def _nmse(ds, control, comparison, running=None, reference_period=None):
"""
Normalized MSE (NMSE) = Normalized Ensemble Variance (NEV) metric.
.. math:: NMSE = NEV = frac{MSE}{\sigma^2_{control} \cdot fac}
Perfect forecast: 0
Climatology forecast: 1
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.
Args:
ds (xarray object): xr.Dataset/xr.DataArray with member dimension.
control (xarray object): xr.Dataset/xr.DataArray of control simulation.
comparison (function): comparison function.
running (int): smoothing of control. Default: None (no smoothing).
reference_period (str): see _control_for_reference_period.
Returns:
nmse_skill (xarray object): skill of NMSE.
"""
supervector_dim = 'svd'
forecast, reference = comparison(ds, supervector_dim)
mse_skill = _mse(forecast, reference, dim=supervector_dim)
var = _get_variance(control,
time_length=running,
reference_period=reference_period)
fac = _get_norm_factor(comparison)
nmse_skill = 1 - mse_skill / var / fac
return nmse_skill
def _nmae(ds, control, comparison, running=None, reference_period=None):
"""
Normalized Ensemble Mean Absolute Error metric.
Formula
-------
NMAE-SS = 1 - mse / var
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.
NOTE: NMSE = - 1 - NEV
"""
supervector_dim = 'svd'
fct, truth = comparison(ds, supervector_dim)
mse_skill = _mse(fct, truth, dim=supervector_dim)
var = _get_variance(control,
time_length=running,
reference_period=reference_period)
fac = _get_norm_factor(comparison)
nmse_skill = 1 - mse_skill / var / fac
return nmse_skill
def _ppp(ds, control, comparison, running=None, reference_period=None):
"""Prognostic Potential Predictability (PPP) metric.
.. math:: PPP = 1 - \frac{MSE}{ \sigma_{control} \cdot fac}
Perfect forecast: 1
Climatology forecast: 0
References:
* 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.
* 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.
Args:
ds (xarray object): xr.Dataset/xr.DataArray with member dimension.
control (xarray object): xr.Dataset/xr.DataArray of control simulation.
comparison (function): comparison function.
running (int): smoothing of control. Default: None (no smoothing).
reference_period (str): see _control_for_reference_period.
Returns:
ppp_skill (xarray object): skill of PPP.
"""
supervector_dim = 'svd'
forecast, reference = comparison(ds, supervector_dim)
mse_skill = _mse(forecast, reference, dim=supervector_dim)
var = _get_variance(control,
time_length=running,
reference_period=reference_period)
fac = _get_norm_factor(comparison)
ppp_skill = 1 - mse_skill / var / fac
return ppp_skill