Metric

KLDivergenceSingleExponential

KL-divergence based forecast error assuming single-exponential errors (KL-DE1).

KL-DE1 uses the Kullback-Leibler divergence between the actual and predicted distributions, assuming double-exponential (Laplace) distributed forecast errors scaled by the rolling standard deviation of the true values. Output is non-negative floating point, lower is better, with 0.0 indicating a perfect forecast.

KL-DE1 differs from KL-DE2 only in the scale estimator: KL-DE1 uses the standard deviation (square root of variance), while KL-DE2 uses the mean absolute deviation (MAD).

For a univariate, non-hierarchical sample of true values \(y_1, \dots, y_n\) and predicted values \(\widehat{y}_1, \dots, \widehat{y}_n\), evaluate or call returns

\[\text{KL-DE1} = \frac{1}{n}\sum_{i=1}^{n} \left[\exp\!\left(-\frac{|e_i|}{\hat\sigma_i}\right) + \frac{|e_i|}{\hat\sigma_i} - 1 \right]\]

where \(e_i = y_i - \widehat{y}_i\) and

\[\hat\sigma_i = \sqrt{ \frac{1}{i-1}\sum_{j=1}^{i-1}(y_j - \bar{y}_{i-1})^2 } \quad (i \ge 2)\]

is the rolling standard deviation of the first \(i-1\) true values.

\(\hat\sigma_1\) is undefined (no prior observations); it is clamped to eps.

Since KL-DE1 is a simple mean of per-index terms, evaluate_by_index returns the per-index KL-DE1 terms directly.

multioutput and multilevel control averaging across variables and hierarchy indices, see below.

Schnellstart

python

from sktime.performance_metrics.forecasting import KLDivergenceSingleExponential

estimator = KLDivergenceSingleExponential(multioutput='uniform_average', multilevel='uniform_average', by_index=False, eps=None, window=None)

Parameter(5)

windowint or None, default=None: Number of prior observations used to estimate the rolling standard deviation.
If None (default), an expanding window is used: all prior observations \(y_1, \dots, y_{i-1}\) contribute to \(\hat\sigma_i\).
If int, a fixed-length rolling window of the given size is used.
epsfloat, default=None: Numerical epsilon used in denominator to avoid division by zero. Values smaller than eps are replaced by eps. If None, defaults to np.finfo(np.float64).eps
multioutput‘uniform_average’ (default), 1D array-like, or ‘raw_values’: Whether and how to aggregate metric for multivariate (multioutput) data.
If 'uniform_average' (default), errors of all outputs are averaged with uniform weight.
If 1D array-like, errors are averaged across variables, with values used as averaging weights (same order).
If 'raw_values', does not average across variables (outputs), per-variable errors are returned.
multilevel{‘raw_values’, ‘uniform_average’, ‘uniform_average_time’}: How to aggregate the metric for hierarchical data (with levels).
If 'uniform_average' (default), errors are mean-averaged across levels.
If 'uniform_average_time', metric is applied to all data, ignoring level index.
If 'raw_values', does not average errors across levels, hierarchy is retained.
by_indexbool, default=False: Controls averaging over time points in direct call to metric object.
If False (default), direct call to the metric object averages over time points, equivalent to a call of the evaluate method.
If True, direct call to the metric object evaluates the metric at each time point, equivalent to a call of the evaluate_by_index method.

Beispiele

>>> import numpy as np
>>> from sktime.performance_metrics.forecasting import (
... KLDivergenceSingleExponential,
... )
>>> y_true = np. array ([3.0, 5.0, 2.0, 7.0, 4.0, 6.0 ])
>>> y_pred = np. array ([3.0, 5.0, 3.0, 6.0, 5.0, 5.5 ])
>>> klde1 = KLDivergenceSingleExponential ()
>>> klde1 (y_true, y_pred) np.float64(0.1285730069501481)

Referenzen

Chen, Z. and Yang, Y. (2004). “Assessing Forecast Accuracy Measures”, Preprint 2004-10, Iowa State University.