ROmondi/STST · Datasets at Hugging Face

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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

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