Metric

MeanAsymmetricError

Calculate mean of asymmetric loss function.

Output is non-negative floating point. The best value is 0.0.

Error values that are less than the asymmetric threshold have left_error_function applied. Error values greater than or equal to asymmetric threshold have right_error_function applied.

Many forecasting loss functions (like those discussed in [1]) assume that over- and under- predictions should receive an equal penalty. However, this may not align with the actual cost faced by users’ of the forecasts. Asymmetric loss functions are useful when the cost of under- and over- prediction are not the same.

Setting asymmetric_threshold to zero, left_error_function to ‘squared’ and right_error_function to ‘absolute` results in a greater penalty applied to over-predictions (y_true - y_pred < 0). The opposite is true for left_error_function set to ‘absolute’ and right_error_function set to ‘squared`.

The left_error_penalty and right_error_penalty can be used to add differing multiplicative penalties to over-predictions and under-predictions.

Quickstart

python

from sktime.performance_metrics.forecasting import MeanAsymmetricError

estimator = MeanAsymmetricError(multioutput='uniform_average', multilevel='uniform_average', asymmetric_threshold=0, left_error_function='squared', right_error_function='absolute', left_error_penalty=1.0, right_error_penalty=1.0, by_index=False)

Parameters(8)

asymmetric_thresholdfloat, default = 0.0: The value used to threshold the asymmetric loss function. Error values that are less than the asymmetric threshold have left_error_function applied. Error values greater than or equal to asymmetric threshold have right_error_function applied.
left_error_function{‘squared’, ‘absolute’}, default=’squared’: Loss penalty to apply to error values less than the asymmetric threshold.
right_error_function{‘squared’, ‘absolute’}, default=’absolute’: Loss penalty to apply to error values greater than or equal to the asymmetric threshold.
left_error_penaltyint or float, default=1.0: An additional multiplicative penalty to apply to error values less than the asymmetric threshold.
right_error_penaltyint or float, default=1.0: An additional multiplicative penalty to apply to error values greater than the asymmetric threshold.
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 MeanAsymmetricError
>>> y_true = np. array ([3, - 0.5, 2, 7, 2 ])
>>> y_pred = np. array ([2.5, 0.0, 2, 8, 1.25 ])
>>> asymmetric_error = MeanAsymmetricError ()
>>> asymmetric_error (y_true, y_pred) np.float64(0.5)
>>> asymmetric_error = MeanAsymmetricError (left_error_function = 'absolute', right_error_function = 'squared')
>>> asymmetric_error (y_true, y_pred) np.float64(0.4625)
>>> y_true = np. array ([[0.5, 1 ], [- 1, 1 ], [7, - 6 ]])
>>> y_pred = np. array ([[0, 2 ], [- 1, 2 ], [8, - 5 ]])
>>> asymmetric_error = MeanAsymmetricError ()
>>> asymmetric_error (y_true, y_pred) np.float64(0.75)
>>> asymmetric_error = MeanAsymmetricError (left_error_function = 'absolute', right_error_function = 'squared')
>>> asymmetric_error (y_true, y_pred) np.float64(0.7083333333333334)
>>> asymmetric_error = MeanAsymmetricError (multioutput = 'raw_values')
>>> asymmetric_error (y_true, y_pred) array([0.5, 1. ])
>>> asymmetric_error = MeanAsymmetricError (multioutput = [0.3, 0.7 ])
>>> asymmetric_error (y_true, y_pred) np.float64(0.85)

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.

[2]

Diebold, Francis X. (2007). “Elements of Forecasting (4th ed.)”, Thomson, South-Western: Ohio, US.