold).astype(float)

    bs = np.mean((m_prob - o_bin) ** 2)
    obs_clim = np.mean(o_bin)
    bs_ref = np.mean((obs_clim - o_bin) ** 2)

    if bs_ref == 0:
        return 0.0
    return 1 - (bs / bs_ref)

CRPS(ensemble, obs, axis=0)

Continuous Ranked Probability Score (CRPS) for ensemble forecasts.

Supports lazy evaluation via Xarray/Dask.

Parameters

ensemble : xarray.DataArray or numpy.ndarray Ensemble forecasts. If DataArray, should have an ensemble dimension. obs : xarray.DataArray or numpy.ndarray Observed values. axis : int or str, optional Axis or dimension corresponding to ensemble members. Default is 0.

Returns

xarray.DataArray or numpy.ndarray CRPS values.

Examples

import numpy as np ens = np.array([[1, 2], [2, 3], [3, 4]]) obs = np.array([2, 3]) CRPS(ens, obs, axis=0) array([0.22222222, 0.22222222])

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def CRPS(
    ensemble: Union[xr.DataArray, np.ndarray],
    obs: Union[xr.DataArray, np.ndarray],
    axis: Union[int, str] = 0,
) -> Union[xr.DataArray, np.ndarray]:
    """
    Continuous Ranked Probability Score (CRPS) for ensemble forecasts.

    Supports lazy evaluation via Xarray/Dask.

    Parameters
    ----------
    ensemble : xarray.DataArray or numpy.ndarray
        Ensemble forecasts. If DataArray, should have an ensemble dimension.
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    axis : int or str, optional
        Axis or dimension corresponding to ensemble members. Default is 0.

    Returns
    -------
    xarray.DataArray or numpy.ndarray
        CRPS values.

    Examples
    --------
    >>> import numpy as np
    >>> ens = np.array([[1, 2], [2, 3], [3, 4]])
    >>> obs = np.array([2, 3])
    >>> CRPS(ens, obs, axis=0)
    array([0.22222222, 0.22222222])
    """

    def _crps_numpy(ens, observation, ens_axis=0):
        ens_sorted = np.sort(ens, axis=ens_axis)
        n = ens.shape[ens_axis]
        # Compute empirical CDFs
        cdf_ens = np.arange(1, n + 1) / n
        shape = [1] * ens.ndim
        shape[ens_axis] = n
        cdf_ens = np.reshape(cdf_ens, shape)
        # Broadcast obs for comparison
        obs_broadcast = np.expand_dims(observation, ens_axis)
        cdf_obs = (ens_sorted >= obs_broadcast).astype(float)
        return np.sum((cdf_ens - cdf_obs) ** 2, axis=ens_axis)

    if isinstance(ensemble, xr.DataArray) and isinstance(obs, xr.DataArray):
        # Determine core dimension
        if isinstance(axis, int):
            ens_dim = ensemble.dims[axis]
        else:
            ens_dim = axis

        res = xr.apply_ufunc(
            _crps_numpy,
            ensemble,
            obs,
            input_core_dims=[[ens_dim], []],
            output_core_dims=[[]],
            kwargs={"ens_axis": -1},
            dask="parallelized",
            output_dtypes=[float],
            dask_gufunc_kwargs={"allow_rechunk": True},
        )
        return _update_history(res, "Continuous Ranked Probability Score (CRPS)")

    return _crps_numpy(np.asarray(ensemble), np.asarray(obs), ens_axis=axis)

EDS(obs, mod, threshold)

Extreme Dependency Score (EDS) for rare event detection.

Typical Use Cases

  • Assessing model performance for rare extreme events (e.g., heavy precipitation).
  • Used when traditional scores like CSI or ETS go to zero as the event becomes rarer.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed field. mod : xarray.DataArray or numpy.ndarray Model field. threshold : float Event threshold to define the extreme event.

Returns

xarray.DataArray or numpy.ndarray or float Extreme Dependency Score.

Examples

import numpy as np obs = np.zeros((10, 10)); obs[5, 5] = 1 mod = np.zeros((10, 10)); mod[5, 5] = 1 EDS(obs, mod, threshold=0.5) 1.0

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def EDS(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    threshold: float,
) -> Union[xr.DataArray, np.ndarray, float]:
    """
    Extreme Dependency Score (EDS) for rare event detection.

    Typical Use Cases
    -----------------
    - Assessing model performance for rare extreme events (e.g., heavy precipitation).
    - Used when traditional scores like CSI or ETS go to zero as the event becomes rarer.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed field.
    mod : xarray.DataArray or numpy.ndarray
        Model field.
    threshold : float
        Event threshold to define the extreme event.

    Returns
    -------
    xarray.DataArray or numpy.ndarray or float
        Extreme Dependency Score.

    Examples
    --------
    >>> import numpy as np
    >>> obs = np.zeros((10, 10)); obs[5, 5] = 1
    >>> mod = np.zeros((10, 10)); mod[5, 5] = 1
    >>> EDS(obs, mod, threshold=0.5)
    1.0
    """
    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        obs, mod = xr.align(obs, mod, join="inner")
        obs_bin = obs >= threshold
        mod_bin = mod >= threshold
        hits = (obs_bin & mod_bin).sum()
        n_obs = obs_bin.sum()
        n_mod = mod_bin.sum()
        n = obs.size

        # Use xr.where for lazy evaluation
        p = n_obs / n
        q = n_mod / n

        # We need to handle the log carefully for dask
        eds = np.log(hits / n) / np.log(p * q)
        # Handle cases where hits=0 or n_obs/n_mod=0 which would result in inf/nan
        # EDS is undefined if p=0 or q=0 or hits=0
        res = xr.where((hits > 0) & (p > 0) & (q > 0), eds, np.nan)
        return _update_history(res, "Extreme Dependency Score (EDS)")

    obs_bin = np.asarray(obs) >= threshold
    mod_bin = np.asarray(mod) >= threshold
    hits = np.logical_and(obs_bin, mod_bin).sum()
    n_obs = obs_bin.sum()
    n_mod = mod_bin.sum()
    n = np.size(obs)
    if hits == 0 or n_obs == 0 or n_mod == 0:
        return np.nan
    p = n_obs / n
    q = n_mod / n
    return np.log(hits / n) / np.log(p * q)

SAL(obs, mod, threshold=None)

Structure-Amplitude-Location (SAL) score for spatial verification.

Note: This metric currently triggers computation for Xarray/Dask inputs as it relies on scipy.ndimage for object identification.

Parameters

obs : xarray.DataArray or numpy.ndarray Observed 2D field. mod : xarray.DataArray or numpy.ndarray Model 2D field. threshold : float, optional Threshold for object identification. If None, uses mean of obs.

Returns

S : float Structure component (-2 to 2, 0 is best). A : float Amplitude component (-2 to 2, 0 is best). L : float Location component (0 to 2, 0 is best).

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def SAL(
    obs: Union[xr.DataArray, np.ndarray],
    mod: Union[xr.DataArray, np.ndarray],
    threshold: Optional[float] = None,
) -> Tuple[float, float, float]:
    """
    Structure-Amplitude-Location (SAL) score for spatial verification.

    Note: This metric currently triggers computation for Xarray/Dask inputs
    as it relies on scipy.ndimage for object identification.

    Parameters
    ----------
    obs : xarray.DataArray or numpy.ndarray
        Observed 2D field.
    mod : xarray.DataArray or numpy.ndarray
        Model 2D field.
    threshold : float, optional
        Threshold for object identification. If None, uses mean of obs.

    Returns
    -------
    S : float
        Structure component (-2 to 2, 0 is best).
    A : float
        Amplitude component (-2 to 2, 0 is best).
    L : float
        Location component (0 to 2, 0 is best).
    """
    import scipy.ndimage as ndi

    if isinstance(obs, xr.DataArray) and isinstance(mod, xr.DataArray):
        # We explicitly compute for now because SAL is inherently non-local
        # and hard to dask-ify without complex overlapping.
        obs_np = obs.values
        mod_np = mod.values
    else:
        obs_np = np.asarray(obs)
        mod_np = np.asarray(mod)

    if threshold is None:
        threshold = np.nanmean(obs_np)

    # Amplitude
    denom_a = np.nanmean(mod_np) + np.nanmean(obs_np)
    A = 2 * (np.nanmean(mod_np) - np.nanmean(obs_np)) / denom_a if denom_a != 0 else 0.0

    # Structure
    def structure(X):
        labeled, n = ndi.label(threshold <= X)
        if n == 0:
            return 0.0, 0.0
        masses = ndi.sum(X, labeled, index=np.arange(1, n + 1))
        max_mass = np.max(masses)
        total_mass = np.sum(masses)
        return max_mass, total_mass

    max_mod, sum_mod = structure(mod_np)
    max_obs, sum_obs = structure(obs_np)
    denom_s = (max_mod / sum_mod + max_obs / sum_obs) if sum_mod > 0 and sum_obs > 0 else 0
    S = 2 * (max_mod / sum_mod - max_obs / sum_obs) / denom_s if denom_s != 0 else 0.0

    # Location
    def centroid(X):
        labeled, n = ndi.label(threshold <= X)
        if n == 0:
            return np.array([np.nan, np.nan])
        centers = np.array(ndi.center_of_mass(X, labeled, index=np.arange(1, n + 1)))
        masses = ndi.sum(X, labeled, index=np.arange(1, n + 1))
        weighted = np.average(centers, axis=0, weights=masses)
        return weighted

    c_mod = centroid(mod_np)
    c_obs = centroid(obs_np)
    dist = np.linalg.norm(c_mod - c_obs)
    max_dist = np.sqrt(obs_np.shape[0] ** 2 + obs_np.shape[1] ** 2)
    L1 = dist / max_dist if max_dist != 0 else 0.0

    # Spread of objects
    def spread(X):
        labeled, n = ndi.label(threshold <= X)
        if n == 0:
            return 0.0
        centers = np.array(ndi.center_of_mass(X, labeled, index=np.arange(1, n + 1)))
        masses = ndi.sum(X, labeled, index=np.arange(1, n + 1))
        c = np.average(centers, axis=0, weights=masses)
        return np.average(np.linalg.norm(centers - c, axis=1), weights=masses)

    r_mod = spread(mod_np)
    r_obs = spread(obs_np)
    L2 = abs(r_mod - r_obs) / max_dist if max_dist != 0 else 0.0
    L = L1 + L2
    return S, A, L

ensemble_mean(ensemble, axis=0)

Calculate the ensemble mean.

Parameters

ensemble : xarray.DataArray or numpy.ndarray Ensemble forecasts. axis : int or str, optional Axis or dimension corresponding to ensemble members. Default is 0.

Returns

xarray.DataArray or numpy.ndarray Ensemble mean.

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def ensemble_mean(
    ensemble: Union[xr.DataArray, np.ndarray],
    axis: Union[int, str] = 0,
) -> Union[xr.DataArray, np.ndarray]:
    """
    Calculate the ensemble mean.

    Parameters
    ----------
    ensemble : xarray.DataArray or numpy.ndarray
        Ensemble forecasts.
    axis : int or str, optional
        Axis or dimension corresponding to ensemble members. Default is 0.

    Returns
    -------
    xarray.DataArray or numpy.ndarray
        Ensemble mean.
    """
    if isinstance(ensemble, xr.DataArray):
        dim = axis
        if isinstance(axis, int):
            dim = ensemble.dims[axis]
        res = ensemble.mean(dim=dim)
        return _update_history(res, "Ensemble Mean")
    return np.mean(ensemble, axis=axis)

ensemble_std(ensemble, axis=0)

Calculate the ensemble standard deviation.

Parameters

ensemble : xarray.DataArray or numpy.ndarray Ensemble forecasts. axis : int or str, optional Axis or dimension corresponding to ensemble members. Default is 0.

Returns

xarray.DataArray or numpy.ndarray Ensemble standard deviation.

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def ensemble_std(
    ensemble: Union[xr.DataArray, np.ndarray],
    axis: Union[int, str] = 0,
) -> Union[xr.DataArray, np.ndarray]:
    """
    Calculate the ensemble standard deviation.

    Parameters
    ----------
    ensemble : xarray.DataArray or numpy.ndarray
        Ensemble forecasts.
    axis : int or str, optional
        Axis or dimension corresponding to ensemble members. Default is 0.

    Returns
    -------
    xarray.DataArray or numpy.ndarray
        Ensemble standard deviation.
    """
    if isinstance(ensemble, xr.DataArray):
        dim = axis
        if isinstance(axis, int):
            dim = ensemble.dims[axis]
        res = ensemble.std(dim=dim)
        return _update_history(res, "Ensemble Standard Deviation")
    return np.std(ensemble, axis=axis)

rank_histogram(ensemble, obs, axis=0)

Calculate the rank histogram counts.

Parameters

ensemble : xarray.DataArray or numpy.ndarray Ensemble forecasts. obs : xarray.DataArray or numpy.ndarray Observed values. axis : int or str, optional Axis or dimension corresponding to ensemble members. Default is 0.

Returns

xarray.DataArray or numpy.ndarray Rank histogram counts.

Examples

import numpy as np ens = np.array([[1, 2], [2, 3], [3, 4]]) obs = np.array([2, 3]) rank_histogram(ens, obs, axis=0) array([0., 0., 2., 0.])

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def rank_histogram(
    ensemble: Union[xr.DataArray, np.ndarray],
    obs: Union[xr.DataArray, np.ndarray],
    axis: Union[int, str] = 0,
) -> Union[xr.DataArray, np.ndarray]:
    """
    Calculate the rank histogram counts.

    Parameters
    ----------
    ensemble : xarray.DataArray or numpy.ndarray
        Ensemble forecasts.
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    axis : int or str, optional
        Axis or dimension corresponding to ensemble members. Default is 0.

    Returns
    -------
    xarray.DataArray or numpy.ndarray
        Rank histogram counts.

    Examples
    --------
    >>> import numpy as np
    >>> ens = np.array([[1, 2], [2, 3], [3, 4]])
    >>> obs = np.array([2, 3])
    >>> rank_histogram(ens, obs, axis=0)
    array([0., 0., 2., 0.])
    """

    def _rank_numpy(ens, observation, ens_axis=0):
        o_exp = np.expand_dims(observation, ens_axis)
        full_ensemble = np.concatenate([ens, o_exp], axis=ens_axis)
        ranks = np.argsort(full_ensemble, axis=ens_axis)
        obs_rank = np.argmax(ranks == ens.shape[ens_axis], axis=ens_axis)
        n_ens = ens.shape[ens_axis]
        hist, _ = np.histogram(obs_rank, bins=np.arange(n_ens + 2))
        return hist.astype(float)

    if isinstance(ensemble, xr.DataArray) and isinstance(obs, xr.DataArray):
        if isinstance(axis, int):
            ens_dim = ensemble.dims[axis]
        else:
            ens_dim = axis

        def _rank_ufunc(ens, observation):
            o_exp = np.expand_dims(observation, -1)
            full_ensemble = np.concatenate([ens, o_exp], axis=-1)
            ranks = np.argsort(full_ensemble, axis=-1)
            return np.argmax(ranks == ens.shape[-1], axis=-1)

        obs_rank = xr.apply_ufunc(
            _rank_ufunc,
            ensemble,
            obs,
            input_core_dims=[[ens_dim], []],
            output_core_dims=[[]],
            dask="parallelized",
            output_dtypes=[int],
        )

        n_ens = ensemble.sizes[ens_dim]
        bins = np.arange(n_ens + 2)
        if hasattr(obs_rank.data, "dask"):
            import dask.array as da

            hist, _ = da.histogram(obs_rank.data, bins=bins)
            res = xr.DataArray(hist, dims="rank", coords={"rank": np.arange(n_ens + 1)})
        else:
            hist, _ = np.histogram(obs_rank.values, bins=bins)
            res = xr.DataArray(hist, dims="rank", coords={"rank": np.arange(n_ens + 1)})
        return _update_history(res, "Rank Histogram")

    return _rank_numpy(np.asarray(ensemble), np.asarray(obs), ens_axis=axis)

spread_error(ensemble, obs, axis=0)

Spread-Error Relationship for ensemble forecasts.

Typical Use Cases

  • Assessing if the ensemble spread is a good proxy for the forecast error.
  • Ideally, mean spread should equal RMSE of the ensemble mean.

Parameters

ensemble : xarray.DataArray or numpy.ndarray Ensemble forecasts. obs : xarray.DataArray or numpy.ndarray Observed values. axis : int or str, optional Axis or dimension corresponding to ensemble members. Default is 0.

Returns

mean_spread : float or xarray.DataArray Mean ensemble spread. mean_error : float or xarray.DataArray Mean absolute error of ensemble mean vs. obs.

Source code in src/monet_stats/spatial_ensemble_metrics.py 
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def spread_error(
    ensemble: Union[xr.DataArray, np.ndarray],
    obs: Union[xr.DataArray, np.ndarray],
    axis: Union[int, str] = 0,
) -> Tuple[Any, Any]:
    """
    Spread-Error Relationship for ensemble forecasts.

    Typical Use Cases
    -----------------
    - Assessing if the ensemble spread is a good proxy for the forecast error.
    - Ideally, mean spread should equal RMSE of the ensemble mean.

    Parameters
    ----------
    ensemble : xarray.DataArray or numpy.ndarray
        Ensemble forecasts.
    obs : xarray.DataArray or numpy.ndarray
        Observed values.
    axis : int or str, optional
        Axis or dimension corresponding to ensemble members. Default is 0.

    Returns
    -------
    mean_spread : float or xarray.DataArray
        Mean ensemble spread.
    mean_error : float or xarray.DataArray
        Mean absolute error of ensemble mean vs. obs.
    """
    if isinstance(ensemble, xr.DataArray) and isinstance(obs, xr.DataArray):
        if isinstance(axis, int):
            dim = ensemble.dims[axis]
        else:
            dim = axis

        spread = ensemble.std(dim=dim)
        ens_mean = ensemble.mean(dim=dim)
        error = abs(ens_mean - obs)

        # We return means over all remaining dimensions as well?
        # The original implementation returned np.mean(spread), np.mean(error)
        # which are scalars.
        m_spread = spread.mean()
        m_error = error.mean()

        return _update_history(m_spread, "Mean Ensemble Spread"), _update_history(m_error, "Mean Ensemble Error")

    ens = np.asarray(ensemble)
    observation = np.asarray(obs)
    spread = np.std(ens, axis=axis)
    ens_mean = np.mean(ens, axis=axis)
    error = np.abs(ens_mean - observation)
    return np.mean(spread), np.mean(error)

Utility Functions

Utility Functions for Statistics

angular_difference(angle1, angle2, units='degrees')

Calculate the smallest angular difference between two angles.

Backend-agnostic (supports NumPy and Xarray/Dask).

Parameters

angle1 : array-like First angle(s). angle2 : array-like Second angle(s). units : str, optional Units of angles ('degrees' or 'radians'). Default is 'degrees'.

Returns

array-like Smallest angular difference between the two angles.

Source code in src/monet_stats/utils_stats.py 
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def angular_difference(angle1: ArrayLike, angle2: ArrayLike, units: str = "degrees") -> Any:
    """
    Calculate the smallest angular difference between two angles.

    Backend-agnostic (supports NumPy and Xarray/Dask).

    Parameters
    ----------
    angle1 : array-like
        First angle(s).
    angle2 : array-like
        Second angle(s).
    units : str, optional
        Units of angles ('degrees' or 'radians'). Default is 'degrees'.

    Returns
    -------
    array-like
        Smallest angular difference between the two angles.
    """
    if units == "degrees":
        max_val = 360.0
    elif units == "radians":
        max_val = 2 * np.pi
    else:
        raise ValueError("units must be 'degrees' or 'radians'")

    if (hasattr(angle1, "attrs") and hasattr(angle1, "coords")) or (
        hasattr(angle2, "attrs") and hasattr(angle2, "coords")
    ):
        if (hasattr(angle1, "attrs") and hasattr(angle1, "coords")) and (
            hasattr(angle2, "attrs") and hasattr(angle2, "coords")
        ):
            angle1, angle2 = xr.align(angle1, angle2, join="inner")
        diff = abs(angle1 - angle2)
        result = xr.where(diff > max_val / 2, max_val - diff, diff)
        return _update_history(result, "angular_difference")

    angle1_arr = np.asarray(angle1)
    angle2_arr = np.asarray(angle2)
    diff = np.abs(angle1_arr - angle2_arr)
    return np.minimum(diff, max_val - diff)

circlebias(b)

Circular bias (wind direction difference, wrapped to [-180, 180] degrees).

Typical Use Cases

  • Calculating the signed difference between two wind directions, accounting for circularity.
  • Used in wind direction bias and error metrics to avoid artificial large errors across 0/360 boundaries.

Parameters

b : array-like Difference between two wind directions (degrees).

Returns

array-like Circularly wrapped difference (degrees).

Source code in src/monet_stats/utils_stats.py 
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def circlebias(b: ArrayLike) -> Any:
    """
    Circular bias (wind direction difference, wrapped to [-180, 180] degrees).

    Typical Use Cases
    -----------------
    - Calculating the signed difference between two wind directions, accounting
      for circularity.
    - Used in wind direction bias and error metrics to avoid artificial large
      errors across 0/360 boundaries.

    Parameters
    ----------
    b : array-like
        Difference between two wind directions (degrees).

    Returns
    -------
    array-like
        Circularly wrapped difference (degrees).
    """
    if hasattr(b, "attrs") and hasattr(b, "coords"):
        res = (b + 180) % 360 - 180
        return _update_history(res, "circlebias")

    return (np.asarray(b) + 180) % 360 - 180

circlebias_m(b)

Circular bias for wind direction (robust to masked arrays).

Typical Use Cases

  • Calculating the signed difference between two wind directions, accounting for circularity, robust to masked arrays.
  • Used in wind direction bias and error metrics for masked or missing data.

Parameters

b : array-like Difference between two wind directions (degrees).

Returns

array-like Circularly wrapped difference (degrees).

Source code in src/monet_stats/utils_stats.py 
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def circlebias_m(b: ArrayLike) -> Any:
    """
    Circular bias for wind direction (robust to masked arrays).

    Typical Use Cases
    -----------------
    - Calculating the signed difference between two wind directions, accounting
      for circularity, robust to masked arrays.
    - Used in wind direction bias and error metrics for masked or missing data.

    Parameters
    ----------
    b : array-like
        Difference between two wind directions (degrees).

    Returns
    -------
    array-like
        Circularly wrapped difference (degrees).
    """
    if hasattr(b, "attrs") and hasattr(b, "coords"):
        res = (b + 180) % 360 - 180
        return _update_history(res, "circlebias_m")

    b_masked = np.ma.masked_invalid(b)
    return (b_masked + 180) % 360 - 180

correlation(x, y, axis=None)

Calculate Pearson correlation coefficient between x and y.

Parameters

x : numpy.ndarray or xarray.DataArray First variable. y : numpy.ndarray or xarray.DataArray Second variable. axis : int, str, or iterable, optional Axis along which to compute correlation.

Returns

Union[np.number, np.ndarray, xr.DataArray] Pearson correlation coefficient.

Raises

ValueError If input arrays are empty.

Source code in src/monet_stats/utils_stats.py 
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def correlation(
    x: Union[np.ndarray, xr.DataArray],
    y: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Calculate Pearson correlation coefficient between x and y.

    Parameters
    ----------
    x : numpy.ndarray or xarray.DataArray
        First variable.
    y : numpy.ndarray or xarray.DataArray
        Second variable.
    axis : int, str, or iterable, optional
        Axis along which to compute correlation.

    Returns
    -------
    Union[np.number, np.ndarray, xr.DataArray]
        Pearson correlation coefficient.

    Raises
    ------
    ValueError
        If input arrays are empty.
    """
    if hasattr(x, "size") and x.size == 0:
        raise ValueError("Input arrays cannot be empty")
    if hasattr(y, "size") and y.size == 0:
        raise ValueError("Input arrays cannot be empty")

    from .correlation_metrics import pearsonr

    res = pearsonr(x, y, axis=axis)
    if hasattr(res, "attrs") and hasattr(res, "coords"):
        return _update_history(res, "correlation")
    return res

mae(obs, mod, axis=None)

Calculate Mean Absolute Error between observations and model.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable, optional Axis along which to compute MAE.

Returns

Union[np.number, np.ndarray, xr.DataArray] Mean absolute error.

Source code in src/monet_stats/utils_stats.py 
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def mae(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Calculate Mean Absolute Error between observations and model.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable, optional
        Axis along which to compute MAE.

    Returns
    -------
    Union[np.number, np.ndarray, xr.DataArray]
        Mean absolute error.
    """
    from .error_metrics import MAE

    res = MAE(obs, mod, axis=axis)
    if hasattr(res, "attrs") and hasattr(res, "coords"):
        return _update_history(res, "mae")
    return res

matchedcompressed(a1, a2)

Return compressed (non-masked) values from two masked arrays with matched masks.

Note: For Xarray DataArrays, this function will trigger a computation if the data is Dask-backed, as it returns NumPy ndarrays. For lazy operations, prefer using Xarray-native methods with skipna=True.

Parameters

a1 : array-like First input array. a2 : array-like Second input array.

Returns

tuple of ndarray Tuple of (a1_compressed, a2_compressed), both 1D arrays of valid values.

Source code in src/monet_stats/utils_stats.py 
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def matchedcompressed(a1: ArrayLike, a2: ArrayLike) -> Tuple[np.ndarray, np.ndarray]:
    """
    Return compressed (non-masked) values from two masked arrays with matched masks.

    Note: For Xarray DataArrays, this function will trigger a computation if
    the data is Dask-backed, as it returns NumPy ndarrays. For lazy operations,
    prefer using Xarray-native methods with `skipna=True`.

    Parameters
    ----------
    a1 : array-like
        First input array.
    a2 : array-like
        Second input array.

    Returns
    -------
    tuple of ndarray
        Tuple of (a1_compressed, a2_compressed), both 1D arrays of valid values.
    """
    # Handle Xarray objects by extracting values (explicitly mentioned as computation-triggering)
    if hasattr(a1, "values") and hasattr(a1, "coords"):
        a1 = a1.values
    if hasattr(a2, "values") and hasattr(a2, "coords"):
        a2 = a2.values

    # Convert to masked arrays to handle existing masks and NaNs
    a1_m = np.ma.masked_invalid(a1)
    a2_m = np.ma.masked_invalid(a2)

    # Handle mismatched shapes for numpy arrays by truncating
    if a1_m.shape != a2_m.shape:
        min_size = min(a1_m.size, a2_m.size)
        a1_m = a1_m.flat[:min_size]
        a2_m = a2_m.flat[:min_size]

    mask = np.ma.getmaskarray(a1_m) | np.ma.getmaskarray(a2_m)
    a1_matched = np.ma.masked_where(mask, a1_m)
    a2_matched = np.ma.masked_where(mask, a2_m)
    return a1_matched.compressed(), a2_matched.compressed()

matchmasks(a1, a2)

Match and combine masks from two masked arrays or align Xarray objects.

Parameters

a1 : array-like First input array. a2 : array-like Second input array.

Returns

tuple Tuple of (a1_matched, a2_matched).

Source code in src/monet_stats/utils_stats.py 
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def matchmasks(a1: ArrayLike, a2: ArrayLike) -> Tuple[Any, Any]:
    """
    Match and combine masks from two masked arrays or align Xarray objects.

    Parameters
    ----------
    a1 : array-like
        First input array.
    a2 : array-like
        Second input array.

    Returns
    -------
    tuple
        Tuple of (a1_matched, a2_matched).
    """
    if isinstance(a1, xr.DataArray) and isinstance(a2, xr.DataArray):
        # Align xarray objects (works for dask-backed as well)
        return xr.align(a1, a2, join="inner")
    else:
        a1_arr = np.asanyarray(a1)
        a2_arr = np.asanyarray(a2)
        mask = np.ma.getmaskarray(a1_arr) | np.ma.getmaskarray(a2_arr)
        return np.ma.masked_where(mask, a1_arr), np.ma.masked_where(mask, a2_arr)

rmse(obs, mod, axis=None)

Calculate Root Mean Square Error between observations and model.

Parameters

obs : numpy.ndarray or xarray.DataArray Observed values. mod : numpy.ndarray or xarray.DataArray Model or predicted values. axis : int, str, or iterable, optional Axis or dimension along which to compute RMSE.

Returns

Union[np.number, np.ndarray, xr.DataArray] Root mean square error.

Source code in src/monet_stats/utils_stats.py 
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def rmse(
    obs: Union[np.ndarray, xr.DataArray],
    mod: Union[np.ndarray, xr.DataArray],
    axis: Optional[Union[int, str, Iterable[Union[int, str]]]] = None,
) -> Union[np.number, np.ndarray, xr.DataArray]:
    """
    Calculate Root Mean Square Error between observations and model.

    Parameters
    ----------
    obs : numpy.ndarray or xarray.DataArray
        Observed values.
    mod : numpy.ndarray or xarray.DataArray
        Model or predicted values.
    axis : int, str, or iterable, optional
        Axis or dimension along which to compute RMSE.

    Returns
    -------
    Union[np.number, np.ndarray, xr.DataArray]
        Root mean square error.
    """
    from .error_metrics import RMSE

    res = RMSE(obs, mod, axis=axis)
    if hasattr(res, "attrs") and hasattr(res, "coords"):
        return _update_history(res, "rmse")
    return res

Contributing to API Documentation

If you find issues with the API documentation or would like to suggest improvements:

  1. Check the GitHub Issues
  2. Submit new issues with clear descriptions
  3. Consider contributing improvements via pull requests

For development documentation, see the Contributing Guide.