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Metric

MedianAbsolutePercentageError

API referenceSource
Multivariate

Median absolute percentage error (MdAPE) or symmetric version.

For a univariate, non-hierarchical sample of true values \(y_1, \dots, y_n\) and predicted values \(\widehat{y}_1, \dots, \widehat{y}_n\), at time indices \(t_1, \dots, t_n\), evaluate or call returns the Median Absolute Percentage Error, \(median(\left|\frac{y_i - \widehat{y}_i}{y_i} \right|)\). (the time indices are not used)

if symmetric is True then calculates symmetric Median Absolute Percentage Error (sMdAPE), defined as \(median(\frac{2|y_i-\widehat{y}_i|}{|y_i|+|\widehat{y}_i|})\).

Both MdAPE and sMdAPE output non-negative floating point which is in fractional units rather than percentage. The best value is 0.0.

MdAPE and sMdAPE are measured in percentage error relative to the test data. Because it takes the absolute value rather than square the percentage forecast error, it penalizes large errors less than MSPE, RMSPE, MdSPE or RMdSPE.

Taking the median instead of the mean of the absolute percentage errors also makes this metric more robust to error outliers since the median tends to be a more robust measure of central tendency in the presence of outliers.

MAPE has no limit on how large the error can be, particularly when y_true values are close to zero. In such cases the function returns a large value instead of inf. While sMAPE is bounded at 2.

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

evaluate_by_index returns, at a time index \(t_i\), the absolute percentage error at that time index, \(\left| \frac{y_i - \widehat{y}_i}{y_i} \right|\), or \(\frac{2|y_i - \widehat{y}_i|}{|y_i| + |\widehat{y}_i|}\), the symmetric version, if symmetric is True, for all time indices \(t_1, \dots, t_n\) in the input.

Quickstart

python
from sktime.performance_metrics.forecasting import MedianAbsolutePercentageError

estimator = MedianAbsolutePercentageError(multioutput='uniform_average', multilevel='uniform_average', symmetric=False, by_index=False, relative_to='y_true', eps=None)

Parameters(6)

symmetricbool, default = False
Whether to calculate the symmetric version of the percentage metric
relative_to{“y_true”, “y_pred”}, default=”y_true”

Determines the denominator of the percentage error.

  • If "y_true", the denominator is the true values,

  • If "y_pred", the denominator is the predicted values.

epsfloat, default=None
Numerical epsilon used in denominator to avoid division by zero. Absolute 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.

Examples

>>> import numpy as np
>>> from sktime.performance_metrics.forecasting import MedianAbsolutePercentageError
>>> y_true = np. array ([3, - 0.5, 2, 7, 2 ])
>>> y_pred = np. array ([2.5, 0.0, 2, 8, 1.25 ])
>>> mdape = MedianAbsolutePercentageError (symmetric = False)
>>> mdape (y_true, y_pred) np.float64(0.16666666666666666)
>>> smdape = MedianAbsolutePercentageError (symmetric = True)
>>> smdape (y_true, y_pred) np.float64(0.18181818181818182)
>>> y_true = np. array ([[0.5, 1 ], [- 1, 1 ], [7, - 6 ]])
>>> y_pred = np. array ([[0, 2 ], [- 1, 2 ], [8, - 5 ]])
>>> mdape (y_true, y_pred) np.float64(0.5714285714285714)
>>> smdape (y_true, y_pred) np.float64(0.39999999999999997)
>>> mdape = MedianAbsolutePercentageError (multioutput = 'raw_values', symmetric = False)
>>> mdape (y_true, y_pred) array([0.14285714, 1. ])
>>> smdape = MedianAbsolutePercentageError (multioutput = 'raw_values', symmetric = True)
>>> smdape (y_true, y_pred) array([0.13333333, 0.66666667])
>>> mdape = MedianAbsolutePercentageError (multioutput = [0.3, 0.7 ], symmetric = False)
>>> mdape (y_true, y_pred) np.float64(0.7428571428571428)
>>> smdape = MedianAbsolutePercentageError (multioutput = [0.3, 0.7 ], symmetric = True)
>>> smdape (y_true, y_pred) np.float64(0.5066666666666666)

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.