ROmondi/STST · Datasets at Hugging Face

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the class of Borel probability measures on Rd such that the 2p-th moments of all univariate margins are
f
inite. The weights wij can be selected to emphasize or depreciate certain pair interactions. For example, in
a spatial context, it can be expected the dependence between pairs decays with the distance: choosing the
weights proportional to the inverse of the distance between locations can increase the signal-to-noise ratio
and improve the discriminatory power of the VS (Scheuerer and Hamill, 2015).
When the pdf f of the probabilistic forecast F is available, multivariate versions of the univariate scoring
rules based on the pdf are available. The multivariate versions of the scoring rules have the same properties
and limitations as their univariate counterpart. The logarithmic score (11) has a natural multivariate version
:
LogS(F,y) = −log(f(y)),
for y such that f(y) > 0. The logarithmic score is strictly proper relative to the class L1(Rd).
The Hyvärinen score (HS; Hyvärinen 2005) was initially proposed in its multivariate form
HS(F,y) = 2∆log(f(y))+\|∇log(f(y))\|2,
for y such that f(y) > 0, where ∆ is the Laplace operator (i.e., the sum of the second-order partial deriva
tives) and ∇ is the gradient operator (i.e., vector of the first-order partial derivatives). In the multivariate
setting, the HS can also be computed if the predicted pdf is known up to a normalizing constant. The HS
is proper relative to the subclass of P(Rd) such that the density f exists, is twice continuously differentiable
and satisfies ∥∇log(f(x))∥ → 0 as ∥x∥ → ∞.
8
The quadratic score and pseudospherical score are directly suited to the multivariate setting :
QuadS(F,y) = ∥f∥2
2 −2f(y);
PseudoS(F,y) = −f(y)α−1/∥f∥α−1
α ,
where ∥f∥α = ( Rd
f(y)αdy)1/α. The quadratic score is strictly proper relative to the class L2(Rd). The
pseudospherical score is strictly proper relative to the class Lα(Rd).
Additionally, other multivariate scoring rules have been proposed among which the marginal-copula score
(Ziel and Berk, 2019) or wavelet-based scoring rules (see, e.g., Buschow et al. 2019). These scoring rules will
be briefly mentioned in Section 4 in light of the proper scoring rule construction framework proposed in this
article. Appendix B provides formulas for the expected multivariate scoring rules presented above.
2.4 Spatial verification tools
Spatial forecasts are a very important group of multivariate forecasts as they are involved in various appli
cations (e.g., weather or renewable energy forecasting). Spatial fields are often characterized by high dimen
sionality and potentially strong correlations between neighboring locations. These characteristics make the
verification of spatial forecasts very demanding in terms of discriminating misspecified dependence struc
tures, for example. In the case of spatial forecasts, it is known that traditional verification methods (e.g.,
gridpoint-by-gridpoint verification) may result in a double penalty. The double-penalty effect was pinned in
Ebert (2008) and refers to the fact that if a forecast presents a spatial (or temporal) shift with respect to
observations, the error made would be penalized twice: once where the event was observed and again where
the forecast predicted it. In particular, high-resolution forecasts are more penalized than less realistic blurry
forecasts. The double-penalty effect may also affect spatio-temporal forecasts in general.
In parallel with the development of scoring rules, various application-focused spatial verification meth
ods have been developed to evaluate weather forecasts. The efforts toward improving spatial verification
methods have been guided by two projects: the intercomparison project (ICP; Gilleland et al. 2009) and its
second phase, called Mesoscale Verification Intercomparison over Complex Terrain (MesoVICT; Dorninger
et al. 2018). These projects resulted in the comparison of spatial verification methods with a particular focus
on understanding their limitations and clarifying their interpretability. Only a few links exist between the
approaches studied in these projects (and the work they induced) and the proper scoring rules framework.
In particular, Casati et al. (2022) noted "a lack of representation of novel spatial verification methods for
ensemble prediction systems". In general, there is a clear lack of methods focusing on the spatial verification
of probabilistic forecasts. Moreover, to help bridging the gap between the two communities, we would like
to recall the approach of spatial verification tools in the light of the scoring rule framework introduced above.
One of the goals of the ICP was to provide insights on how to develop methods robust to the double
penalty effect. In particular, Gilleland et al. (2009) proposed a classification of spatial verification tools
updated later in Dorninger et al. (2018) resulting in a five-category classification. The classes differ in the
computing principle they rely on. Not all spatial verification tools mentioned in these studies can be applied
to probabilistic forecasts, some of them can solely be applied to deterministic forecasts. In the following
description of the classes, we try to focus on methods suited to probabilistic forecasts or at least the special
case of ensemble forecasts.
Neighborhood-based methods consist of applying a smoothing filter to the forecast and observation fields
to prevent the double-penalty effect. The smoothing filter can take various forms (e.g., a minimum, a
maximum, a mean, or a Gaussian filter) and be applied over a given neighborhood. For example, Stein
and Stoop (2022) proposed a neighborhood-based CRPS for ensemble forecasts gathering forecasts and
observations made within the neighborhood of the location considered. The use of a neighborhood prevents
the double-penalty effect from taking place at scales smaller than that of the neighborhood. In this general
definition, neighborhood-based methods can lead to proper scoring rules, in particular, see the notion of
patches in Section 4.
Scale-separation techniques denote methods for which the verification is obtained after comparing forecast
and observation fields across different scales. The scale-separation process can be seen as several single
bandpass spatial filters (e.g., projection onto a base of wavelets as wavelet-based scoring rules; Buschow et al.
9
2019). However, in order to obtain proper scoring rules, the comparison of the scale-specific characteristics
needs to be performed using a proper scoring rule. Section 4 provides a discussion on wavelet-based scoring
rules and their propriety.
Object-based methods rely on the identification of objects of interest and the comparison of the objects
obtained in the forecast and observation fields. Object identification is application-dependent and can
take the form of objects that forecasters are familiar with (e.g., storm cells for precipitation forecasts). A
well-known verification tool within this class is the structure-amplitude-location (SAL; Wernli et al. 2008)
method which has been generalized to ensemble forecasts in Radanovics et al. (2018). The three components
of the ensemble SAL do not lead to proper scoring rules. They rely on the mean of the forecast within
scoring functions inconsistent with the mean. Thus, the ideal forecast does not minimize the expected value.
Nonetheless, the three components of the SAL method could be adapted to use proper scoring rules sensitive
to the misspecification of the same features.
Field-deformation techniques consist of deforming the forecasts field into the observation field (the simi
larity between the fields can be ensured by a metric of interest). The field of distortion associated with the
morphing of the forecast field into the observation field becomes a measure of the predictive performance of
the forecast (see, e.g., Han and Szunyogh 2018).
Distance measures between binary images, such as exceedance of a threshold of interest, of the forecast
and observation fields. These methods are inspired by development in image processing (e.g., Baddeley’s
delta measure Gilleland 2011).
These five categories are partially overlapping as it can be argued that some methods belong to multiple
categories (e.g., some distance measures techniques can be seen as a mix of field-deformation and object
based). They define different principles that can be used to build verification tools that are not subject
to the double-penalty effect. The reader may refer to Dorninger et al. (2018) and references therein for
details on the classification and the spatial verification methods not used thereafter. The frontier between
the aforementioned spatial verification methods and the proper scoring rules framework is porous with,
for example, wavelet-based scoring rules belonging to both. It appears that numerous spatial verification
methods seek interpretability and we believe that this is not incompatible with the use of proper scoring

the class of Borel probability measures on Rd such that the 2p-th moments of all univariate margins are

f

inite. The weights wij can be selected to emphasize or depreciate certain pair interactions. For example, in

a spatial context, it can be expected the dependence between pairs decays with the distance: choosing the

weights proportional to the inverse of the distance between locations can increase the signal-to-noise ratio

and improve the discriminatory power of the VS (Scheuerer and Hamill, 2015).

When the pdf f of the probabilistic forecast F is available, multivariate versions of the univariate scoring

rules based on the pdf are available. The multivariate versions of the scoring rules have the same properties

and limitations as their univariate counterpart. The logarithmic score (11) has a natural multivariate version

:

LogS(F,y) = −log(f(y)),

for y such that f(y) > 0. The logarithmic score is strictly proper relative to the class L1(Rd).

The Hyvärinen score (HS; Hyvärinen 2005) was initially proposed in its multivariate form

HS(F,y) = 2∆log(f(y))+|∇log(f(y))|2,

for y such that f(y) > 0, where ∆ is the Laplace operator (i.e., the sum of the second-order partial deriva

tives) and ∇ is the gradient operator (i.e., vector of the first-order partial derivatives). In the multivariate

setting, the HS can also be computed if the predicted pdf is known up to a normalizing constant. The HS

is proper relative to the subclass of P(Rd) such that the density f exists, is twice continuously differentiable

and satisfies ∥∇log(f(x))∥ → 0 as ∥x∥ → ∞.

8

The quadratic score and pseudospherical score are directly suited to the multivariate setting :

QuadS(F,y) = ∥f∥2

2 −2f(y);

PseudoS(F,y) = −f(y)α−1/∥f∥α−1

α ,

where ∥f∥α = ( Rd

f(y)αdy)1/α. The quadratic score is strictly proper relative to the class L2(Rd). The

pseudospherical score is strictly proper relative to the class Lα(Rd).

Additionally, other multivariate scoring rules have been proposed among which the marginal-copula score

(Ziel and Berk, 2019) or wavelet-based scoring rules (see, e.g., Buschow et al. 2019). These scoring rules will

be briefly mentioned in Section 4 in light of the proper scoring rule construction framework proposed in this

article. Appendix B provides formulas for the expected multivariate scoring rules presented above.

2.4 Spatial verification tools

Spatial forecasts are a very important group of multivariate forecasts as they are involved in various appli

cations (e.g., weather or renewable energy forecasting). Spatial fields are often characterized by high dimen

sionality and potentially strong correlations between neighboring locations. These characteristics make the

verification of spatial forecasts very demanding in terms of discriminating misspecified dependence struc

tures, for example. In the case of spatial forecasts, it is known that traditional verification methods (e.g.,

gridpoint-by-gridpoint verification) may result in a double penalty. The double-penalty effect was pinned in

Ebert (2008) and refers to the fact that if a forecast presents a spatial (or temporal) shift with respect to

observations, the error made would be penalized twice: once where the event was observed and again where

the forecast predicted it. In particular, high-resolution forecasts are more penalized than less realistic blurry

forecasts. The double-penalty effect may also affect spatio-temporal forecasts in general.

In parallel with the development of scoring rules, various application-focused spatial verification meth

ods have been developed to evaluate weather forecasts. The efforts toward improving spatial verification

methods have been guided by two projects: the intercomparison project (ICP; Gilleland et al. 2009) and its

second phase, called Mesoscale Verification Intercomparison over Complex Terrain (MesoVICT; Dorninger

et al. 2018). These projects resulted in the comparison of spatial verification methods with a particular focus

on understanding their limitations and clarifying their interpretability. Only a few links exist between the

approaches studied in these projects (and the work they induced) and the proper scoring rules framework.

In particular, Casati et al. (2022) noted "a lack of representation of novel spatial verification methods for

ensemble prediction systems". In general, there is a clear lack of methods focusing on the spatial verification

of probabilistic forecasts. Moreover, to help bridging the gap between the two communities, we would like

to recall the approach of spatial verification tools in the light of the scoring rule framework introduced above.

One of the goals of the ICP was to provide insights on how to develop methods robust to the double

penalty effect. In particular, Gilleland et al. (2009) proposed a classification of spatial verification tools

updated later in Dorninger et al. (2018) resulting in a five-category classification. The classes differ in the

computing principle they rely on. Not all spatial verification tools mentioned in these studies can be applied

to probabilistic forecasts, some of them can solely be applied to deterministic forecasts. In the following

description of the classes, we try to focus on methods suited to probabilistic forecasts or at least the special

case of ensemble forecasts.

Neighborhood-based methods consist of applying a smoothing filter to the forecast and observation fields

to prevent the double-penalty effect. The smoothing filter can take various forms (e.g., a minimum, a

maximum, a mean, or a Gaussian filter) and be applied over a given neighborhood. For example, Stein

and Stoop (2022) proposed a neighborhood-based CRPS for ensemble forecasts gathering forecasts and

observations made within the neighborhood of the location considered. The use of a neighborhood prevents

the double-penalty effect from taking place at scales smaller than that of the neighborhood. In this general

definition, neighborhood-based methods can lead to proper scoring rules, in particular, see the notion of

patches in Section 4.

Scale-separation techniques denote methods for which the verification is obtained after comparing forecast

and observation fields across different scales. The scale-separation process can be seen as several single

bandpass spatial filters (e.g., projection onto a base of wavelets as wavelet-based scoring rules; Buschow et al.

9

2019). However, in order to obtain proper scoring rules, the comparison of the scale-specific characteristics

needs to be performed using a proper scoring rule. Section 4 provides a discussion on wavelet-based scoring

rules and their propriety.

Object-based methods rely on the identification of objects of interest and the comparison of the objects

obtained in the forecast and observation fields. Object identification is application-dependent and can

take the form of objects that forecasters are familiar with (e.g., storm cells for precipitation forecasts). A

well-known verification tool within this class is the structure-amplitude-location (SAL; Wernli et al. 2008)

method which has been generalized to ensemble forecasts in Radanovics et al. (2018). The three components

of the ensemble SAL do not lead to proper scoring rules. They rely on the mean of the forecast within

scoring functions inconsistent with the mean. Thus, the ideal forecast does not minimize the expected value.

Nonetheless, the three components of the SAL method could be adapted to use proper scoring rules sensitive

to the misspecification of the same features.

Field-deformation techniques consist of deforming the forecasts field into the observation field (the simi

larity between the fields can be ensured by a metric of interest). The field of distortion associated with the

morphing of the forecast field into the observation field becomes a measure of the predictive performance of

the forecast (see, e.g., Han and Szunyogh 2018).

Distance measures between binary images, such as exceedance of a threshold of interest, of the forecast

and observation fields. These methods are inspired by development in image processing (e.g., Baddeley’s

delta measure Gilleland 2011).

These five categories are partially overlapping as it can be argued that some methods belong to multiple

categories (e.g., some distance measures techniques can be seen as a mix of field-deformation and object

based). They define different principles that can be used to build verification tools that are not subject

to the double-penalty effect. The reader may refer to Dorninger et al. (2018) and references therein for

details on the classification and the spatial verification methods not used thereafter. The frontier between

the aforementioned spatial verification methods and the proper scoring rules framework is porous with,

for example, wavelet-based scoring rules belonging to both. It appears that numerous spatial verification

methods seek interpretability and we believe that this is not incompatible with the use of proper scoring