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Correlation Metrics

Statistical correlation and skill score calculations for model evaluation.

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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
        return _update_history(result, "AC")
    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)))
        result = p1 / p2
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        # 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
        return _update_history(result, "CCC")
    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
        result = numerator / denominator
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "E1")
    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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "E1_prime")
    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 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")
        dim = _resolve_axis_to_dim(obs, 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)
        return _update_history(result, "IOA")
    else:
        obs_m = np.ma.masked_invalid(obs)
        mod_m = np.ma.masked_invalid(mod)
        obs_mean = np.ma.mean(obs_m, axis=axis, keepdims=True)
        num = np.ma.sum((obs_m - mod_m) ** 2, axis=axis)
        denom = np.ma.sum((np.ma.abs(mod_m - obs_mean) + np.ma.abs(obs_m - obs_mean)) ** 2, axis=axis)
        with np.errstate(divide="ignore", invalid="ignore"):
            result = 1.0 - (num / denom)
        return result.item() if hasattr(result, "item") and 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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "IOA_prime")
    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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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
        return _update_history(result, "KGE")
    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
    """
    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

        # Use native xarray correlation for speed and laziness (Aero Protocol)
        r = xr.corr(obs, mod, dim=dim)
        result = r**2
        return _update_history(result, "R2")
    else:
        from scipy.stats import pearsonr

        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)
            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).

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")
        dim = _resolve_axis_to_dim(obs, axis)
        result = ((mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        return _update_history(result, "RMSE")
    else:
        result = np.ma.sqrt(np.ma.mean((np.subtract(mod, obs)) ** 2, axis=axis))
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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).

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).

    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
    """
    res = _vectorized_regression_stats(obs, mod, axis=axis, mode="RMSEs")
    return _update_history(res, "RMSEs")

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).

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).

    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
    """
    res = _vectorized_regression_stats(obs, mod, axis=axis, mode="RMSEu")
    return _update_history(res, "RMSEu")

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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
        return _update_history(result, "WDAC")
    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)
        )
        result = numerator / denominator
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "WDIOA")
    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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "WDIOA_m")
    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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        result = (circlebias(mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        return _update_history(result, "WDRMSE")
    else:
        result = np.ma.sqrt(np.ma.mean((circlebias(np.subtract(mod, obs))) ** 2, axis=axis))
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        result = (circlebias_m(mod - obs) ** 2).mean(dim=dim, keep_attrs=True) ** 0.5
        return _update_history(result, "WDRMSE_m")
    else:
        result = np.ma.sqrt(np.ma.mean((circlebias_m(np.subtract(mod, obs))) ** 2, axis=axis))
        return result.item() if hasattr(result, "item") and np.ndim(result) == 0 else result

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)

        return _update_history(result, "d1")
    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 tau correlation coefficient (Aero Protocol: Standardized).

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 tau correlation coefficient (Aero Protocol: Standardized).

    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

    # Standardize to DataArray to leverage Xarray's apply_ufunc vectorization
    is_numpy = not isinstance(obs, xr.DataArray)
    if is_numpy:
        obs = xr.DataArray(obs)
        mod = xr.DataArray(mod)
        if axis is not None:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, Iterable) and not isinstance(axis, str):
                dim = [obs.dims[i] for i in axis]
            else:
                dim = axis
        else:
            dim = obs.dims
    else:
        if 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

    if axis is None:
        obsc, modc = matchedcompressed(obs, mod)
        if len(obsc) < 2:
            return np.nan
        return _kendalltau(obsc, modc)[0]

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

    # Use apply_ufunc to eliminate manual loops and support Dask
    icd = [dim] if isinstance(dim, (str, int)) else list(dim)
    result = xr.apply_ufunc(
        _kendalltau_onlytau,
        obs,
        mod,
        input_core_dims=[icd] * 2,
        output_core_dims=[[]],
        vectorize=True,
        dask="parallelized",
        dask_gufunc_kwargs={"allow_rechunk": True},
        output_dtypes=[float],
    )

    if is_numpy:
        return result.values
    return _update_history(result, "kendalltau")

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
    """
    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

        # Use native xarray correlation for speed and laziness (Aero Protocol)
        result = xr.corr(obs, mod, dim=dim)
        return _update_history(result, "pearsonr")
    else:
        from scipy.stats import pearsonr as _pearsonr

        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 with pairwise deletion
            obs = np.asanyarray(obs)
            mod = np.asanyarray(mod)
            mask = np.isnan(obs) | np.isnan(mod)
            obs = np.where(mask, np.nan, obs)
            mod = np.where(mask, np.nan, mod)

            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 (Aero Protocol: Vectorized).

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 (Aero Protocol: Vectorized).

    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
    """
    # Handle Xarray/NumPy alignment and dimension resolution
    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
    else:
        dim = axis

    if axis is None:
        obsc, modc = matchedcompressed(obs, mod)
        if len(obsc) < 2:
            return np.nan
        from scipy.stats import spearmanr as _spearmanr

        return _spearmanr(obsc, modc)[0]

    # Spearman is Pearson on ranks. Use rankdata along the axis.
    from scipy.stats import rankdata

    # Apply pairwise masking to ensure ranks are computed on the same set of points
    mask = np.isnan(obs) | np.isnan(mod)
    obs = xr.where(mask, np.nan, obs) if isinstance(obs, xr.DataArray) else np.where(mask, np.nan, obs)
    mod = xr.where(mask, np.nan, mod) if isinstance(mod, xr.DataArray) else np.where(mask, np.nan, mod)

    def _rank_wrapper(data, axis):
        # Handle all-NaN slices to avoid Scipy warnings or errors
        if np.all(np.isnan(data)):
            return np.full(data.shape, np.nan)
        return rankdata(data, axis=axis, nan_policy="omit")

    # Use apply_ufunc for rankdata to support both NumPy and Xarray/Dask
    if isinstance(obs, xr.DataArray):
        icd = [dim] if isinstance(dim, (str, int)) else list(dim)
        obs_ranked = xr.apply_ufunc(
            _rank_wrapper,
            obs,
            input_core_dims=[icd],
            output_core_dims=[icd],
            kwargs={"axis": -1},
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
            keep_attrs=True,
        )
        mod_ranked = xr.apply_ufunc(
            _rank_wrapper,
            mod,
            input_core_dims=[icd],
            output_core_dims=[icd],
            kwargs={"axis": -1},
            dask="parallelized",
            dask_gufunc_kwargs={"allow_rechunk": True},
            output_dtypes=[float],
            keep_attrs=True,
        )
        return pearsonr(obs_ranked, mod_ranked, axis=dim)
    else:
        obs_ranked = _rank_wrapper(obs, axis=axis)
        mod_ranked = _rank_wrapper(mod, axis=axis)
        return pearsonr(obs_ranked, mod_ranked, axis=axis)

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:
            if isinstance(axis, int):
                dim = obs.dims[axis]
            elif isinstance(axis, (list, tuple)):
                dim = [obs.dims[d] if isinstance(d, int) else d for d in axis]
            else:
                dim = axis
        else:
            dim = obs.dims

        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)
        return _update_history(result, "taylor_skill")
    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