climpred.metrics._msess¶

climpred.metrics._msess(forecast, verif, dim=None, **metric_kwargs)[source]¶

Mean Squared Error Skill Score (MSESS).

$MSESS = 1 - \frac{MSE}{\sigma^2_{ref} \cdot fac} = 1 - \frac{\overline{(f - o)^{2}}}{\sigma^2_{ref} \cdot fac},$

where $fac$ is 1 when using comparisons involving the ensemble mean (m2e, e2c, e2o) and 2 when using comparisons involving individual ensemble members (m2c, m2m, m2o). See _get_norm_factor().

This skill score can be intepreted as a percentage improvement in accuracy. I.e., it can be multiplied by 100%.

Note

climpred uses a single-valued internal reference forecast for the MSSS, in the terminology of Murphy 1988. I.e., we use a single climatological variance of the verification data within the experimental window for normalizing MSE.

Parameters

forecast (xarray object) – Forecast.
verif (xarray object) – Verification data.
dim (str) – Dimension(s) to perform metric over.
comparison (str) – Name comparison needed for normalization factor fac, see _get_norm_factor() (Handled internally by the compute functions)
metric_kwargs (dict) – see mse()

Details:

minimum	-∞
maximum	1.0
perfect	1.0
orientation	positive
better than climatology	> 0.0
equal to climatology	0.0
worse than climatology	< 0.0

Reference:

Griffies, S. M., and K. Bryan. “A Predictability Study of Simulated North Atlantic Multidecadal Variability.” Climate Dynamics 13, no. 7–8 (August 1, 1997): 459–87. https://doi.org/10/ch4kc4.
Murphy, Allan H. “Skill Scores Based on the Mean Square Error and Their Relationships to the Correlation Coefficient.” Monthly Weather Review 116, no. 12 (December 1, 1988): 2417–24. https://doi.org/10/fc7mxd.
Pohlmann, Holger, Michael Botzet, Mojib Latif, Andreas Roesch, Martin Wild, and Peter Tschuck. “Estimating the Decadal Predictability of a Coupled AOGCM.” Journal of Climate 17, no. 22 (November 1, 2004): 4463–72. https://doi.org/10/d2qf62.
Bushuk, Mitchell, Rym Msadek, Michael Winton, Gabriel Vecchi, Xiaosong Yang, Anthony Rosati, and Rich Gudgel. “Regional Arctic Sea–Ice Prediction: Potential versus Operational Seasonal Forecast Skill. Climate Dynamics, June 9, 2018. https://doi.org/10/gd7hfq.

Example

>>> HindcastEnsemble.verify(metric='msess', comparison='e2o', alignment='same_verifs',
...     dim='init')
<xarray.Dataset>
Dimensions:  (lead: 10)
Coordinates:
  * lead     (lead) int32 1 2 3 4 5 6 7 8 9 10
    skill    <U11 'initialized'
Data variables:
    SST      (lead) float64 0.8268 0.8175 0.783 0.7691 ... 0.4753 0.3251 0.2268