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 HindcastEnsemble and PerfectModelEnsemble. See terminology for clarification on the differences between these two experimental setups.

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

Hindcast Ensembles#

In HindcastEnsemble, 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. [Boer et al., 2016]

keyword: "e2o", "e2r"

_e2o(initialized, verif, metric)

Compare the ensemble mean forecast to the verification data.

keyword: "m2o", "m2r"

_m2o(initialized, verif, metric)

Compare each ensemble member individually to the verification data.

Perfect Model Ensembles#

In PerfectModelEnsemble, there are many more ways of verifying forecasts. Séférian et al. [2018] 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 Griffies and Bryan [1997].

keyword: "m2e"

_m2e(initialized[, metric, verif])

Compare all members to ensemble mean while leaving out the verif in ensemble mean.

keyword: "m2c"

_m2c(initialized, metric[, verif])

Compare all other member forecasts to a single member verification.

keyword: "m2m"

_m2m(initialized, metric[, verif])

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

keyword: "e2c"

_e2c(initialized[, metric, verif])

Compare ensemble mean forecast to single member verification.


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 [Séférian et al., 2018] to accomodate different comparison styles. For example, metric="nrmse" gets smaller 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 [Bushuk et al., 2018], but also returns a mean forecast skill over all initializations.

The supervector approach shown in Bushuk et al. [2018] and just calculating a distance-based metric like rmse over the member dimension as in Griffies and Bryan [1997] 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" or dim=[] 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 climpred.comparisons.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 probabilistic metrics 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, M2M_MEMBER_DIM

def _my_m2median_comparison(initialized, metric=None):
    """Identical to m2e but median."""
    observations_list = []
    forecast_list = []
    supervector_dim = "member"
    for m in initialized.member.values:
        forecast = initialized.drop_sel(member=m).median("member")
        observations = initialized.sel(member=m).squeeze()
    observations = xr.concat(observations_list, M2M_MEMBER_DIM)
    forecast = xr.concat(forecast_list, M2M_MEMBER_DIM)
    forecast[M2M_MEMBER_DIM] = np.arange(forecast[M2M_MEMBER_DIM].size)
    observations[M2M_MEMBER_DIM] = np.arange(observations[M2M_MEMBER_DIM].size)
    return forecast, observations

Then initialize this comparison function with climpred.comparisons.Comparison:

__my_m2median_comparison = Comparison(

Finally, compute skill based on your own 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.



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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. doi:10/gd7hfq.


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


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