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Efficiency Metrics

Model efficiency and performance measures for evaluating forecast skill.

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