climpred.metrics._nrmse

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

Normalized Root Mean Square Error (NRMSE).

Root Mean Square Error (rmse) normalized by the standard deviation of the verification data.

NRMSE = \frac{RMSE}{\sigma_{o}\cdot\sqrt{fac}}
      = \sqrt{\frac{MSE}{\sigma^{2}_{o}\cdot fac}}
      = \sqrt{ \frac{\overline{(f - o)^{2}}}{ \sigma^2_{o}\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().

Note

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

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 rmse()

Details:

minimum

0.0

maximum

perfect

0.0

orientation

negative

better than climatology

0.0 - 1.0

worse than climatology

> 1.0

Reference:
  • 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.

  • Hawkins, Ed, Steffen Tietsche, Jonathan J. Day, Nathanael Melia, Keith Haines, and Sarah Keeley. “Aspects of Designing and Evaluating Seasonal-to-Interannual Arctic Sea-Ice Prediction Systems.” Quarterly Journal of the Royal Meteorological Society 142, no. 695 (January 1, 2016): 672–83. https://doi.org/10/gfb3pn.

  • 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.

Example

>>> HindcastEnsemble.verify(metric='nrmse', 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 0.4161 0.4272 0.4658 0.4806 ... 0.7244 0.8215 0.8793