MeanSquaredError

class MeanSquaredError(multioutput='uniform_average', multilevel='uniform_average', square_root=False, by_index=False)[source]

Mean squared error (MSE) or root mean squared error (RMSE).

MSE and RMSE output is non-negative floating point. MSE has units of the input data squared, while RMSE is of the same unit as the input data. Lower is better, and the lowest possible value is 0.0.

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:

if square_root is False, the Mean Squared Error, \(\frac{1}{n}\sum_{i=1}^n \left(y_i - \widehat{y}_i\right)^2\)
if square_root is True, the Root Mean Squared Error, \(\sqrt{\frac{1}{n}\sum_{i=1}^n \left(y_i - \widehat{y}_i\right)^2}\)

MSE and RMSE are both non-negative floating point, lower values are better. The lowest possible value is 0.0.

multioutput and multilevel control averaging across variables and hierarchy indices, see below. If square_root is True, averages are taken over square roots of squared errors.

evaluate_by_index returns, at a time index \(t_i\):

if square_root is False, the squared error at that time index, \(\left(y_i - \widehat{y}_i\right)^2\), for all time indices \(t_1, \dots, t_n\) in the input.
if square_root is True, the jackknife pseudo-value of the RMSE at that time index, \(n * \bar{\varepsilon} - (n-1) * \varepsilon_i\), where \(\bar{\varepsilon}\) is the RMSE over all time indices, and \(\varepsilon_i\) is the RMSE with the i-th time index removed, i.e., using values \(y_1, \dots, y_{i-1}, y_{i+1}, \dots, y_n\), and \(\widehat{y}_1, \dots, \widehat{y}_{i-1}, \widehat{y}_{i+1}, \dots, \widehat{y}_n\).

MSE is measured in squared units of the input data, and RMSE is on the same scale as the data. Because MSE and RMSE square the forecast error rather than taking the absolute value, they penalize large errors more than MAE.

Parameters:

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.

square_rootbool, default = False

Whether to take the square root of the metric

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.

References

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

Examples

>>> import numpy as np
>>> from sktime.performance_metrics.forecasting import MeanSquaredError
>>> y_true = np.array([3, -0.5, 2, 7, 2])
>>> y_pred = np.array([2.5, 0.0, 2, 8, 1.25])
>>> mse = MeanSquaredError()
>>> mse(y_true, y_pred)
np.float64(0.4125)
>>> y_true = np.array([[0.5, 1], [-1, 1], [7, -6]])
>>> y_pred = np.array([[0, 2], [-1, 2], [8, -5]])
>>> mse(y_true, y_pred)
np.float64(0.7083333333333334)
>>> rmse = MeanSquaredError(square_root=True)
>>> rmse(y_true, y_pred)
np.float64(0.8227486121839513)
>>> rmse = MeanSquaredError(multioutput='raw_values')
>>> rmse(y_true, y_pred)
array([0.41666667, 1.        ])
>>> rmse = MeanSquaredError(multioutput='raw_values', square_root=True)
>>> rmse(y_true, y_pred)
array([0.64549722, 1.        ])
>>> rmse = MeanSquaredError(multioutput=[0.3, 0.7])
>>> rmse(y_true, y_pred)
np.float64(0.825)
>>> rmse = MeanSquaredError(multioutput=[0.3, 0.7], square_root=True)
>>> rmse(y_true, y_pred)
np.float64(0.8936491673103708)

Methods

__call__(y_true, y_pred, **kwargs)

Calculate metric value using underlying metric function.

__call__(y_true, y_pred, **kwargs)[source]

Calculate metric value using underlying metric function.

Parameters:

y_truetime series in sktime compatible data container format.

Ground truth (correct) target values.

Individual data formats in sktime are so-called mtype specifications, each mtype implements an abstract scitype.

Series scitype = individual time series, vanilla forecasting. pd.DataFrame, pd.Series, or np.ndarray (1D or 2D)
Panel scitype = collection of time series, global/panel forecasting. pd.DataFrame with 2-level row MultiIndex (instance, time), 3D np.ndarray (instance, variable, time), list of Series typed pd.DataFrame
Hierarchical scitype = hierarchical collection, for hierarchical forecasting. pd.DataFrame with 3 or more level row MultiIndex (hierarchy_1, ..., hierarchy_n, time)

For further details on data format, see glossary on mtype. For usage, see forecasting tutorial examples/01_forecasting.ipynb

y_predtime series in sktime compatible data container format

Predicted values to evaluate against ground truth. Must be of same format as y_true, same indices and columns if indexed.

y_pred_benchmarkoptional, time series in sktime compatible data container format

Benchmark predictions to compare y_pred to, used for relative metrics. Required only if metric requires benchmark predictions, as indicated by tag requires-y-pred-benchmark. Otherwise, can be passed to ensure interface consistency, but is ignored. Must be of same format as y_true, same indices and columns if indexed.

y_trainoptional, time series in sktime compatible data container format

Training data used to normalize the error metric. Required only if metric requires training data, as indicated by tag requires-y-train. Otherwise, can be passed to ensure interface consistency, but is ignored. Must be of same format as y_true, same columns if indexed, but not necessarily same indices.

sample_weightoptional, 1D array-like, or callable, default=None

Sample weights for each time point.

If None, the time indices are considered equally weighted.
If an array, must be 1D. If y_true and y_pred``are a single time series, ``sample_weight must be of the same length as y_true. If the time series are panel or hierarchical, the length of all individual time series must be the same, and equal to the length of sample_weight, for all instances of time series passed.
If a callable, it must follow SampleWeightGenerator interface, or have one of the following signatures: y_true: pd.DataFrame -> 1D array-like, or y_true: pd.DataFrame x y_pred: pd.DataFrame -> 1D array-like.

Returns:

lossfloat, np.ndarray, or pd.DataFrame

Calculated metric, averaged or by variable. Weighted by sample_weight if provided.

float if multioutput="uniform_average" or array-like, and ``multilevel="uniform_average" or “uniform_average_time”``. Value is metric averaged over variables and levels (see class docstring)
np.ndarray of shape (y_true.columns,) if multioutput=”raw_values”` and multilevel="uniform_average" or "uniform_average_time". i-th entry is the, metric calculated for i-th variable
pd.DataFrame if multilevel="raw_values". of shape (n_levels, ), if multioutput="uniform_average"; of shape (n_levels, y_true.columns) if multioutput="raw_values". metric is applied per level, row averaging (yes/no) as in multioutput.