# Source code for pastas.stats.metrics

```
"""The following methods may be used to describe the fit between the model simulation
and the observations.
Examples
========
These methods may be used as follows:
>>> ps.stats.rmse(sim, obs)
or directly from a Pastas model:
>>> ml.stats.rmse()
"""
from logging import getLogger
# Type Hinting
from typing import Optional
from numpy import abs as npabs
from numpy import average, log, nan, sqrt
from pandas import Series
from pastas.stats.core import _get_weights, mean, std, var
__all__ = [
"rmse",
"sse",
"mae",
"nse",
"evp",
"rsq",
"bic",
"aic",
"pearsonr",
"kge_2012",
]
logger = getLogger(__name__)
# Absolute Error Metrics
[docs]def mae(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Mean Absolute Error (MAE).
Parameters
----------
sim: pandas.Series, optional
Series with the simulated values.
obs: pandas.Series, optional
The Series with the observed values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
weighted: bool, optional
Weight the values by the normalized time step to account for irregular time
series. Default is True.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
Notes
-----
The Mean Absolute Error (MAE) between the observed (:math:`y_o`) and simulated
(:math:`y_s`) time series is computed as follows:
.. math:: \\text{MAE} = \\sum_{i=1}^{N} w_i |y_s - y_o|
where :math:`N` is the number of observations in the observed time series.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
w = _get_weights(err, weighted=weighted, max_gap=max_gap)
return (w * npabs(err.to_numpy())).sum()
[docs]def rmse(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Root Mean Squared Error (RMSE).
Parameters
----------
sim: pandas.Series, optional
Series with the simulated values.
obs: pandas.Series, optional
The Series with the observed values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
weighted: bool, optional
Weight the values by the normalized time step to account for irregular time
series. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
Notes
-----
Computes the Root Mean Squared Error (RMSE) as follows:
.. math:: \\text{RMSE} = \\sqrt{\\sum_{i=1}^{N} w_i \\epsilon_i^2}
where :math:`N` is the number of error :math:`\\epsilon`.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
w = _get_weights(err, weighted=weighted, max_gap=max_gap)
return sqrt((w * err.to_numpy() ** 2).sum())
[docs]def sse(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
) -> float:
"""Compute the Sum of the Squared Errors (SSE).
Parameters
----------
sim: pandas.Series, optional
Series with the simulated values.
obs: pandas.Series, optional
The Series with the observed values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
Notes
-----
The Sum of the Squared Errors (SSE) is calculated as follows:
.. math:: \\text{SSE} = \\sum(\\epsilon^2)
where :math:`\\epsilon` are the errors.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
return (err.to_numpy() ** 2).sum()
# Percentage Error Metrics
[docs]def pearsonr(
obs: Series,
sim: Series,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Pearson correlation (r).
Parameters
----------
sim: pandas.Series
The Series with the simulated values.
obs: pandas.Series
The Series with the observed values.
missing: str, optional
string with the rule to deal with missing values in the observed series. Only
"drop" is supported now.
weighted: bool, optional
Weight the values by the normalized time step to account for irregular time
series. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
Notes
-----
The Pearson correlation (r) is computed as follows:
.. math:: r = \\frac{\\sum_{i=1}^{N}w_i (y_{o,i} - \\bar{y_o})(y_{s,i} - \\bar{
y_s})} {\\sqrt{\\sum_{i=1}^{N} w_i(y_{o,i}-\\bar{y_o})^2 \\sum_{i=1}^{N}w_i(
y_{s,i} -\\bar{y_s})^2}}
Where :math:`y_o` is observed time series, :math:`y_s` the simulated time series,
and :math:`N` the number of observations in the observed time series.
"""
if missing == "drop":
obs = obs.dropna()
w = _get_weights(obs, weighted=weighted, max_gap=max_gap)
sim = sim.reindex(obs.index).dropna().to_numpy()
# Return nan if the time indices of the sim and obs don't match
if sim.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
sim = sim - average(sim, weights=w)
obs = obs.to_numpy() - average(obs.to_numpy(), weights=w)
r = (w * sim * obs).sum() / sqrt((w * sim**2).sum() * (w * obs**2).sum())
return r
[docs]def evp(
obs: Series,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Explained Variance Percentage (EVP).
Parameters
----------
obs: pandas.Series
Series with the observed values.
sim: pandas.Series, optional
The Series with the simulated values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
weighted: bool, optional
If weighted is True, the variances are computed using the time step between
observations as weights. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
Notes
-----
Commonly used goodness-of-fit metric groundwater level models as computed in
:cite:t:`von_asmuth_groundwater_2012`.
.. math:: \\text{EVP} = \\frac{\\sigma_h^2 - \\sigma_r^2}{\\sigma_h^2}
* 100
where :math:`\\sigma_h^2` is the variance of the observations and
:math:`\\sigma_r^2` is the variance of the errors. The returned value is
bounded between 0% and 100%.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
if obs.var() == 0.0:
return 100.0
else:
return (
max(
0.0,
(
1
- var(err, weighted=weighted, max_gap=max_gap)
/ var(obs, weighted=weighted, max_gap=max_gap)
),
)
* 100
)
[docs]def nse(
obs: Series,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Nash-Sutcliffe Efficiency (NSE).
Parameters
----------
obs: pandas.Series
Series with the observed values.
sim: pandas.Series, optional
The Series with the simulated values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
weighted: bool, optional
If weighted is True, the variances are computed using the time step between
observations as weights. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
Notes
-----
NSE computed according to :cite:t:`nash_river_1970`
.. math:: \\text{NSE} = 1 - \\frac{\\sum(h_s-h_o)^2}{\\sum(h_o-\\mu_{h,o})}
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
w = _get_weights(err, weighted=weighted, max_gap=max_gap)
mu = average(obs.to_numpy(), weights=w)
return 1 - (w * err.to_numpy() ** 2).sum() / (w * (obs.to_numpy() - mu) ** 2).sum()
[docs]def rsq(
obs: Series,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
nparam: Optional[int] = None,
) -> float:
"""Compute R-squared, possibly adjusted for the number of free parameters.
Parameters
----------
obs: pandas.Series
Series with the observed values.
sim: pandas.Series, optional
The Series with the simulated values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
weighted: bool, optional
If weighted is True, the variances are computed using the time step between
observations as weights. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the weights.
All time steps larger than max_gap are replace with the max_gap value.
Default value is 30 days.
nparam: int, optional
number of calibrated parameters.
Notes
-----
.. math:: \\rho_{adj} = 1- \\frac{n-1}{n-n_{param}}*\\frac{rss}{tss}
Where n is the number of observations, :math:`n_{param}` the number of free
parameters, rss the sum of the squared errors, and tss the total sum of
squared errors.
When nparam is provided, the :math:`\\rho` is adjusted for the number of
calibration parameters.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
w = _get_weights(err, weighted=weighted, max_gap=max_gap)
mu = average(obs.to_numpy(), weights=w)
rss = (w * err.to_numpy() ** 2.0).sum()
tss = (w * (obs.to_numpy() - mu) ** 2.0).sum()
if nparam:
return 1.0 - (obs.size - 1.0) / (obs.size - nparam) * rss / tss
else:
return 1.0 - rss / tss
[docs]def bic(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
nparam: int = 1,
) -> float:
"""Compute the Bayesian Information Criterium (BIC).
Parameters
----------
obs: pandas.Series, optional
Series with the observed values.
sim: pandas.Series, optional
The Series with the simulated values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
nparam: int, optional
number of calibrated parameters.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
Notes
-----
The Bayesian Information Criterium (BIC) :cite:p:`akaike_bayesian_1979` is
computed as follows:
.. math:: \\text{BIC} = -2 log(L) + n_{param} * log(N)
where :math:`n_{param}` is the number of calibration parameters.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
n = err.index.size
return n * log((err.to_numpy() ** 2.0).sum() / n) + nparam * log(n)
[docs]def aic(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
nparam: int = 1,
) -> float:
"""Compute the Akaike Information Criterium (AIC).
Parameters
----------
obs: pandas.Series, optional
Series with the observed values.
sim: pandas.Series, optional
The Series with the simulated values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
nparam: int, optional
number of calibrated parameters.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
Notes
-----
The Akaike Information Criterium (AIC) :cite:p:`akaike_new_1974` is computed as
follows:
.. math:: \\text{AIC} = -2 log(L) + 2 nparam
where :math:`n_{param}` is the number of calibration parameters and L is the
likelihood function for the model.
"""
err = _compute_err(obs=obs, sim=sim, res=res, missing=missing)
# Return nan if the time indices of the sim and obs don't match
if err.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
n = err.index.size
return n * log((err.to_numpy() ** 2.0).sum() / n) + 2.0 * nparam
# Forecast Error Metrics
[docs]def kge_2012(
obs: Series,
sim: Series,
missing: str = "drop",
weighted: bool = False,
max_gap: int = 30,
) -> float:
"""Compute the (weighted) Kling-Gupta Efficiency (KGE).
Parameters
----------
sim: pandas.Series
Series with the simulated values.
obs: pandas.Series
The Series with the observed values.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is
supported now.
weighted: bool, optional
Weight the values by the normalized time step to account for
irregular time series. Default is False.
max_gap: int, optional
maximum allowed gap period in days to use for the computation of the
weights. All time steps larger than max_gap are replace with the
max_gap value. Default value is 30 days.
Notes
-----
The (weighted) Kling-Gupta Efficiency :cite:t:`kling_runoff_2012` is
computed as follows:
.. math:: \\text{KGE} = 1 - \\sqrt{(r-1)^2 + (\\beta-1)^2 - (\\gamma-1)^2}
where :math:`\\beta = \\bar{x} / \\bar{y}` and :math:`\\gamma =
\\frac{\\bar{\\sigma}_x / \\bar{x}}{\\bar{\\sigma}_y / \\bar{y}}`. If
weighted equals True, the weighted mean, variance and pearson
correlation are used.
"""
if missing == "drop":
obs = obs.dropna()
sim = sim.reindex(obs.index).dropna()
# Return nan if the time indices of the sim and obs don't match
if sim.index.size == 0:
logger.warning("Time indices of the sim and obs don't match.")
return nan
r = pearsonr(obs=obs, sim=sim, weighted=weighted, max_gap=max_gap)
mu_sim = mean(sim, weighted=weighted, max_gap=max_gap)
mu_obs = mean(obs, weighted=weighted, max_gap=max_gap)
beta = mu_sim / mu_obs
gamma = (std(sim, weighted=weighted, max_gap=max_gap) / mu_sim) / (
std(obs, weighted=weighted, max_gap=max_gap) / mu_obs
)
kge = 1 - sqrt((r - 1) ** 2 + (beta - 1) ** 2 + (gamma - 1) ** 2)
return kge
def _compute_err(
obs: Optional[Series] = None,
sim: Optional[Series] = None,
res: Optional[Series] = None,
missing: str = "drop",
):
"""
Parameters
----------
Parameters
----------
sim: pandas.Series, optional
Series with the simulated values.
obs: pandas.Series, optional
The Series with the observed values.
res: pandas.Series, optional
The Series with the residual values. If time series for the residuals are
provided, the sim and obs arguments are ignored. Note that the residuals
must be computed as `obs - sim` here.
missing: str, optional
string with the rule to deal with missing values. Only "drop" is supported now.
Returns
-------
err: pandas.Series
The pandas.Series with the errors, computed as
"""
if (obs is not None) and (sim is not None):
err = sim.subtract(obs)
elif res is not None:
err = -res
else:
msg = (
"Either the residuals, or the simulation and observations have to be "
"provided. Please provide one of these two input options."
)
logger.error(msg)
raise ValueError(msg)
if missing == "drop":
err = err.dropna()
return err
```