Comparisons

Forecasts have to be verified against some product to evaluate their performance. However, when verifying against a product, there are many different ways one can compare the ensemble of forecasts. Here we cover the comparison options for both hindcast and perfect model ensembles. See terminology for clarification on the differences between these two experimental setups.

All high-level functions like verify() and bootstrap() (for both HindcastEnsemble and PerfectModelEnsemble objects) take a comparison='' keyword to select the comparison style. See below for a detailed description on the differences between these comparisons.

Hindcast Ensembles

In hindcast ensembles, the ensemble mean forecast (comparison='e2o') is expected to perform better than individual ensemble members (comparison='m2o') as the chaotic component of forecasts is expected to be suppressed by this averaging, while the memory of the system sustains. [Boer2016]

keyword: 'e2o', 'e2r'

_e2o(hind, verif[, metric])

Compare the ensemble mean forecast to the verification data for a HindcastEnsemble setup.

keyword: 'm2o', 'm2r'

_m2o(hind, verif[, metric])

Compares each ensemble member individually to the verification data for a HindcastEnsemble setup.

Perfect Model Ensembles

In perfect-model frameworks, there are many more ways of verifying forecasts. [Seferian2018] uses a comparison of all ensemble members against the control run (comparison='m2c') and all ensemble members against all other ensemble members (comparison='m2m'). Furthermore, the ensemble mean forecast can be verified against one control member (comparison='e2c') or all members (comparison='m2e') as done in [Griffies1997].

keyword: 'm2e'

_m2e(ds[, metric])

Compare all members to ensemble mean while leaving out the reference in

keyword: 'm2c'

_m2c(ds[, metric])

Compare all other member forecasts to a single member verification, which is the first member.

keyword: 'm2m'

_m2m(ds[, metric])

Compare all members to all others in turn while leaving out the verification member.

keyword: 'e2c'

_e2c(ds[, metric])

Compare ensemble mean forecast to single member verification.

Normalization

The goal of a normalized distance metric is to get a constant or comparable value of typically 1 (or 0 for metrics defined as 1 - metric) when the metric saturates and the predictability horizon is reached (see metrics).

A factor is added in the normalized metric formula (see [Seferian2018]) to accomodate different comparison styles. For example, nrmse gets smalled in comparison m2e than m2m by design, since the ensembe mean is always closer to individual members than the ensemble members to each other. In turn, the normalization factor is 2 for comparisons m2c, m2m, and m2o. It is 1 for m2e, e2c, and e2o.

Interpretation of Results

When HindcastEnsemble skill is computed over all initializations dim='init' of the hindcast, the resulting skill is a mean forecast skill over all initializations.

PerfectModelEnsemble skill is computed over a supervector comprised of all initializations and members, which allows the computation of the ACC-based skill [Bushuk2018], but also returns a mean forecast skill over all initializations.

The supervector approach shown in [Bushuk2018] and just calculating a distance-based metric like rmse over the member dimension as in [Griffies1997] yield very similar results.

Compute over dimension

The argument dim defines over which dimension a metric is computed. We can apply a metric over all dimensions from the initialized dataset expect lead. The resulting skill is then reduced by this dim. Therefore, applying a metric over dim='member' creates a skill for all initializations individually. This can show the initial conditions dependence of skill. Likewise when computing skill over 'init', we get skill for each member. This dim argument is different from the comparison argument which just specifies how forecast and observations are defined.

However, this above logic applies to deterministic metrics. Probabilistic metrics need to be applied to the member dimension and comparison from ['m2c', 'm2m'] in PerfectModelEnsemble verify() and 'm2o' comparison in HindcastEnsemble verify().

dim should not contain member when the comparison already computes ensemble means as in ['e2o', 'e2c'].

User-defined comparisons

You can also construct your own comparisons via the Comparison class.

Comparison(name, function, hindcast, ...[, ...])

Master class for all comparisons.

First, write your own comparison function, similar to the existing ones. If a comparison should also be used for probabilistic metrics, make sure that metric.probabilistic returns forecast with member dimension and observations without. For deterministic metrics, return forecast and observations with identical dimensions but without an identical comparison:

from climpred.comparisons import Comparison, _drop_members

def _my_m2median_comparison(ds, metric=None):
    """Identical to m2e but median."""
    observations_list = []
    forecast_list = []
    supervector_dim = 'member'
    for m in ds.member.values:
        forecast = _drop_members(ds, rmd_member=[m]).median('member')
        observations = ds.sel(member=m).squeeze()
        forecast_list.append(forecast)
        observations_list.append(observations)
    observations = xr.concat(observations_list, supervector_dim)
    forecast = xr.concat(forecast_list, supervector_dim)
    forecast[supervector_dim] = np.arange(forecast[supervector_dim].size)
    observations[supervector_dim] = np.arange(observations[supervector_dim].size)
    return forecast, observations

Then initialize this comparison function with Comparison:

__my_m2median_comparison = Comparison(
    name='m2me',
    function=_my_m2median_comparison,
    probabilistic=False,
    hindcast=False)

Finally, compute skill based on your own comparison:

skill = compute_perfect_model(ds, control,
                              metric='rmse',
                              comparison=__my_m2median_comparison)

Once you come up with an useful comparison for your problem, consider contributing this comparison to climpred, so all users can benefit from your comparison, see contributing.

References

Boer2016

Boer, G. J., D. M. Smith, C. Cassou, F. Doblas-Reyes, G. Danabasoglu, B. Kirtman, Y. Kushnir, et al. “The Decadal Climate Prediction Project (DCPP) Contribution to CMIP6.” Geosci. Model Dev. 9, no. 10 (October 25, 2016): 3751–77. https://doi.org/10/f89qdf.

Bushuk2018(1,2)

Mitchell Bushuk, 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 2018. https://doi.org/10/gd7hfq.

Griffies1997(1,2)
    1. Griffies and K. Bryan. A predictability study of simulated North Atlantic multidecadal variability. Climate Dynamics, 13(7-8):459–487, August 1997. https://doi.org/10/ch4kc4.

Seferian2018(1,2)

Roland Séférian, Sarah Berthet, and Matthieu Chevallier. Assessing the Decadal Predictability of Land and Ocean Carbon Uptake. Geophysical Research Letters, March 2018. https://doi.org/10/gdb424.