median_squared_percentage_error(y_true, y_pred, horizon_weight=None, multioutput='uniform_average', square_root=False, symmetric=False, **kwargs)[source]#

Median squared percentage error (MdSPE) or square root version.

If square_root is False then calculates MdSPE and if square_root is True then calculates root median squared percentage error (RMdSPE). If symmetric is True then calculates sMdSPE or sRMdSPE. Output is non-negative floating point. The best value is 0.0.

MdSPE is measured in squared percentage error relative to the test data. RMdSPE is measured in percentage error relative to the test data. Because the calculation takes the square rather than absolute value of the percentage forecast error, large errors are penalized more than MAPE, sMAPE, MdAPE or sMdAPE.

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.

There is no limit on how large the error can be, particulalrly when y_true values are close to zero. In such cases the function returns a large value instead of inf.

y_truepd.Series, pd.DataFrame or np.array of shape (fh,) or (fh, n_outputs) where fh is the forecasting horizon

Ground truth (correct) target values.

y_predpd.Series, pd.DataFrame or np.array of shape (fh,) or (fh, n_outputs) where fh is the forecasting horizon

Forecasted values.

horizon_weightarray-like of shape (fh,), default=None

Forecast horizon weights.

multioutput{‘raw_values’, ‘uniform_average’} or array-like of shape (n_outputs,), default=’uniform_average’

Defines how to aggregate metric for multivariate (multioutput) data. If array-like, values used as weights to average the errors. If ‘raw_values’, returns a full set of errors in case of multioutput input. If ‘uniform_average’, errors of all outputs are averaged with uniform weight.

square_rootbool, default=False

Whether to take the square root of the mean squared error. If True, returns root mean squared error (RMSPE) If False, returns mean squared error (MSPE)

symmetricbool, default=False

Calculates symmetric version of metric if True.


MdSPE or RMdSPE loss. If multioutput is ‘raw_values’, then MdSPE or RMdSPE is returned for each output separately. If multioutput is ‘uniform_average’ or an ndarray of weights, then the weighted average MdSPE or RMdSPE of all output errors is returned.


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


>>> from sktime.performance_metrics.forecasting import         median_squared_percentage_error
>>> y_true = np.array([3, -0.5, 2, 7, 2])
>>> y_pred = np.array([2.5, 0.0, 2, 8, 1.25])
>>> median_squared_percentage_error(y_true, y_pred, symmetric=False)
>>> median_squared_percentage_error(y_true, y_pred, square_root=True,     symmetric=False)
>>> y_true = np.array([[0.5, 1], [-1, 1], [7, -6]])
>>> y_pred = np.array([[0, 2], [-1, 2], [8, -5]])
>>> median_squared_percentage_error(y_true, y_pred, symmetric=False)
>>> median_squared_percentage_error(y_true, y_pred, square_root=True,     symmetric=False)
>>> median_squared_percentage_error(y_true, y_pred, multioutput='raw_values',     symmetric=False)
array([0.02040816, 1.        ])
>>> median_squared_percentage_error(y_true, y_pred, multioutput='raw_values',     square_root=True, symmetric=False)
array([0.14285714, 1.        ])
>>> median_squared_percentage_error(y_true, y_pred, multioutput=[0.3, 0.7],     symmetric=False)
>>> median_squared_percentage_error(y_true, y_pred, multioutput=[0.3, 0.7],     square_root=True, symmetric=False)