# geometric_mean_squared_error#

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

Geometric mean squared error (GMSE) or Root geometric mean squared error (RGMSE).

If square_root is False then calculates GMSE and if square_root is True then RGMSE is calculated. Both GMSE and RGMSE return non-negative floating point. The best value is approximately zero, rather than zero.

Like MSE and MdSE, GMSE is measured in squared units of the input data. RMdSE is on the same scale as the input data like RMSE and RdMSE. Because GMSE and RGMSE square the forecast error rather than taking the absolute value, they penalize large errors more than GMAE.

Parameters:
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 geometric mean squared error (RGMSE) If False, returns geometric mean squared error (GMSE)

Returns:
lossfloat

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

Notes

The geometric mean uses the product of values in its calculation. The presence of a zero value will result in the result being zero, even if all the other values of large. To partially account for this in the case where elements of y_true and y_pred are equal (zero error), the resulting zero error values are replaced in the calculation with a small value. This results in the smallest value the metric can take (when y_true equals y_pred) being close to but not exactly zero.

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 aeon.performance_metrics.forecasting import     geometric_mean_squared_error as gmse
>>> y_true = np.array([3, -0.5, 2, 7, 2])
>>> y_pred = np.array([2.5, 0.0, 2, 8, 1.25])
>>> gmse(y_true, y_pred)
2.80399089461488e-07
>>> gmse(y_true, y_pred, square_root=True)
0.000529527232030127
>>> y_true = np.array([[0.5, 1], [-1, 1], [7, -6]])
>>> y_pred = np.array([[0, 2], [-1, 2], [8, -5]])
>>> gmse(y_true, y_pred)
0.5000000000115499
>>> gmse(y_true, y_pred, square_root=True)
0.5000024031086919
>>> gmse(y_true, y_pred, multioutput='raw_values')
array([2.30997255e-11, 1.00000000e+00])
>>> gmse(y_true, y_pred, multioutput='raw_values',     square_root=True)
array([4.80621738e-06, 1.00000000e+00])
>>> gmse(y_true, y_pred, multioutput=[0.3, 0.7])
0.7000000000069299
>>> gmse(y_true, y_pred, multioutput=[0.3, 0.7],     square_root=True)
0.7000014418652152
```