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Metric

MSEnormalizedBySD

API-ReferenzQuellcode
Multivariate

Normalized mean squared error (NMSE).

NMSE normalizes the root mean squared error by the standard deviation of the true values, making it location-scale invariant. Output is non-negative floating point, lower is better, with 0.0 indicating a perfect forecast.

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

\[\text{NMSE} = \sqrt{ \frac{\sum_{i=1}^{n}(y_i - \widehat{y}_i)^2} {\sum_{i=1}^{n}(y_i - \bar{y})^2} }\]

where \(\bar{y} = \frac{1}{n}\sum_{i=1}^n y_i\).

Note that the squared NMSE equals one minus the coefficient of determination: \(\text{NMSE}^2 = 1 - R^2\). A model no better than predicting the mean yields \(\text{NMSE} = 1\).

If the variance of the true values is zero (constant series), the denominator is clamped to eps to avoid division by zero.

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

evaluate_by_index returns jackknife pseudo-values of the NMSE, at each time index \(t_i\), computed as \(n \cdot \text{NMSE} - (n-1) \cdot \text{NMSE}_{-i}\), where \(\text{NMSE}_{-i}\) is the NMSE with the i-th observation removed.

Schnellstart

python
from sktime.performance_metrics.forecasting import MSEnormalizedBySD

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

Parameter(4)

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 MSEnormalizedBySD
>>> y_true = np. array ([3, - 0.5, 2, 7, 2 ])
>>> y_pred = np. array ([2.5, 0.0, 2, 8, 1.25 ])
>>> nmse = MSEnormalizedBySD ()
>>> nmse (y_true, y_pred) np.float64(0.2630806138733395)

Referenzen

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