Back to models
Metric

RMSEnormalizedByIQR

API referenceSource
Multivariate

RMSE normalized by interquartile range (IQR).

Normalizes the root mean squared error (RMSE) by the interquartile range (IQR) 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

\[\text{IQR} = \frac{ \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \widehat{y}_i)^2} }{Q_3(y) - Q_1(y)}\]

where \(Q_1(y)\) and \(Q_3(y)\) are the 25th and 75th percentiles of the true values.

If the interquartile range is zero (e.g. more than half the values are identical), it 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 IQR error, at each time index \(t_i\), computed as \(n \cdot \text{IQR} - (n-1) \cdot \text{IQR}_{-i}\), where \(\text{IQR}_{-i}\) is the metric with the i-th observation removed.

Quickstart

python
from sktime.performance_metrics.forecasting import RMSEnormalizedByIQR

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

Parameters(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.

Examples

>>> import numpy as np
>>> from sktime.performance_metrics.forecasting import RMSEnormalizedByIQR
>>> y_true = np. array ([3, - 0.5, 2, 7, 2 ])
>>> y_pred = np. array ([2.5, 0.0, 2, 8, 1.25 ])
>>> iqre = RMSEnormalizedByIQR ()
>>> iqre (y_true, y_pred) np.float64(0.6422616289332564)

References

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