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Metric

GeometricMeanAbsoluteError

API referenceSource
Multivariate

Geometric mean absolute error (GMAE).

For a univariate, non-hierarchical sample of true values \(y_1, \dots, y_n\) and predicted values \(\widehat{y}_1, \dots, \widehat{y}_n\) (in \(mathbb{R}\)), at time indices \(t_1, \dots, t_n\), evaluate or call returns the Geometric Mean Absolute Error, \(\left(\prod_{i=1}^n |y_i - \widehat{y}_i| \right)^{\frac{1}{n}}\). (the time indices are not used)

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

evaluate_by_index returns, at a time index \(t_i\), jackknife pseudo-samples of the GMAE at that time index, \(n * \bar{\varepsilon} - (n-1) * \varepsilon_i\), where \(\bar{\varepsilon}\) is the GMAE over all time indices, and \(\varepsilon_i\) is the GMAE with the i-th time index removed.

GMAE output is non-negative floating point. The best value is approximately zero, rather than zero.

Like MAE and MdAE, GMAE is measured in the same units as the input data. Because GMAE takes the absolute value of the forecast error rather than squaring it, MAE penalizes large errors to a lesser degree than squared error variants like MSE, RMSE or GMSE or RGMSE.

Quickstart

python
from sktime.performance_metrics.forecasting import GeometricMeanAbsoluteError

estimator = GeometricMeanAbsoluteError(multioutput='uniform_average', multilevel='uniform_average', by_index=False)

Parameters(3)

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.

Examples

>>> import numpy as np
>>> from sktime.performance_metrics.forecasting import GeometricMeanAbsoluteError
>>> y_true = np. array ([3, - 0.5, 2, 7, 2 ])
>>> y_pred = np. array ([2.5, 0.0, 2, 8, 1.25 ])
>>> gmae = GeometricMeanAbsoluteError ()
>>> gmae (y_true, y_pred) np.float64(0.000529527232030127)
>>> y_true = np. array ([[0.5, 1 ], [- 1, 1 ], [7, - 6 ]])
>>> y_pred = np. array ([[0, 2 ], [- 1, 2 ], [8, - 5 ]])
>>> gmae (y_true, y_pred) np.float64(0.5000024031086919)
>>> gmae = GeometricMeanAbsoluteError (multioutput = 'raw_values')
>>> gmae (y_true, y_pred) array([4.80621738e-06, 1.00000000e+00])
>>> gmae = GeometricMeanAbsoluteError (multioutput = [0.3, 0.7 ])
>>> gmae (y_true, y_pred) np.float64(0.7000014418652152)

References

  1. Hyndman, R. J and Koehler, A. B. (2006). “Another look at measures of forecast accuracy”, International Journal of Forecasting, Volume 22, Issue 4.