# -*- coding: utf-8 -*-
"""some measures for evaluation of prediction, tests and model selection
Created on Tue Nov 08 15:23:20 2011
Updated on Wed Jun 03 10:42:20 2020
Authors: Josef Perktold & Peter Prescott
License: BSD-3
"""
import numpy as np
from statsmodels.tools.validation import array_like
[docs]def mse(x1, x2, axis=0):
"""mean squared error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
mse : ndarray or float
mean squared error along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass, for example
numpy matrices will silently produce an incorrect result.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.mean((x1 - x2) ** 2, axis=axis)
[docs]def rmse(x1, x2, axis=0):
"""root mean squared error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
rmse : ndarray or float
root mean squared error along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass, for example
numpy matrices will silently produce an incorrect result.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.sqrt(mse(x1, x2, axis=axis))
[docs]def rmspe(y, y_hat, axis=0, zeros=np.nan):
"""
Root Mean Squared Percentage Error
Parameters
----------
y : array_like
The actual value.
y_hat : array_like
The predicted value.
axis : int
Axis along which the summary statistic is calculated
zeros : float
Value to assign to error where y is zero
Returns
-------
rmspe : ndarray or float
Root Mean Squared Percentage Error along given axis.
"""
y_hat = np.asarray(y_hat)
y = np.asarray(y)
error = y - y_hat
loc = y != 0
loc = loc.ravel()
percentage_error = np.full_like(error, zeros)
percentage_error.flat[loc] = error.flat[loc] / y.flat[loc]
mspe = np.nanmean(percentage_error ** 2, axis=axis) * 100
return np.sqrt(mspe)
[docs]def maxabs(x1, x2, axis=0):
"""maximum absolute error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
maxabs : ndarray or float
maximum absolute difference along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.max(np.abs(x1 - x2), axis=axis)
[docs]def meanabs(x1, x2, axis=0):
"""mean absolute error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
meanabs : ndarray or float
mean absolute difference along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.mean(np.abs(x1 - x2), axis=axis)
[docs]def bias(x1, x2, axis=0):
"""bias, mean error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
bias : ndarray or float
bias, or mean difference along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.mean(x1 - x2, axis=axis)
[docs]def vare(x1, x2, ddof=0, axis=0):
"""variance of error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
vare : ndarray or float
variance of difference along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.var(x1 - x2, ddof=ddof, axis=axis)
[docs]def stde(x1, x2, ddof=0, axis=0):
"""standard deviation of error
Parameters
----------
x1, x2 : array_like
The performance measure depends on the difference between these two
arrays.
axis : int
axis along which the summary statistic is calculated
Returns
-------
stde : ndarray or float
standard deviation of difference along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they need to broadcast.
This uses ``numpy.asanyarray`` to convert the input. Whether this is the
desired result or not depends on the array subclass.
"""
x1 = np.asanyarray(x1)
x2 = np.asanyarray(x2)
return np.std(x1 - x2, ddof=ddof, axis=axis)
[docs]def iqr(x1, x2, axis=0):
"""
Interquartile range of error
Parameters
----------
x1 : array_like
One of the inputs into the IQR calculation.
x2 : array_like
The other input into the IQR calculation.
axis : {None, int}
axis along which the summary statistic is calculated
Returns
-------
irq : {float, ndarray}
Interquartile range along given axis.
Notes
-----
If ``x1`` and ``x2`` have different shapes, then they must broadcast.
"""
x1 = array_like(x1, "x1", dtype=None, ndim=None)
x2 = array_like(x2, "x1", dtype=None, ndim=None)
if axis is None:
x1 = x1.ravel()
x2 = x2.ravel()
axis = 0
xdiff = np.sort(x1 - x2, axis=axis)
nobs = x1.shape[axis]
idx = np.round((nobs - 1) * np.array([0.25, 0.75])).astype(int)
sl = [slice(None)] * xdiff.ndim
sl[axis] = idx
iqr = np.diff(xdiff[tuple(sl)], axis=axis)
iqr = np.squeeze(iqr) # drop reduced dimension
return iqr
# Information Criteria
# ---------------------
[docs]def aic(llf, nobs, df_modelwc):
"""
Akaike information criterion
Parameters
----------
llf : {float, array_like}
value of the loglikelihood
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
aic : float
information criterion
References
----------
https://en.wikipedia.org/wiki/Akaike_information_criterion
"""
return -2.0 * llf + 2.0 * df_modelwc
[docs]def aicc(llf, nobs, df_modelwc):
"""
Akaike information criterion (AIC) with small sample correction
Parameters
----------
llf : {float, array_like}
value of the loglikelihood
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
aicc : float
information criterion
References
----------
https://en.wikipedia.org/wiki/Akaike_information_criterion#AICc
"""
return -2.0 * llf + 2.0 * df_modelwc * nobs / (nobs - df_modelwc - 1.0)
[docs]def bic(llf, nobs, df_modelwc):
"""
Bayesian information criterion (BIC) or Schwarz criterion
Parameters
----------
llf : {float, array_like}
value of the loglikelihood
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
bic : float
information criterion
References
----------
https://en.wikipedia.org/wiki/Bayesian_information_criterion
"""
return -2.0 * llf + np.log(nobs) * df_modelwc
[docs]def hqic(llf, nobs, df_modelwc):
"""
Hannan-Quinn information criterion (HQC)
Parameters
----------
llf : {float, array_like}
value of the loglikelihood
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
hqic : float
information criterion
References
----------
Wikipedia does not say much
"""
return -2.0 * llf + 2 * np.log(np.log(nobs)) * df_modelwc
# IC based on residual sigma
[docs]def aic_sigma(sigma2, nobs, df_modelwc, islog=False):
r"""
Akaike information criterion
Parameters
----------
sigma2 : float
estimate of the residual variance or determinant of Sigma_hat in the
multivariate case. If islog is true, then it is assumed that sigma
is already log-ed, for example logdetSigma.
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
aic : float
information criterion
Notes
-----
A constant has been dropped in comparison to the loglikelihood base
information criteria. The information criteria should be used to compare
only comparable models.
For example, AIC is defined in terms of the loglikelihood as
:math:`-2 llf + 2 k`
in terms of :math:`\hat{\sigma}^2`
:math:`log(\hat{\sigma}^2) + 2 k / n`
in terms of the determinant of :math:`\hat{\Sigma}`
:math:`log(\|\hat{\Sigma}\|) + 2 k / n`
Note: In our definition we do not divide by n in the log-likelihood
version.
TODO: Latex math
reference for example lecture notes by Herman Bierens
See Also
--------
References
----------
https://en.wikipedia.org/wiki/Akaike_information_criterion
"""
if not islog:
sigma2 = np.log(sigma2)
return sigma2 + aic(0, nobs, df_modelwc) / nobs
[docs]def aicc_sigma(sigma2, nobs, df_modelwc, islog=False):
"""Akaike information criterion (AIC) with small sample correction
Parameters
----------
sigma2 : float
estimate of the residual variance or determinant of Sigma_hat in the
multivariate case. If islog is true, then it is assumed that sigma
is already log-ed, for example logdetSigma.
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
aicc : float
information criterion
Notes
-----
A constant has been dropped in comparison to the loglikelihood base
information criteria. These should be used to compare for comparable
models.
References
----------
https://en.wikipedia.org/wiki/Akaike_information_criterion#AICc
"""
if not islog:
sigma2 = np.log(sigma2)
return sigma2 + aicc(0, nobs, df_modelwc) / nobs
[docs]def bic_sigma(sigma2, nobs, df_modelwc, islog=False):
"""Bayesian information criterion (BIC) or Schwarz criterion
Parameters
----------
sigma2 : float
estimate of the residual variance or determinant of Sigma_hat in the
multivariate case. If islog is true, then it is assumed that sigma
is already log-ed, for example logdetSigma.
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
bic : float
information criterion
Notes
-----
A constant has been dropped in comparison to the loglikelihood base
information criteria. These should be used to compare for comparable
models.
References
----------
https://en.wikipedia.org/wiki/Bayesian_information_criterion
"""
if not islog:
sigma2 = np.log(sigma2)
return sigma2 + bic(0, nobs, df_modelwc) / nobs
[docs]def hqic_sigma(sigma2, nobs, df_modelwc, islog=False):
"""Hannan-Quinn information criterion (HQC)
Parameters
----------
sigma2 : float
estimate of the residual variance or determinant of Sigma_hat in the
multivariate case. If islog is true, then it is assumed that sigma
is already log-ed, for example logdetSigma.
nobs : int
number of observations
df_modelwc : int
number of parameters including constant
Returns
-------
hqic : float
information criterion
Notes
-----
A constant has been dropped in comparison to the loglikelihood base
information criteria. These should be used to compare for comparable
models.
References
----------
xxx
"""
if not islog:
sigma2 = np.log(sigma2)
return sigma2 + hqic(0, nobs, df_modelwc) / nobs
# from var_model.py, VAR only? separates neqs and k_vars per equation
# def fpe_sigma():
# ((nobs + self.df_model) / self.df_resid) ** neqs * np.exp(ld)
__all__ = [
maxabs,
meanabs,
medianabs,
medianbias,
mse,
rmse,
rmspe,
stde,
vare,
aic,
aic_sigma,
aicc,
aicc_sigma,
bias,
bic,
bic_sigma,
hqic,
hqic_sigma,
iqr,
]