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rules. We propose the following framework to facilitate the construction of interpretable proper scoring rules. |
3 Aframework for interpretable proper scoring rules |
Wedefine a framework to design proper scoring rules for multivariate forecasts. Its definition is motivated by |
remarks on the multivariate forecasts literature and operational use. There seems to be a growing consensus |
around the fact that no single verification method has it all (see, e.g., Bjerregård et al. 2021). Most of the |
studies comparing forecast verification methods highlight that verification procedures should not be reduced |
to the use of a single method and that each procedure needs to be well suited to the context (see, e.g., |
Scheuerer and Hamill 2015; Thorarinsdottir and Schuhen 2018). Moreover, from a more theoretical point |
of view, (strict) propriety does not ensure discrimination ability and different (strictly) proper scoring rules |
can lead to different rankings of sub-efficient forecasts. |
Standard verification procedures gradually increase the complexity of the quantities verified. Procedures |
often start by verifying simple quantities such as quantiles, mean, or binary events (e.g., prediction of |
dry/wet events for precipitation). If multiple forecasts have a satisfying performance for these quantities, |
marginal distributions of the multivariate forecast can be verified using univariate scoring rules. Finally, |
multivariate-related quantities, such as the dependence structure, can be verified through multivariate scoring |
rules. Forecasters rely on multiple verification methods to evaluate a forecast and ideally, the verification |
method should be interpretable by targeting specific aspects of the distribution or thanks to the forecaster’s |
experience. This type of verification procedure allows the forecaster to understand what characterizes the |
predictive performance of a forecast instead of directly looking at a strictly proper scoring rule giving an |
encapsulated summary of the predictive performance. |
Various multivariate forecast calibration methods rely on the calibration of univariate quantities obtained |
by dimension reduction techniques. As the general principle of multivariate calibration leans on studying the |
calibration of quantities obtained by pre-rank functions, Allen et al. (2024) argue that calibration procedures |
10 |
should not rely on a single pre-rank function and should instead use multiple simple pre-rank functions and |
leverage the interpretability of the PIT/rank histograms associated. A similar principle can be applied to |
increase the interpretability of verification methods based on scoring rules. |
As general multivariate strictly proper scoring rules fail to discriminate forecasts with respect to arbitrary |
misspecifications and they may lead to different ranking of sub-efficient forecasts, multivariate verification |
could benefit from using multiple proper scoring rules targeting specific aspects of the forecasts. Thereby, |
forecasters know which aspect of the observations are well-predicted by the forecast and can update their |
forecast or select the best forecast among others in the light of this better understanding of the forecast. |
To facilitate the construction of interpretable proper scoring rules, we define a framework based on two |
principles: transformation and aggregation. |
The transformation principle consists of transforming both forecast and observation before applying a |
scoring rule. Heinrich-Mertsching et al. (2021) introduced this general principle in the context of point |
processes. In particular, they present scoring rules based on summary statistics targeting the clustering |
behavior or the intensity of the processes. In a more general context, the use of transformations was |
disseminated in the literature for several years (see Section 4). Proposition 1 shows how transformations can |
be used to construct proper scoring rules. |
Proposition 1. Let F ⊂ P(Rd) be a class of Borel probability measure on Rd and let F ∈ F be a forecast |
and y ∈ Rd an observation. Let T : Rd → Rk be a transformation and let S be a scoring rule on Rk that is |
proper relative to T(F) = {L(T(X)),X ∼ F ∈ F}. Then, the scoring rule |
ST(F,y) = S(T(F),T(y)) |
is proper relative to F. If S is strictly proper relative to T(F) and T is injective, then the resulting scoring |
rule ST is strictly proper relative to F. |
To gain interpretability, it is natural to have dimension-reducing transformations (i.e., k < d), which gen |
erally leads to T not being injective and ST not being strictly proper. Nonetheless, as expressed previously, |
interpretability is important and it can mostly be leveraged if the transformation simplifies the multivariate |
quantities. Particularly, it is generally preferred to choose k = 1 to make the quantity easier to interpret and |
focus on specific information contained in the forecast or the observation. Straightforward transformations |
can be projections on a k-dimensional margin or a summary statistic relevant to the forecast type such as |
the total over a domain in the case of precipitations. Simple transformations may be preferred for their |
interpretability and their potential lack of discriminatory power can be made up for via the use of multiple |
simpler transformations. Numerous examples of transformations are presented, discussed, and linked to the |
literature in Section 4. The proof of Proposition 1 is provided in Appendix C.1. |
The second principle is the aggregation of scoring rules. Aggregation can be used on scoring rules in order |
to combine them and obtain a single scoring rule summarizing the evaluation. It can be used to operate on |
scoring rules acting on different spaces, times or locations. Note that Dawid and Musio (2014) introduced |
the notion of composite score which is related to the aggregation principle but is closer to the combined |
application of both principles. Proposition 2 presents a general aggregation principle to build proper scoring |
rules. This principle has been known since proper scoring rules have been introduced. |
Proposition 2. Let S = {Si}1≤i≤m be a set of proper scoring rules relative to F ⊂ P(Rd). Let w = |
{wi}1≤i≤m be nonnegative weights. Then, the scoring rule |
m |
SS,w(F,y) = |
i=1 |
wiSi(F,y) |
is proper relative to F. If at least one scoring rule Si is strictly proper relative to F and wi > 0, the aggregated |
scoring rule SS,w is strictly proper relative to F. |
It is worth noting that Proposition 2 does not specify any strict condition for the scoring rules used. For |
example, the scoring rules aggregated do not need to be the same or do not need to be expressed in the |
11 |
same units. Aggregated scoring rules can be used to summarize the evaluation of univariate probabilistic |
forecasts (e.g., aggregation of CRPS at different locations) or to summarize complementary scoring rules |
(e.g., aggregation of Brier score and a threshold-weighted CRPS). Unless stated otherwise, for simplicity, we |
will restrict ourselves to cases where the aggregated scoring rules are of the same type. Bolin and Wallin |
(2023) showed that the aggregation of scoring rules can lead to unintuitive behaviors. For the aggregation |
of univariate scoring rules, they showed that scoring rules do not necessarily have the same dependence on |
the scale of the forecasted phenomenon: this leads to scoring rules putting more (or less) emphasis on the |
forecasts with larger scales. They define and propose local scale-invariant scoring rules to make scale-agnostic |
scoring rules. When performing aggregation, it is important to be aware of potential preferences or biases |
of the scoring rules. |
We only consider aggregation of proper scoring rules through a weighted sum. To conserve (strict) pro |
priety of scoring rules, aggregations can take, more generally, the form of (strictly) isotonic transformations, |
such as a multiplicative structure when positive scoring rules are considered (Ziel and Berk, 2019). |
The two principles of Proposition 1 and Proposition 2 can be used simultaneously to create proper scoring |
rules based on both transformations and aggregation as presented in Corollary 1. |
Corollary 1. Let T = {Ti}1≤i≤m be a set of transformations from Rd to Rk. Let ST = {STi |
}1≤i≤m be a |
set of proper scoring rules where S is proper relative to Ti(F), for all 1 ≤ i ≤ m. Let w = {wi}1≤i≤m be |
nonnegative weights. Then, the scoring rule |
m |
SST,w(F,y) = |
is proper relative to F. |
i=1 |
wiSTi |
(F,y) |
Strict propriety relative to F of the resulting scoring rule is obtained as soon as there exists 1 ≤ i ≤ m |
such that S is strictly proper relative to Ti(F), Ti is injective and wi > 0. The result of Corollary 1 can be |
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