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Multivariate Forecasting Evaluation: On Sensitive and Strictly Proper Scoring Rules
\maketitle
\lhead{\nouppercase{\leftmark}}
\begin{abstract}
In recent years, probabilistic forecasting is an emerging topic, which is why there is a growing need of suitable methods for the evaluation of multivariate predictions. We analyze the sensitivity of the most common scoring rules, especially regarding quality of the forecasted dependency structures. Additionally, we propose scoring rules based on the copula, which uniquely describes the dependency structure for every probability distribution with continuous marginal distributions. Efficient estimation of the considered scoring rules and evaluation methods such as the Diebold-Mariano test are discussed. In detailed simulation studies, we compare the performance of the renowned scoring rules and the ones we propose. Besides extended synthetic studies based on recently published results we also consider a real data example. We find that the energy score, which is probably the most widely used multivariate scoring rule, performs comparably well in detecting forecast errors, also regarding dependencies. This contradicts other studies. The results also show that a proposed copula score provides very strong distinction between models with correct and incorrect dependency structure. We close with a comprehensive discussion on the proposed methodology.
\end{abstract}
{\bf Keywords:}
Scoring rules,
Multivariate forecasting,
Prediction evaluation,
Ensemble forecasting,
Energy score,
Copula score,
Strictly proper,
Diebold-Mariano test
\section{Introduction} \label{Introduction}
Forecasting evaluation is still highly discussed in different forecasting communities, like in meteorology, energy, economics, business or social and natural sciences. Also, in recent years, the term \emph{probabilistic forecasting} (essentially density forecasting or quantile forecasting) became popular across all areas of application (\cite{gneiting2014probabilistic}, \cite{reich2015probabilistic}, \cite{hong2016probabilistic}, \cite{lerch2017forecaster}). It is clear that the evaluation of forecasts is crucial for assessing the quality of different forecasts.
However, for 1-step ahead forecasts the discussion is much more comprehensive than for multiple step ahead predictions, say $H$-step ahead forecasts.
The evaluation of forecasted marginal distributions of $H$-step ahead forecasts have been treated in literature using so called \emph{univariate scoring rules}. Essentially it breaks down to the 1-step ahead forecasting case for each marginal distribution. In contrast, the (fully) multivariate case where the complete $H$-dimensional distribution is usually not covered, in the forecasting itself such as in the evaluation. Obviously, this causes problems because the multivariate $H$-step ahead distribution is not fully explained by reporting $H$-step ahead marginal distributions or quantiles.
A main reason for these circumstances is the lack of suitable evaluation measures that can judge the quality of forecasts adequately.
We require approaches to assess not only the performance of a prediction with respect to the marginals but also to the dependency structures, we need suitable \emph{multivariate scoring rules}.
The most well known multivariate scoring rule, the energy score, was introduced by \cite{gneiting2007strictly}. For instance, in \cite{sari2016statistical, spath2015time, gneiting2008assessing, pinson2012adaptive, berrocal2008probabilistic, pinson2012evaluating, moller2013multivariate, baran2015joint, yang2015multi, moller2015spatially} the energy score is used for evaluation. This is mainly in the area of meteorologic forecasting, primarily wind speed and wind power forecasting. In the wind power forecasting review of \cite{zhang2014review} it is the only mentioned multivariate forecasting evaluation method.
The energy score is a strictly proper scoring, meaning that only the true model optimizes the corresponding score.
However, in many empirical applications the energy score is still avoided. In the study of \cite{pinson2013discrimination}, the discrimination ability of the energy score is checked in a simulation study while considering the bivariate normal distribution. They conclude that the energy score can not separate differences in the dependency structure well, which made the researchers and practitioners skeptical in using it. In this paper, we will discuss this study in detail and draw somewhat different conclusions.
In contrast, another multivariate scoring rule were introduced to overcome the reported problem in energy score.
\cite{scheuerer2015variogram} developed the variogram score, that is sensitive to changes in the correlation. However, it is only a proper scoring rule, but not a strictly proper one. Thus, it can not identify the true underlying model.
Another reported plausible candidate is the multivariate log-score. However, it requires that we have multivariate density forecasts which is not the case in many applications.
As discussed in \cite{lerch2017forecaster}, this density may be approximated. But these approximation methods suffer efficiency in higher dimensions. The results depend crucially on the chosen approximation method.
Additionally, we have the Dawid-Sebastiani score (see e.g. \cite{gneiting2007strictly}), which evaluates the mean and covariance matrix of the distributions
and corresponds to the log-score in the multivariate Gaussian settings. Moreover, a characterization only by the first two moments is not sufficient for many applications.
With all these considerations in mind, we introduce a new scoring rule that is sensitive to dependency changes in the distribution using copula theory. By Sklar's theorem, we are able to extract the dependency structure of our forecast (in the form of the copula) and construct a more sensitive measure for evaluating the dependencies. Afterwards, we combine the score of the copula with a score of the marginal distributions to obtain a proper and even strictly proper scoring rules. The approach is somehow flexible with respect to the copula and marginal scores that are used.
We start with the introduction of the aforementioned famous scoring rules in section~\ref{sec:scores}. Afterwards, we introduce our concept of marginal-copula scores. In section~\ref{sec:reporting} we give a short overview on how to report multivariate forecasts. Efficient estimation of scores is crucial in practice, we describe some approaches for different scoring rules in section~\ref{sec:estim}. Since the sensitivity of a score value alone does not indicate very much about the goodness-of-fit or the discrimination ability, we introduce the concept of the Diebold-Mariano test in section~\ref{sec:eval}. In section~\ref{sec:studies}, we apply the renowned scoring rules and the marginal-copula scores in three different synthetic case studies, compare and discuss the results. A real data example using airline passenger data is discussed in section~\ref{sec:realdata}. We conclude the results in section~\ref{sec:conclusion}.
\section{Scoring rules} \label{sec:scores}
In this section we are going to introduce a concept of scoring rules to adequately assess the goodness-of-fit of multivariate probabilistic forecasts. In order to do that, we are going to introduce some well known scoring rules in a first step. Additionally, we present a new score which combines univariate scoring of the forecasted marginal distributions as well as multivariate scoring for the forecasted copula in a way that desirable preferences like (strict) propriety are preserved. We start with some notations and basics on scoring rules.
\subsection{Notations and basics}
Let $\boldsymbol Y = (Y_1,\ldots, Y_H)'$ be an $H$-dimensional random variable with multivariate (cumulative) distribution function $\boldsymbol F_{\boldsymbol Y}$ which we want to forecast. The most standard example would be
that $\boldsymbol Y$ is a $H$-step ahead forecast of a univariate time series, hence $H$ is the forecasting horizon. From $\boldsymbol Y$ we observe the realized respectively materialized vector $\boldsymbol y = (y_1,\ldots, y_H)'$.
Let $\boldsymbol X = (X_1,\ldots, X_H)$ be the forecast random vector for $\boldsymbol Y$. In practice we have to think about
the reporting of these forecasts (see Section \ref{sec:reporting}).
Here, we simply assume that we have the full multivariate (cumulative) distribution function $\boldsymbol F_{\boldsymbol X}$ as reported forecast available.
Given a forecast $\boldsymbol F_{\boldsymbol X}$ and a random variable $\boldsymbol Y$ we can define a scoring rule $\text{S}(\boldsymbol F_{\boldsymbol X}, \boldsymbol Y)$ that maps into ${\mathbb R}$ (or $\ov{{\mathbb R}}$).
They are designed in such a way that a good forecast yields small values (positively orientated), that is if $\boldsymbol F_{\boldsymbol Y}$ is close (or closely related) to $\boldsymbol F_{\boldsymbol X}$.
Such a scoring rule is \emph{proper} if $\text{S}(\boldsymbol F_{\boldsymbol Y}, \boldsymbol Y) \leq \text{S}(\boldsymbol F_{\boldsymbol X}, \boldsymbol Y) $ holds for arbitrary random vectors $\boldsymbol X$, and
\emph{strictly proper} if equality holds only if $\boldsymbol F_{\boldsymbol X} = \boldsymbol F_{\boldsymbol Y}$. Hence the true distribution can be well separated.
In forecasting applications the random variable $\boldsymbol Y$ is usually observed (or realized / materialized), therefore we often write directly $\text{S}(\boldsymbol F_{\boldsymbol X}, \boldsymbol y)$.
Note in some literature, see e.g. \cite{gneiting2007strictly}, scoring rules are defined with inverse orientation. Obviously, all the theory holds as well. Here, we use the notation that is popular in applications,
that means the smaller the score the better the forecast.
As pointed out in the introduction, at the moment there is basically only a limited amount of scoring rules available. Here we recall the the major important ones, as
we will work with them in the latter part of this paper. We start with the most popular univariate scoring rule.
\subsection{Continuous ranked probability score}
The continuous ranked probability score (CRPS) for a random variable $X$ with distribution function $F_{X}$ and given forecast $y$ is defined by
\begin{align}
\text{CRPS}(F_{X}, y) = \int_{-\infty}^{\infty} (F_{X}(z) - \mathbbm{1}\{y < z\})^2 \,d z = {\mathbb E} | X - y | - \frac{1}{2} {\mathbb E}| X -\widetilde{X}|
\label{eq_crps}
\end{align}
where $\widetilde{X}$ is an iid copy of $X$.
It is a special case of the energy score, which we will introduce in the next section, for one dimension and $\beta=1$.
It is a strictly proper scoring rule with respect to the distribution of $Y$.
Another univariate score is for example the quantile loss or pinball score on a grid of quantiles ($0\leq\alpha_1<\alpha_2<\ldots <\alpha_N\leq 1$). Note that for an equidistant grid for $|\alpha_i - \alpha_{i+1}| \to 0$ the corresponding score converges to the CRPS, see e.g. \cite{nowotarski2017recent}.
If $Y$ respectively $X$ has a density, the logarithmic score is also a candidate for univariate scoring. However, in practical applications
we have the problem that for sophisticated models we do not have an explicit formula for the forecasted density available.
Still, it may be approximated \cite{lerch2017forecaster}.
For more details on univariate scoring rules we recommend \cite{gneiting2007strictly}.
In the subsequent sections, we will introduce some multivariate scores.
\subsection{Energy score}
For the $H$-dimensional random variable $\boldsymbol X$ and
the observation vector $\boldsymbol y$ of the target distribution $\boldsymbol Y$ the (Euclidean) \textbf{energy score} (see \cite{gneiting2007strictly}) is given by
\begin{equation}
\text{ES}_\beta ( \boldsymbol F_{\boldsymbol X}, \boldsymbol y) =
{\mathbb E}\left( \left\| \boldsymbol X - \boldsymbol y \right\|_2^\beta \right) -
\frac{1}{2} {\mathbb E}\left( \| \boldsymbol X- \widetilde{\boldsymbol X} \|_2^\beta \right)
\label{eq_energy_score}
\end{equation}
where $\widetilde{\boldsymbol X}$ is an i.i.d. copy of $\boldsymbol X$, so it is drawn independently from the same distribution
$\boldsymbol F_{\boldsymbol X}$ as $\boldsymbol X$. Moreover, $\beta \in (0,2)$ and $\|\cdot\|_2$ is the the Euclidean norm.
As described in \cite{gneiting2007strictly}, $\text{ES}_\beta ( \boldsymbol F_{\boldsymbol X}, \boldsymbol y)$ is a strictly proper evaluation score for all $\beta$,
but $\beta=1$ seems to be the standard choice in application. \cite{szekely2013energy} points out that for heavy tailed data
the choice of small $\beta$ values should be favored to guarantee that the corresponding moments exists.
\subsection{Variogram score}
The second multivariate score of interest is the \textbf{variogram score} (see \cite{scheuerer2015variogram}) which is motivated by the variogram that is a popular tool in geostatistics, see \cite{cressie1985fitting}.
The variogram score is defined by
\begin{equation}
\text{VS}_{\boldsymbol W,p}( \boldsymbol F_{\boldsymbol X} , \boldsymbol y )
= \sum_{i=1}^H \sum_{j=1}^H w_{i,j} ( |y_{i}-y_{j}|^p - {\mathbb E}| X_{i} - X_{j}|^p )^2
\label{eq_variogram_score}
\end{equation}
with $p>0$ and weight matrix $\boldsymbol W = (w_{i,j})_{i,j}$.
In applications standard cases for $p$ are $p=0.5$ and $p=1$, the weight matrix is usually chosen with $w_{i,j}=1$.
It is designed to capture differences in the correlation structure.
\subsection{Dawid-Sebastiani score}
Another alternative is motivated by the multivariate \textbf{logarithmic score} (or log-score) which is available if $\boldsymbol X$ has a density $\boldsymbol f_{\boldsymbol X}$. It is given by
$$ \text{LogS}( \boldsymbol F_{\boldsymbol X} , \boldsymbol y) = \log(\boldsymbol f_{\boldsymbol X}( \boldsymbol y )) .$$
If $\boldsymbol Y$ has a density, this score is strictly proper as well. Moreover, it has locality properties which might be an advantage, depending on the application.
As mentioned in the introduction, in many applications a density forecast is not available.
Therefore the multivariate log-score has a limited range of applications, even though the density might be approximated, see \cite{lerch2017forecaster}.
Under the normality assumption for $\boldsymbol X$ (up to some dropped constants) the log-score yields the \textbf{Dawid-Sebastiani score}, which is given by
\begin{equation}
\text{DSS}( \boldsymbol F_{\boldsymbol X} , \boldsymbol y) =
\log( \det( \boldsymbol \Sigma_{\boldsymbol X}) ) + (\boldsymbol y - \boldsymbol \mu_{\boldsymbol X})'\boldsymbol \Sigma_{\boldsymbol X}^{-1}(\boldsymbol y - \boldsymbol \mu_{\boldsymbol X})
\label{eq_dawid_sebastiani_score}
\end{equation}
with $\boldsymbol \mu_{\boldsymbol X}$ and $\boldsymbol \Sigma_{\boldsymbol X}$ as mean and covariance matrix of $\boldsymbol X$ and $\det$ as determinant.
It is clear this score is only proper but not strictly proper as it only matches the mean and the covariance matrix of the forecast with the observations.
\subsection{Marginal-copula score}
After we introduced the most well known and adapted scoring rules, we present a new scoring approach using copulas in the subsequent part. The idea is to describe a given multivariate distribution by its marginals and the copula for the dependency structure. Hence, we have to combine a scoring rule for the marginals with one for the copula. We start with the description of the marginal score.
\subsubsection{Marginal score}
Let $\text{MS}_h$ be a scoring rule for the random variable $Y_h$, the \emph{$h$-th marginal score}.
$\text{MS}_h$ is a function depending on the forecast $X_h$ of $Y_h$ and the observation $y_h$,
so precisely $\text{MS}_h = \text{MS}_h(F_{X_h}, y_h)$ with $F_{X_h}$ as forecasted (cumulative) distribution function of $X_h$, which is the $h$-th marginal distribution of $\boldsymbol X$.
Following \cite{gneiting2007strictly}, we could use every plausible scoring rule for univariate distributions, as the aforementioned CRPS.
We construct a marginal score as follows. Let $\text{\textbf{MS}} = (\text{MS}_1, \ldots, \text{MS}_H)'$ denote the vector of marginal scores for each dimension and $\boldsymbol a = (a_1,\ldots, a_H)'$ be a coefficient vector with $a_h>0$, then we define the (joint)
\emph{marginal score} by
$$\text{MS}(\boldsymbol a) = \boldsymbol a' \text{\textbf{MS}} = \sum_{h=1}^H a_h \text{MS}_h .$$
It is obvious, that if $\text{MS}_h$ is a (strictly) proper scoring rule for $Y_h$ then the resulting marginal score $\text{MS}(\boldsymbol a)$ is (strictly) proper for $\boldsymbol Y$ for any choice of $\boldsymbol a$.
However, the most intuitive choice is clearly $\boldsymbol a = \boldsymbol 1_H$ which equally weights the marginals. This is usually a plausible choice as all the $i$-th step ahead forecasts live on the same scale.
In many applications $\boldsymbol a = \boldsymbol 1_H /H$ or $\boldsymbol a = \boldsymbol 1_H$ is used by default, e.g. in the Global Energy Forecasting Competition (GEFCom2014) (\cite{hong2016probabilistic}). Still, other weighting schemes for evaluation the marginals are plausible as well.
Any marginal score can correctly identify the true marginal distribution.
However, the major drawback of those scores is that they are not feasible for identification of the correlation structure of the distribution. In order to account for that, we combine the marginal score with another score for the dependency structure. Here the copula comes into play.
\subsubsection{Copula score}
First, we briefly recall some copula theory.
The core of all copula theory is Sklar's theorem. It states that any random vector $\boldsymbol Y = (Y_1,\ldots,Y_H)'$ can be represented
by its marginal distributions $F_{Y_1}, \ldots, F_{Y_H}$ and the copula function $\boldsymbol C_{\boldsymbol Y}$ defined on a $H$-dimensional unit cube.
Vice versa, given marginal distribution and a copula function, they describe the joint distribution function.
If the marginal distributions $F_{Y_1}, \ldots, F_{Y_H}$ are continuous this representation is unique.
As $F_{Y_1}, \ldots, F_{Y_H}$ characterize the marginal distributions,
the copula $\boldsymbol C_{\boldsymbol Y}$ represents the full dependency structure of $\boldsymbol Y$. We want make use of this characteristic, as e.g. the energy score shows weaknesses in discriminating well between dependency structures. Further, remember that the copula $\boldsymbol C_{\boldsymbol Y}$ is a (cumulative) distribution function - so we can apply all characteristics to them.
Let $\text{CS}$ denote a \emph{copula score}, i.e.~an evaluation score for the copula $\boldsymbol C_{\boldsymbol Y}$, which is a multivariate (cumulative) distribution function.
If the $\text{CS}$ is a strictly proper scoring rule it can identify the true dependency structure of $\boldsymbol Y$.
In the forecasting evaluation literature there is no specific focus on the copulas, even though their are often used to link marginal distribution forecast with a copula forecast to represent the full multivariate distribution or adjust ensemble forecasts (e.g. \cite{moller2013multivariate}, \cite{madadgar2014towards}).
However, as every copula is just a special multivariate distribution we can use the forecasting evaluation and scoring rule literature on multivariate forecasting evaluation
to construct copula scores. Essentially we have three possible options, the energy score, the variogram score and the Dawid-Sebastiani score. Note that only the former one is a strictly proper scoring rule and thus ad-hoc preferable. Still, we will discuss all options within the paper.
\subsubsection{Combining marginal and copula score}
Considering that we have the marginal score $\text{MS}$ and the copula score $\text{CS}$ we have to link both to create a proper scoring rule.
Therefore, we need a transformation $g:{\mathbb R}\times{\mathbb R} \to {\mathbb R}$ that preserves the properties of $\text{MS}$ and $\text{CS}$.
It is easy to observe that the functions $g$ should satisfy some monotonicity conditions, e.g. $g$ is strictly monotone in both components.
In more detail, we need that $g$ is strictly isotonic (also known as $2$-monotonic) on the supports. Thus, we require $g$ such that
$g(x_1,y_1) - g(x_1,y_2) - g(x_2,y_1) + g(x_2,y_2) > 0 $ holds for $x_1,x_2\in\text{supp}(\text{MS})$ and $y_1,y_2\in\text{supp}(\text{CS})$ with $x_1<x_2$ and $y_1<y_2$.
The choice $g(x,y) = x+y$ is strictly isotonic on ${\mathbb R}\times {\mathbb R}$, while $g(x,y)=xy$ works on $(0,\infty)\times(0,\infty)$.
Other options would be feasible as well, but for practical application a simple transformation $g$ is favored.
The construction of a combined score is done as follows: Let $\text{MS}(\boldsymbol a)$ be the marginal score and $\text{CS}$ be the copula score, then we propose the \emph{marginal-copula score} by
\begin{align}
\text{MCS}(\boldsymbol a) = \text{MS}(\boldsymbol a) \cdot \text{CS} = \text{CS} \sum_{h=1}^H a_h \text{MS}_h \ \ .
\label{eq_msc}
\end{align}
Hence, formally we choose $g(\text{MS}(\boldsymbol a),\text{CS}) = \text{MS}(\boldsymbol a) \cdot \text{CS} $ which is a multiplicative structure.
Note that the additive option $g(\text{MS}, \text{CS}) = \lambda \text{MS} + (1-\lambda)\text{CS}$ with $\lambda\in (0,1)$ looks appealing at the beginning as well. However, it turns out to be impractical
in application as $\text{MS}$ and $\text{CS}$ live on different scales.
Consider for example a forecast for $\boldsymbol Y$ and $c \boldsymbol Y$ with $c>0$. For most marginal scores
it follows that $\text{MS}(F_{\boldsymbol Y}, \boldsymbol Y ; \boldsymbol a) \neq \text{MS}(F_{c\boldsymbol Y}, c\boldsymbol Y ; \boldsymbol a)$. For the popular CRPS we even have $\text{MS}(F_{\boldsymbol Y}, \boldsymbol Y ; \boldsymbol a) = \frac{1}{c} \text{MS}(F_{c\boldsymbol Y}, c\boldsymbol Y ; \boldsymbol a)$.
On the other hand, for the copula it holds $\text{CS}(\boldsymbol C_{\boldsymbol Y}, \boldsymbol U_{\boldsymbol Y} ) = \text{CS}(\boldsymbol C_{c\boldsymbol Y}, \boldsymbol U_{c\boldsymbol Y} )$ with $\boldsymbol U_{\boldsymbol Y} \sim \boldsymbol C_{\boldsymbol Y}$ and
$\boldsymbol U_{c\boldsymbol Y} \sim \boldsymbol C_{c\boldsymbol Y}$. So the marginal score changes with its scale but the copula
does not change at all. Thus the additive approach does not seem to be practical for empirical application. One could solve this problem with an adequate scaling for the marginal score but since in general practice the real distribution is unknown it is difficult to find proper bounds.
Moreover, the multiplicative structure of \eqref{eq_msc} can be justified by looking at the 2-dimensional normal distribution as
in the energy score study in \cite{pinson2013discrimination}. For the bivariate normal distribution of $\boldsymbol Y = (Y_1,Y_2)'$ only the correlation $\rho$ determines the copula.
If $Y_1$ and $Y_2$ have a certain unit (e.g. $\$$, $kWh$ or $m^2/s$), then $\mu_1$, $\mu_2$, $\sigma_1$ and $\sigma_2$ have the same unit, but $\rho$
has no unit at all. Usually the marginal scores inherits the units (e.g. $\$$, $kWh$ or $m^2/s$) but the copula score can never have a unit. Therefore
the definition $c \text{MS} + (1-c)\text{CS}$ provides interpretation troubles.
Here we want to summarize everything discussed above using the corollary that follows directly from the theorem of Sklar:
\begin{theorem}
Let $\text{MS}_h$ be a strictly proper score for $Y_h$ and $\text{CS}$ be a
strictly proper score for the copula $\boldsymbol C_{\boldsymbol Y}$ of $\boldsymbol Y = (Y_1,\ldots, Y_H)'$. Then
the marginal-copula score
$$ \text{MCS}(\boldsymbol F_{\boldsymbol X}, \boldsymbol y ; \boldsymbol a)
= \text{MS}( (F_{X_1},\ldots,F_{X_H})', \boldsymbol y ; \boldsymbol a) \cdot \text{CS}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol y} ) = \text{CS}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol y} ) \sum_{h=1}^H a_h \text{MS}_h(F_{X_h}, y_h) $$
with $\boldsymbol F_{\boldsymbol X}$ being the cumulative distribution function, continuous marginals $F_{X_1},\ldots,F_{X_H}$, copula $\boldsymbol C_{\boldsymbol X}$ of forecasts $\boldsymbol X = (X_1,\ldots, X_H)'$,
observation vector $\boldsymbol y$, copula observations
$\boldsymbol u_{\boldsymbol Y} = (u_{\boldsymbol Y,1}, \ldots, u_{\boldsymbol Y,H})' = (F_{Y_1}(y_1), \ldots, F_{Y_H}(y_H))'$ and weight vector
$\boldsymbol a $ with $a_h>0$
is a strictly proper scoring rule.
\end{theorem}
Unfortunately, this holds only for continuous random variables. In case of non continuous marginals the theorem holds for normal (but not strict) propriety at least. Still, many applications are covered.
Another feature that we want to point out is that
the copula is invariant against strictly monotonic transformations.
So if we have strictly monotonic transformations $g_h$ then the copula of $(g_1(Y_1), \ldots,g_H(Y_H))'$
has the same copula as $\boldsymbol Y = (Y_1,\ldots, Y_H)'$. Thus, only the marginal score components $\text{MS}_h$ get affected by
strictly monotonic transformations; especially the scoring rule remains strictly proper after monotonic transformations of the
random variables $\boldsymbol Y$ of interest.
In the next part, we discuss in more details possible choices for the copula score $\text{CS}$.
We discuss the energy score, the variogram score and the Dawid-Sebastiani score applied to copulas.
Remember that latter score evaluates only the first two moments.
As all copulas have uniform marginals, they have all the same mean and variance structure, thus
only the correlation
structure would be evaluated.
The log-score is not suitable for application as we would require a copula density forecast which is rarely available in practice.
We are interested in evaluating the fit of the copula $\boldsymbol C_{\boldsymbol X}$ belonging to the forecast $\boldsymbol F_{\boldsymbol X}$ to the copula $\boldsymbol C_{\boldsymbol Y}$ of $\boldsymbol Y$. Usually we require for forecasting evaluation the observed value $\boldsymbol y=(y_1,\ldots, y_H)'$ of $\boldsymbol Y$.
In the copula case we apply the probability integral transformation of each component to receive
$$\boldsymbol U_{\boldsymbol Y} = (U_{\boldsymbol Y,1}, \ldots, U_{\boldsymbol Y,H})' = (F_{Y_1}(Y_1), \ldots, F_{Y_H}(Y_H))' = \boldsymbol F_{\boldsymbol Y} (\boldsymbol Y) $$
which has uniform marginals
and define the (pseudo) \emph{copula observations} as
$$\boldsymbol u_{\boldsymbol Y} = (u_{\boldsymbol Y,1}, \ldots, u_{\boldsymbol Y,H})' = (F_{Y_1}(y_1), \ldots, F_{Y_H}(y_H))' = \boldsymbol F_{\boldsymbol Y} (\boldsymbol y) .$$
Note that in application we can never observe $\boldsymbol u_{\boldsymbol Y}$ as the required marginal distributions $F_{Y_h}$ are unknown. We will further address this issue in section \ref{sec:copobs}.
\subsubsection{Copula energy score}
Now, we define the copula energy score ($\text{CES}$) as the energy score of the copula scaled by $H^{-\frac12}$
\begin{align}
\text{CES}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y} )
&= \frac{1}{\sqrt{H}} \text{ES}(\boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y}) \nonumber \\
&=
\frac{1}{\sqrt{H}}\left({\mathbb E}\left( \left\| \boldsymbol U_{\boldsymbol X} - \boldsymbol u_{\boldsymbol Y} \right\|_2 \right) -
\frac{1}{2} {\mathbb E}\left( \| \boldsymbol U_{\boldsymbol X} - \widetilde{\boldsymbol U}_{\boldsymbol X} \|_2 \right) - \text{lb}_{\text{CES}}\right)
\label{eq_copula_energy_score}
\end{align}
where $\widetilde{\boldsymbol U}_{\boldsymbol X}$ is an iid copy of $\boldsymbol U_{\boldsymbol X}$, so $\boldsymbol U_{\boldsymbol X}, \widetilde{\boldsymbol U}_{\boldsymbol X} \stackrel{\text{iid}}{\sim} \boldsymbol C_{\boldsymbol X}$.
The additional $ \text{lb}_{\text{CES}}$ term is inserted because it corresponds to the lower bound where we choose $ \text{lb}_{\text{CES}} = \frac{1}{4}- \frac{1}{2 \sqrt{6}}$ and discussed below.
Note that in \cite{gneiting2007strictly} a more general version of the energy score is introduced which involves an additional parameter $\beta$ which
is chosen here (but also in other studies like \cite{pinson2013discrimination}) to be $1$.
Furthermore we consider the energy-score with negative
orientation, thus its optimum is a minimum (for more details see \cite{gneiting2007strictly}). This is in line with many applications.
In \eqref{eq_copula_energy_score} we defined the copula energy score scaled by $H^{-\frac12}$ to adjust for the dependency in the dimension $H$. This allows a better comparison of score values for different forecast horizons, for instance.
Clearly, it would be nice to have a copula score which is scaled in such a way that it allows easy interpretation,
like the correlation which is a bounded measure and takes values
between -1 and 1 and allows for interpretation of the linear dependency structure. However, such a scaling is not feasible for the copula energy score
as its outcome will always depend on the true distribution $\boldsymbol F_{\boldsymbol Y}$, especially the lower and upper bounds will change.
Still, both the lower an upper bound grow in $H$ always with a rate of $\sqrt{H}$. Hence, we suggest to scale the outcome with $H^{-\frac12}$.
Since we use negatively orientated scoring rules,
the lower bound is the score of the optimal forecast.
We know that for a given copula $\boldsymbol C_{\boldsymbol X}$ the energy score $\text{CES}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y} )$
is minimized if the forecasted distribution $\boldsymbol X$ equals the true distribution $\boldsymbol Y$ since the energy score is strictly proper.
We studied the lower bound of the energy score of the copula. However, the authors were not able to proof a strict lower bound. Additional research is ongoing.
Instead, we consider the lower bound $\text{lb}_{\text{CES}} = \frac{1}{4}- \frac{1}{2}\frac{1}{ \sqrt{6}}$ where the $\frac{1}{4}$ corresponds to a lower bound of the first term
${\mathbb E}\left( \| \boldsymbol U_{\boldsymbol X} - \boldsymbol u_{\boldsymbol Y} \|_2 \right)$
in \ref{eq_copula_energy_score}, and the $\frac{1}{ \sqrt{6}}$ to an upper bound of second term ${\mathbb E}\left( \| \boldsymbol U_{\boldsymbol X} - \widetilde{\boldsymbol U}_{\boldsymbol X} \|_2 \right)$.
In the Appendix we show upper and lower bounds for both terms in Lemma \ref{lemma_1_appendix} and Lemma \ref{lemma_2_appendix}.
\subsubsection{Copula variogram score}
As mentioned, we can apply the variogram score as a copula score $\boldsymbol C_X$ as well.
We define the variogram copula score as a scaled variogram score of the copula:
\begin{align}
\text{CVS}_{p}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y}; \boldsymbol W )
&= \frac{1}{\boldsymbol 1' \boldsymbol W \boldsymbol 1} \text{CS}_{p}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y}; \boldsymbol W ) \\
&= \sum_{i=1}^H \sum_{j=1}^H w_{i,j} ( |u_{\boldsymbol Y,i}-u_{\boldsymbol Y,j}|^p - {\mathbb E}| U_{\boldsymbol X,i} - U_{\boldsymbol X,j}|^p )^2
\end{align}
with $\boldsymbol U_{\boldsymbol X} = (U_{\boldsymbol X,1},\ldots,U_{\boldsymbol X,H})' \sim \boldsymbol C_{\boldsymbol X}$,
$p>0$ and weight matrix $\boldsymbol W = (w_{i,j})_{i,j}$.
The standard cases are $p=0.5$, $1$ and $2$. For the weight weight matrix we assume $w_{i,j}=1$ which gives
a scaling coefficient of $\boldsymbol 1' \boldsymbol W \boldsymbol 1= H^2$.
Similarly to the energy score the scaling is motivated by the fact to adjust the dependency on the dimension of the resulting score.
As for the energy score case we can derive an upper scaling bound
\begin{align}
\text{VS}_{\boldsymbol W, p}( \boldsymbol C_{\boldsymbol X}, \boldsymbol u_{\boldsymbol Y} )
&=
\sum_{i=1}^H \sum_{j=1}^H w_{i,j} ( |u_{\boldsymbol Y,i}-u_{\boldsymbol Y,j}|^p - {\mathbb E}| U_{\boldsymbol X,i} - U_{\boldsymbol X,j}|^p )^2 \\
&\leq
\sum_{i=1}^H \sum_{j=1}^H w_{i,j} ( 1^p - 0 )^2 = \boldsymbol 1' \boldsymbol W \boldsymbol 1,
\end{align}
which justifies the scaling constant.
Obviously, the lower bound of $\text{CVS}_{p}$ is zero, as for is holds
$ \text{VS}_{\boldsymbol W,p}( \boldsymbol M_{H}, \boldsymbol U_{\boldsymbol Y}) =0$ if $\boldsymbol U_{\boldsymbol Y} \sim \boldsymbol M_H $.
As the variogram score is only a proper scoring rule, the resulting MS-$\text{CVS}_{p}$ score can not be strictly proper.
However, from practical perspective the use of MS-$\text{CVS}_{p}$ can be more appealing as the disadvantages occur mainly
in the marginal distributions. There is no very simple example where a MS-$\text{CVS}_{p}$ score is not strictly proper if MS is strictly proper.
\subsubsection{Copula Dawid-Sebastiani score}
Finally, we can also consider the Dawid-Sebastiani score (DSS). As mentioned above, the DSS can only evalutate the correlation structure in the forecast,
so it is only appealing if we are interested in evaluating the linearly dependency structure in the copula of the forecasts, but not recommended in general.
\begin{align}
\text{CDSS}( \boldsymbol C_{\boldsymbol X} , \boldsymbol u_{\boldsymbol y})
&=
\text{DSS}( \boldsymbol C_{\boldsymbol X} , \boldsymbol u_{\boldsymbol y}) \nonumber \\
&=
\log( \det( \boldsymbol \Sigma_{\boldsymbol U_{\boldsymbol X}} ) ) + (\boldsymbol u_{\boldsymbol y} - \boldsymbol \mu_{\boldsymbol U_{\boldsymbol X}})'\boldsymbol \Sigma_{\boldsymbol U_{\boldsymbol X}}^{-1}(\boldsymbol u_{\boldsymbol y} - \boldsymbol \mu_{\boldsymbol U_{\boldsymbol X}})
\end{align}
with $\boldsymbol \mu_{\boldsymbol U_{\boldsymbol X}}$ and $\boldsymbol \Sigma_{\boldsymbol U_{\boldsymbol X}}$ as mean and covariance matrix of $\boldsymbol U_{\boldsymbol X}\sim \boldsymbol C_{\boldsymbol X}$.
From properties of the uniform distribution it follows that
$\boldsymbol \mu_{\boldsymbol U_{\boldsymbol X}} = \frac{1}{2}\boldsymbol 1$ and $\boldsymbol \Sigma_{\boldsymbol U_{\boldsymbol X}} = \boldsymbol S \boldsymbol R_{\boldsymbol U_{\boldsymbol X}} \boldsymbol S$ where $\boldsymbol S = \frac{1}{\sqrt{12}} \boldsymbol I$ and correlation matrix $\boldsymbol R_{\boldsymbol U_{\boldsymbol X}}$.
So we receive
\begin{align}
\text{CDSS}( \boldsymbol C_{\boldsymbol X} , \boldsymbol u_{\boldsymbol y}) = -H \log( 12 \det( \boldsymbol R_{\boldsymbol U_{\boldsymbol X}} ) ) + \frac{1}{12}\left(\boldsymbol u_{\boldsymbol y} - \frac{1}{2}\boldsymbol 1\right)'\boldsymbol R_{\boldsymbol U_{\boldsymbol X}}^{-1}\left(\boldsymbol u_{\boldsymbol y} - \frac{1}{2}\boldsymbol 1\right)
\label{eq_CDSS_2}
\end{align}
as $ \det( \boldsymbol \Sigma_{\boldsymbol U_{\boldsymbol X}}) = 12^{-H} \det( \boldsymbol R_{\boldsymbol U_{\boldsymbol X}} ) $. Hence, due to \eqref{eq_CDSS_2} a scaling by $\frac{1}{H}$ might be considered
for the Copula Dawid-Sebastiani score to adjust for impact of the dimension.
In contrast to the copula energy score and copula variogram score we do not apply any scaling constant. The reason is that DSS is an unbounded score.
For instance, when considering a copula with the the constant correlation matrix $\boldsymbol R_{\boldsymbol U_{\boldsymbol X}}(\delta) = (1-\delta) \boldsymbol I + \delta \boldsymbol 1\boldsymbol 1'$
then it holds for the limit
$ \lim_{\delta \to 1} \det( \boldsymbol R_{\boldsymbol U_{\boldsymbol X}}(\delta) ) = 0$. Additionally, the second term is a quadratic form and minimal if $\boldsymbol u_{\boldsymbol y} - \frac{1}{2}\boldsymbol 1$.
Thus, the copula Dawid-Sebastiani score has no lower bound and can take negative values.
\section{Reporting multivariate forecasts and evaluation design} \label{sec:reporting}
Remember, we want to forecast $\boldsymbol Y$ by $\boldsymbol X$.
In statistical communities, often $\widehat{\boldsymbol Y}$ is usually used instead of $\boldsymbol X$ to emphasize relationship to the target $\boldsymbol Y$.
In practice, there are two options for describing the forecasting distribution $\boldsymbol F_{\boldsymbol X}$ of $\boldsymbol X$ or to report this forecast:
\begin{enumerate}
\item \label{item_mdistr} Reporting the forecasting distribution $\boldsymbol F_{\boldsymbol X}$ explicitly
\item \label{item_simul} Providing a independent large sample of the (forecasting) model of $\boldsymbol X$ of the target distribution $\boldsymbol Y$,
we refer to this large sample as \emph{ensemble} and the method as \emph{ensemble forecasting}.
\end{enumerate}
If we are going for option \ref{item_mdistr}, then we have to report the forecasted distribution $\boldsymbol F_{\boldsymbol Y}$ explicitly or an equivalent representation.
A popular way of providing the equivalent information is by reporting an estimate $\boldsymbol f_{\boldsymbol X}$
of the joint density $\boldsymbol f_{\boldsymbol Y}$ which exists for (absolutely) continuous distribution.
Alternatively, $\boldsymbol F_{\boldsymbol X}$ can be described using copulas, in detail we have to report estimates
$f_{X_{1}},\ldots, f_{X_{H}}$ of
the marginal distributions
$f_{Y_1},\ldots, f_{Y_H}$ of $\boldsymbol f_{\boldsymbol Y}$ and a copula estimate $\boldsymbol C_{\boldsymbol X}$ for the copula $\boldsymbol C_{\boldsymbol Y}$. For our copula-score based evaluation approach
this is a very appealing way of reporting, unfortunately not a very common one.
The last option we want to mention is to report an estimator $\boldsymbol \varphi_{\boldsymbol X}$ of the characteristic function defined through
$\boldsymbol \varphi_{\boldsymbol Y}(\boldsymbol z) = {\mathbb E}(\exp(i\boldsymbol z'\boldsymbol Y))$ which determines uniquely $\boldsymbol F_{\boldsymbol Y}$.
From the evaluation point of view we must be able to draw from the reported distribution on a computer; or solve some characteristics (esp. some expected values)
of the reported distribution which is only feasible in limited cases.
Unfortunately for sophisticated forecasting models, an explicit description of $\boldsymbol F_{\boldsymbol X}$ (or an equivalent counterpart)
is either very challenging or not feasible. Still, option \ref{item_simul} can be applied in such cases. Therefore option \ref{item_simul} turned into the standard alternative in application.
In the second reporting option \ref{item_simul} we provide $M$ independent simulations resp. draws from the underlying model of $\boldsymbol X$
which essentially describes $\boldsymbol F_{\boldsymbol X}$. We denote the paths/trajectories of the ensemble by ${\mathcal{X}} = \boldsymbol X^{(1)},\ldots,\boldsymbol X^{(M)}$, so $\boldsymbol X^{(j)}$ is an independent and identically distributed (iid) copy of $\boldsymbol X$.
This methodology is basically a Monte-Carlo simulation, but it also known as ensemble forecasting, ensemble simulation, path simulation, trajectory simulation, scenario simulation or scenario generation (but depending on the community the meaning might be differ slightly as well).
Within this paper we assume that the sample size denoted by $M$ is large, so the true underlying distribution function is described well
by the given simulated sample. This approximation goes back to the multivariate Glivenko-Cantelli theorem which characterize that the multivariate
empirical (cumulative) distribution function is converging almost surely to the drawn distribution.
For comparing forecasts in a scientific way it is not sufficient to do forecasting evaluation based on a single forecast
$\boldsymbol Y$ of $\boldsymbol X$. We require multiple forecasts in a (pseudo-)out-of-sample study design to make significance statements later on.
Therefore we assume to have $N$ forecasts
of $\boldsymbol Y$
available. How these
forecasted values are generated is theoretically not relevant as long only information is taken into account that was available at the time point of generating the forecast.
Now, we assume to have a univariate time series $(Y_t)_{t\in {\mathbb Z}}$ that we are interested to forecast.
We denote these multiple targets within the forecasting window by
$\boldsymbol Y_{ i} = (Y_{T+s_i+1},\ldots, Y_{T+s_i+H})'$ with $s_i< s_{i+1}$ ($1\leq i<N$).
So $\boldsymbol Y_{1},\ldots, \boldsymbol Y_{N}$ are ordered in time. The forecast of $\boldsymbol Y_{1},\ldots, \boldsymbol Y_{N}$ are
denoted by $\boldsymbol X_{1},\ldots, \boldsymbol X_{N}$ in accordance with the notation of the previous section.
Note, that the forecasting periods of $\boldsymbol Y_{i}$ and $\boldsymbol Y_{i+1}$ my overlap, so if e.g. $s_1=0$, $s_2=1$ with $H=24$ we have
forecasts $\boldsymbol X_{1} = (X_{T+1},\ldots, X_{T+24}) $ and $\boldsymbol X_{2} = (X_{T+2},\ldots, X_{T+25} )$ of
$\boldsymbol Y_{1} = (Y_{T+1},\ldots, Y_{T+24}) $ and $\boldsymbol Y_{2} = (Y_{T+2},\ldots, Y_{T+25} )$.
Obviously, this design should be tailored in accordance with the corresponding application.
From the statistical point of view is useful to have nice properties of both the forecasts $\boldsymbol X_{i}$ and the true $\boldsymbol Y_{i}$ in the evaluation window
like stationarity, ergodicity or finite variance. However, we can only influence the forecasting model, so $\boldsymbol X_{i}$.
For forecasting models which evaluate historic data we recommend the usage of a \emph{rolling window forecasting study} using
$N$ equidistant windows. It is the most suitable way to evaluate forecasts.
In general there are several options for the design of a rolling window forecasting study. But they all share together that the in-sample data length is fixed and
the in-sample window moves equally distant across the time range. So the window shift are $s_i = K(i-1)$. If $K<H$ then we have overlapping forecasting horizons.
If $K\geq H$ the forecasting horizons in the forecasting study do not overlap which is favorable from the statistical point of view - but does not necessarily meet practical applications.
Here
the special case $K=H$ is usually chosen.
A slightly different design of an rolling window forecasting study with
$s_i = 0.25 H(i-1)$ and $H=12$ is visualized in Figure \ref{fig_sample_air} of the real data application section along with forecasting reporting option reporting option \ref{item_simul}.
In many practical applications such a design is clearly favorable.
Note that for forecasting evaluation expanding windows forecasting studies are not suitable. More details on the design of forecasting studies, can be found in \cite{diebold2015comparing}.
Finally, we require for the evaluation that we have observations of the full forecasting window of $\boldsymbol Y_{1},\ldots, \boldsymbol Y_{N}$
available. We denote these observations by $\boldsymbol y_{1}, \ldots, \boldsymbol y_{N}$.
This is important for the evaluation method as well. So we do not want to compare the forecasted distributions
of $\boldsymbol X_{1},\ldots, \boldsymbol X_{N}$
with the true underlying $\boldsymbol Y_{1},\ldots, \boldsymbol Y_{N}$ but with the observations $\boldsymbol y_{1}, \ldots, \boldsymbol y_{N}$.
So from the statistical point of view we require a so called one sample testing framework for comparing distributions.
\section{Estimation} \label{sec:estim}
So far we had only a look at the theoretical properties score of a forecasting method.
However, as always in statistics, the theoretical scores involve objects that are unknown in practice.
Therefore we assume that we are in a forecasting study framework, where we have the observations $\boldsymbol y_1,\ldots,\boldsymbol y_N$ of $\boldsymbol Y_1,\ldots, \boldsymbol Y_N$ such as the ensemble forecasts ${\mathcal{X}}_i = (\boldsymbol X_{i}^{(1)},\ldots, \boldsymbol X_{i}^{(M)})'$ of the forecasting distribution $\boldsymbol X_i$ for $\boldsymbol Y_i$ available.
Subsequently, we give an overview on estimators for the standard evaluation methods, namely the energy score, the variogram score and the Dawid-Sebastiani score. Afterwards we deal with the problem of estimating copula observations and the marginal copula score.
\subsection{CRPS}
When estimating \eqref{eq_crps}, we have essentially two options, matching the two possible representations.
The most standard form is to utilise the first one. It solves the corresponding integral numerically by replacing the unknown cdf by an ecdf.
$\widehat{\text{CRPS}}_{i,h} = \int_{-\infty}^{\infty} (\widehat{F}_{Y_{i,h}}(z) - \mathbbm{1}\{y_h < z\})^2 \,d z $
with standard estimator for $F_{Y_{i,h}}$ is the empirical distribution function (ecdf) of $h$-th coordinate of the ensemble ${\mathcal{X}}_i$.
This is given by
$\widehat{F}_{Y_{i,h}}(z) = \widehat{F}_{Y_{i,h}}(z; {\mathcal{X}}_i) = \frac{1}{M} \sum_{j=1}^M \mathbbm{1}\{X_{i,h}^{(j)} \leq z\}$.
Alternatively, we can utilize the second term in \eqref{eq_crps} and estimate the two expected values. This is e.g. done in
\cite{taieb2016forecasting} in a load forecasting context. It additionally helps for interpretation. We discuss this resulting estimator as a special case
of the estimation of the energy score in the subsequent subsection.
Given the scores $\widehat{\text{\textbf{CRPS}}}_{i} = (\widehat{\text{CRPS}}_{i,1},\ldots, \widehat{\text{CRPS}}_{i,H})$ for each marginal distribution we can compute the corresponding marginal score
by $ \widehat{\text{CRPS}}_i(\boldsymbol a) = \boldsymbol a'\widehat{\text{\textbf{CRPS}}}_{i}$
with weight vector $\boldsymbol a = (a_1,\ldots, a_H)'$.
\subsection{Energy score}
First, we can rewrite the energy score definition \eqref{eq_energy_score} as
\begin{align}
\text{ES}_{i,\beta} ( \boldsymbol F_{\boldsymbol X_i}, \boldsymbol y_i)
&= \text{ED}_{\beta,i}(\boldsymbol X_i, \boldsymbol y_i) -
\frac{1}{2} \text{EI}_{\beta,i}(\boldsymbol X_i, \boldsymbol y_i)
\end{align}
If reporting option \ref{item_mdistr} is used, we might be able to compute the two terms in \eqref{eq_energy_score} explicitly,
e.g. under the normality assumption see \cite{pinson2013discrimination}.
If we are able to do use explicit or numeric approximation formulas for the two terms in \eqref{eq_energy_score}, we should prefer them.
They are usually more precise and much faster than the simulation approaches.
However, in the standard case of reporting option \ref{item_simul} we have to estimate the first term in $\text{ED}_{\beta,i}$ and the second term $\text{EI}_{\beta,i}$ in
\eqref{eq_energy_score} given the $M$ samples $\boldsymbol X_{i}^{(1)},\ldots,\boldsymbol X_{i}^{(M)}$ of the distribution $\boldsymbol F_{\boldsymbol X}$.
The estimation of the first term $\text{ED}_{\beta,i}$ is straight forward, we simply estimate the expectation by its sample mean.
So we have
\begin{equation}
\widehat{\text{ED}}_{i,\beta} =
\frac{1}{M} \sum_{j=1}^M \left\| \boldsymbol X_{i}^{(j)} - \boldsymbol y_{i} \right\|_2^\beta .
\end{equation}
The second term $\text{EI}_{\beta,i}$ has multiple plausible options for the estimation.
The reason is that the definition in \eqref{eq_energy_score} of $\text{EI}_{\beta,i}$ suggests we require these independent copies $\widetilde{\boldsymbol X}_{i}$
of our forecasting distribution. As $\boldsymbol X_{i}^{(1)},\ldots,\boldsymbol X_{i}^{(M)}$ are independently drawn,
we can simply use the first have of the data set as draws from $\boldsymbol X$ in \eqref{eq_energy_score} and the second half as draws from the iid copy $\widetilde{\boldsymbol X}$.
Hence we can simply use the estimator
\begin{equation}
\widehat{\text{EI}}^{\text{iid}}_{i,\beta} = \frac{1}{2M} \sum_{j=1}^{0.5M} \left\| \boldsymbol X_{i}^{(j)} - \boldsymbol X_{i}^{(0.5M + j)} \right\|_2^\beta
\end{equation}
where we assume without loss of generality that $M$ is even.
Note that the sum in $\widehat{\text{EI}}^{\text{iid}}_{i,\beta}$ contains only $M/2$ summands. But these summands have nice statistical properties, as they are iid.
Also \cite{moller2013multivariate} and \cite{junk2014comparison} mention the resulting estimator for the energy score as a plausible option.
Still, we can use an alternative estimator that uses more summands and provides in average higher accuracy.
Such an estimator is given by
\begin{equation}
\widehat{\text{EI}}^{K\text{band}}_{i,\beta} = \frac{1}{M} \sum_{j=1}^{M} \sum_{k=j}^K \left\| \boldsymbol X_{i}^{(j)} - \boldsymbol X_{i}^{(j+k)} \right\|_2^\beta
\label{eq_est_energy_score_alt2_second_term}
\end{equation}
for an integer $K$ with $1\leq K\leq M$ where we set for notational purpose $\boldsymbol X_{i}^{(M+k)} = \boldsymbol X_{i}^{(k)}$.
The expression $\widehat{\text{EI}}^{K\text{band}}_{i,\beta}$ has the advantage by using a larger
amount of summands for approximating the sum. This should increase the precision in general. However, as the elements of the sum become pairwise dependent, this improvement is weaker as if they would be all independent.
For the important case $K=1$ we have $M$ summands across the summation diagonal, for $K=M$ have $M(M-1)/2$ different pairs $\boldsymbol X_{i}^{(j)} - \boldsymbol X_{i}^{(l)}$ that are summed up.
The latter cases uses the maximal amount of information for estimating $\text{EI}_{\beta,i}$.
In the \texttt{R} packages \cite{Rpackage_scoringRules} and \cite{Rpackage_energy}, only this estimator is provided. It has the mentioned advantage of high accuracy but a high computation demand for large values of $M$
as the amount of summands is increasing quadratically in $M$. Thus, if $M$ is large (which is recommended to choose), we suggest to use a small $K$ to keep the computational complexity manageable. Then the energy score is estimated by
$$ \widehat{\text{ES}}^{K\text{band}}_{i,\beta} = \widehat{\text{ED}}_{i,\beta} - 0.5 \widehat{\text{EI}}^{K\text{band}}_{i,\beta} .$$
\subsection{Variogram score}
The computation is similarly to the energy score. If we apply the reporting option \ref{item_mdistr}, we might be able to solve some special cases explicitly, like the case of multivariate normality.
If we consider the simulation reporting option \ref{item_simul}, we have can estimate the expected values in \eqref{eq_variogram_score} by the sample means.
Thus, we have
\begin{align}
\widehat{\text{VS}}_{i,\boldsymbol W,p}
= \sum_{j=1}^H \sum_{k=1}^H w_{j,k} ( |y_{i,j}-y_{i,k}|^p - \sum_{m=1}^M | X_{i, j}^{(m)} - X_{i,k}^{(m)}|^p )^2.
\label{eq_variogram_score_est}
\end{align}
for a given $\boldsymbol W$ and $p$. Due to symmetries $y_{i,j}-y_{i,k} = y_{i,k}-y_{i,j}$ and $X_{i,j}-X_{i,k} = X_{i,k}-X_{i,j}$ the computational cost can be halved in
the implementation. Thus it holds
\begin{align}
\widehat{\text{VS}}_{i,\boldsymbol W,p}
= 2\sum_{j=1}^H \sum_{k=j+1}^H w_{j,k} ( |y_{i,j}-y_{i,k}|^p - \sum_{m=1}^M | X_{i, j}^{(m)} - X_{i,k}^{(m)}|^p )^2.
\label{eq_variogram_score_estalt}
\end{align}
\subsection{Dawid-Sebastiani score}
The Dawid-Sebastiani score depends only on the first two moments. Under reporting option \ref{item_mdistr}, we are usually able to compute the corresponding moments explicitly.
Under the reporting option \ref{item_simul}, we estimate the moments by their sample counterparts:
Then we have for the mean and covariance matrix estimates
\begin{align}
\widehat{\boldsymbol \mu}_{i,\boldsymbol X} &= \frac{1}{M} \sum_{m=1}^M \boldsymbol X_{i}^{(m)} \\
\widehat{\boldsymbol \Sigma}_{i,\boldsymbol X} &= \frac{1}{M-1} \sum_{m=1}^M (\boldsymbol X_{i}^{(m)} - \widehat{\boldsymbol \mu}_{i,\boldsymbol X} ) (\boldsymbol X_{i}^{(m)} - \widehat{\boldsymbol \mu}_{i,\boldsymbol X} )'
\end{align}
where the latter one is the unbiased sample covariance matrix.
The resulting plug-in estimator for the Dawid-Sebastiani score is
\begin{equation}
\widehat{\text{DSS}}_i =
\log( \det( \widehat{\boldsymbol \Sigma}_{i,\boldsymbol X} ) ) + (\boldsymbol y_i - \widehat{\boldsymbol \mu}_{i,\boldsymbol X})'\widehat{\boldsymbol \Sigma}_{i,\boldsymbol X}^{-1}(\boldsymbol y_i - \widehat{\boldsymbol \mu}_{i,\boldsymbol X}) .
\label{eq_dawid_sebastiani_score_est}
\end{equation}
Note that we require $M>H$ to receive a positive semidefinite sample covariance matrix.
\subsection{Copula observations}
\label{sec:copobs}
Remember that for all copula scores we are interested in evaluating
$\boldsymbol U_{\boldsymbol Y_i} = (U_{\boldsymbol Y_i,1}, \ldots, U_{\boldsymbol Y_i,H})' = (F_{Y_{i,1}}(Y_{i,1}), \ldots, F_{Y_{i,H}}(Y_{i,H}))' $.
There we defined the
copula observations $\boldsymbol u_{\boldsymbol Y_i} = (u_{\boldsymbol y_i,1}, \ldots, u_{\boldsymbol y_i,H})' = (F_{Y_{i,1}}(y_{i,1}), \ldots, F_{Y_{i,H}}(y_{i,H}))'$.
However, as mentioned already above
a crucial problem here is that the definition of $\boldsymbol u_{\boldsymbol Y_i}$ includes the marginals $F_{Y_{i,h}}$ of the true underlying distribution
which is unknown in practice.
Thus, the only way around this to estimate $F_{Y_{i,h}}$ to get an estimator for $\boldsymbol u_{\boldsymbol Y_i}$.
Of course the standard estimator for $F_{Y_{i,h}}$ is the empirical distribution function (ecdf) of the ensemble ${\mathcal{X}}_i$.
However, we can use the standard estimator
$\widehat{F}_{Y_{i,h}}(z) = \widehat{F}_{Y_{i,h}}(z; {\mathcal{X}}_i) = \frac{1}{M} \sum_{j=1}^M \mathbbm{1}\{\boldsymbol x_i^{(j)} \leq z\}$.\footnote{
The mid-point rule
$\widehat{F}_{Y_{i,h}}^{\text{mid}}(z) = \frac{1}{2M} \sum_{j=1}^M \mathbbm{1}\{\boldsymbol x_i^{(j)} \leq z\} + \mathbbm{1}\{\boldsymbol x_i^{(j)} < z\}$
might be even a better choice.}
Then we receive \emph{estimated copula observations}
$$\widehat{\boldsymbol u}_{\boldsymbol Y_i} = \widehat{\boldsymbol u}_{\boldsymbol Y_i}({\mathcal{X}}_i) = (\widehat{F}_{Y_{i,1}}(y_{i,1}; {\mathcal{X}}_i), \ldots, \widehat{F}_{Y_{i,H}}(y_{i,H}; {\mathcal{X}}_i))'.$$
Intuitively, the use of $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$ instead of $\boldsymbol u_{\boldsymbol Y_i}$ should not be a problem if $\widehat{F}_{Y_{i,h}}$ is a good estimator for $F_{Y_{i,1}}$;
and here intuition is right.
Unfortunately it is not a good choice if this is not the case.
This quickly leads to problems concerning the application of $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$ for the evaluation of copula based scores.
There are situations with a misspecified forecasting marginal distribution and the use of $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$ as copula observation estimator
where the corresponding copula score of the misspecified forecast is smaller than the true forecasting distribution.
If for example $X_{i,h}$ of the forecast $\boldsymbol X_i$ has a variance that is too large then $\widehat{u}_{\boldsymbol Y_i,h}({\mathcal{X}}_i)$ takes too many values around the center (usually around 0.5) and too less values in
the tails around $0$ and $1$.
This is a problem for practical application and restricts the potential applications of $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$ to the comparison of forecasts with the same marginals which is far away from practical needs.
Fortunately, there is a relatively simple way to avoid the problem with the misspecified marginals.
We can force the misspecified marginal distribution to be a (plausible) draw from a copula, so that the general monotonic ordering across is preserved,
but such that they have uniform marginals.
To perform such an adjustment we have to learn about the distribution of the estimated
copula observations $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$. Given a single $\widehat{\boldsymbol u}_{\boldsymbol Y_i}$ this is impossible. Therefore, we explore the distributional structure across
the full out-of-sample period $\widehat{\boldsymbol u}_{\boldsymbol Y_1},\ldots, \widehat{\boldsymbol u}_{\boldsymbol Y_N}$. The most obvious adjustment is to adjust the
copula observations on each marginal $\widehat{u}_{\boldsymbol Y_1,h},\ldots, \widehat{u}_{\boldsymbol Y_N,h}$ so that they follow a perfect uniform distribution on $(0,1)$.
This can be easily done by ranking the estimated observations $\widehat{u}_{\boldsymbol Y_1,h},\ldots, \widehat{u}_{\boldsymbol Y_N,h}$ for each $h$.
Hence, denote $R_{i,h}$ the rank of $\widehat{u}_{\boldsymbol Y_i,h}$ within $\widehat{u}_{\boldsymbol Y_1,h},\ldots, \widehat{u}_{\boldsymbol Y_N,h}$.
Then we define the \emph{adjusted estimated copula observations} by
\begin{align}
\widehat{u}^{*}_{\boldsymbol Y_i,h} = \frac{2R_{i,h} - 1}{2N} .
\label{eq_est_copula_obs_adj}
\end{align}
Note that for the ranks $R_{i,h}$ there can appear ties due to the stepwise structure of the ecdf.
To preserve the optimal distribution, we highly suggest break the ties in the ranks at random (uniform sampling).
This ranking procedure guarantees that $\widehat{u}^{*}_{\boldsymbol Y_1,h},\ldots, \widehat{u}^{*}_{\boldsymbol Y_N,h}$ have perfect uniform marginals, i.e. they take the values $\frac{1}{2N},\ldots, \frac{2N-1}{2N}$.
Perfect means here that for these values the corresponding ecdf minimizes the Kolmogorov-Smirnov distance (but also the L\'evy distance or L\'evy metric which characterizes convergence in distribution)
with respect to the uniform distribution. This procedure assumes that across the sample the comonotonicity within the implied copula structure is preserved.
For the forecasted copula $\boldsymbol C_{\boldsymbol X_i}$ of $\boldsymbol X_i$ we simply consider the empirical copula of the ensemble ${\mathcal{X}}_i$ with elements $\boldsymbol x_i^{(j)} = (x_{i,1}^{(j)},\ldots, x_{i,H}^{(j)})'$
as a corresponding estimator.
Here we suggest to consider
\begin{equation}
\widehat{\boldsymbol C}_{\boldsymbol X_i} (u_1,\ldots, u_H) = \frac{1}{M} \sum_{j=1}^M \mathbbm{1}\{ \widetilde{R}_{i,j,1}/M \leq u_1, \ldots, \widetilde{R}_{i,j,H}/M \leq u_H \}
\label{eq_est_copula}
\end{equation}
with the ranks
$$ \widetilde{R}_{i,j,h} = \frac{1}{2} \sum_{k=1}^M \mathbbm{1}\{ x_{i,h}^{(k)} \leq x_{i,h}^{(j)} \} + \mathbbm{1}\{ x_{i,h}^{(k)} < x_{i,h}^{(j)} \}$$
where we consider the mid-point rule for the ecdf. Similarly as above potential ties should be broken at random (uniform sampling).
\subsection{Marginal copula score}
Whenever, we want to estimate a marginal copula score \eqref{eq_msc} we apply the plug-in principle.
Still, remember that \eqref{eq_msc} has a multiplicative structure and it holds ${\mathbb E}(X){\mathbb E}(Y) ={\mathbb E}( XY) - \operatorname{{\mathbb C} ov}(X,Y)$.
Thus, we get the plug-in estimator
\begin{align}
\widehat{\text{MS}(\boldsymbol a)\text{-}\text{CS}}_i = \widehat{\text{MS}}_i(\boldsymbol a) \widehat{\text{CS}}_i - \widehat{\sigma}_{\text{MS},\text{CS}}
\label{eq_msc_est}
\end{align}
where $\widehat{\text{MS}}_i(\boldsymbol a)$ and $\widehat{\text{CS}}_i$ are the sample means across the marginal and copula score.
$\widehat{\sigma}_{\text{MS},\text{CS}} $ is the covariance between $\text{MS}$ and $\text{CS}$ across $i$. Here we assume implicitly some covariance stationarity across the out-of-sample window study
between the marginal and copula score.
For applications the estimated marginal scores $\widehat{\text{MS}}_i$ can be estimated by the CRPS and the copula score can be
either the estimated energy score $\widehat{\text{ES}}_{i,\beta}$, the estimated variogram score $\widehat{\text{VS}}_{i,\boldsymbol W, p}$ or the estimated Dawid-Sebastiani score $\widehat{\text{DSS}}_{i}$
of the copula observations and the copula of the forecast.
Here, we suggest the estimates $\widehat{u}^{*}_{\boldsymbol Y_i,h}$ (eqn. \eqref{eq_est_copula_obs_adj}) and $\widehat{\boldsymbol C}_{\boldsymbol X_i}$ (eqn. \eqref{eq_est_copula}) for the copula observations and the copula as derived in the previous section.
As the Copula Dawid-Sebastiani score $\widehat{\text{DSS}}_{i}$ is not bonded from below, the resulting scoring rule is not strictly proper. In fact, it results in a useless rule
for applications.
\section{Evaluation} \label{sec:eval}
Let $\text{SC}_i$ be a score of the $i$-th forecasting experiment for $i=1,\ldots,N$ of the considered forecasting model. Further let $\text{SC}^*$ be the score of the true model, so $\boldsymbol Y \sim \boldsymbol F_{\boldsymbol X}$
with draws $\boldsymbol y$.
All covered (strictly) proper scoring rules are negatively oriented. Thus, it holds: the smaller the score the better the forecasting performance.
Obviously, for practical application the corresponding sample mean
\begin{equation}
\ov{\text{SC}} = \frac{1}{N} \sum_{i=1}^N \text{SC}_i .
\label{eq_score_average}
\end{equation}
will be the most relevant criterion when comparing the predictive performance of two forecasts.
In this section we shortly introduce two further evaluation methods, based on the scores.
First, we consider the relative change in scores as used for deriving conclusions in \cite{pinson2013discrimination}.
Secondly, we consider the Diebold-Mariano test, which allows for significance statements.
\subsection{Relative change in scores}
In \cite{pinson2013discrimination} the relative change in the score with respect to the best forecast is considered:
\begin{equation}
\text{RelCh}(\text{SC}) = \frac{\ov{\text{SC}} - \ov{\text{SC}}^*}{\ov{\text{SC}}^*}
\label{eq_relch}
\end{equation}
where $\ov{\text{SC}}^* = 1/N \sum_{i=1}^N \text{SC}^*_i$.
Obviously, it holds $\text{RelCh}(\text{SC}^*) = 0$.
The idea is to measure the sensitivity in the scores with respect to some biased non-optimal forecast in a relative manner.
\subsection{Significance evaluation: the Diebold-Mariano test}
The Diebold-Mariano (DM) test was designed for the point forecast evaluation in \cite{diebold1995comparing}.
However, as pointed out in e.g. \cite{diebold2015comparing} the test design is very general and allows for several generalizations.
For instance, in \cite{ziel2018day} and\cite{uniejewski2018variance} it was applied in multivariate settings.
In \cite{moller2015spatially} it is considered in an application for the energy score.
The DM-test requires a (pseudo-)out-of-sample forecasting study which aims to forecast $\boldsymbol Y_{1},\ldots,\boldsymbol Y_{N}$ by $\boldsymbol X_{1},\ldots,\boldsymbol X_{N}$
as described in section \ref{sec:reporting}. Its target is to compare the forecasting accuracy of two forecasts ${\mathbb A}$ and ${\mathbb B}$ on the same (pseudo-)out-of-sample environment
based on a scoring rule $\text{SC}$.
The DM-test checks whether the scores $\ov{\text{SC}}^{{\mathbb A}}$ is significantly different from $\ov{\text{SC}}^{{\mathbb B}}$, see \eqref{eq_score_average}.
Therefore, score differences are required.
Given the losses $\text{SC}_1^{{\mathbb A}}, \ldots, \text{SC}_N^{{\mathbb A}}$ and $\text{SC}_1^{{\mathbb B}}, \ldots, \text{SC}_N^{{\mathbb B}}$, we define the loss differences by
$$\Delta_i^{{\mathbb A},{\mathbb B}} = \text{SC}_i^{{\mathbb A}} - \text{SC}_i^{{\mathbb B}}.$$
Now, the key idea of the DM-test is to check if the mean loss (or score) difference
$$\ov{\Delta}^{{\mathbb A},{\mathbb B}}
= \frac{1}{N} \sum_{i=1}^N \Delta^{{\mathbb A},{\mathbb B}}_i$$ is significantly
different from zero or not.
Remember, scoring rules $\text{SC}_i$ have a certain direction, so that either a positive loss or a negative loss represents a
better forecasting accuracy. As we consider only negatively oriented scores,
a $\ov{\Delta}^{{\mathbb A},{\mathbb B}}$ which is significantly smaller than zero lead to the conclusion that ${\mathbb A}$ is significantly better than ${\mathbb B}$ with respect to the considered scoring rule.
In general, the distribution of $\boldsymbol X_{i}$ is different from $\boldsymbol X_{j}$ for $i\neq j$,
in the consequence of the distribution
of $\Delta_i^{{\mathbb A},{\mathbb B}}$ is usually different from $\Delta_j^{{\mathbb A},{\mathbb B}}$ as well.
So $\ov{\Delta}^{{\mathbb A},{\mathbb B}}$ is the sum of $N$ different distributions. Moreover,
$\Delta_i^{{\mathbb A},{\mathbb B}}$ and $\Delta_j^{{\mathbb A},{\mathbb B}}$ exhibit a specific dependency structure.
Usually they are not even independent or uncorrelated.
Hence, from the statistical point of view
it is not possible to make further statements about $\ov{\Delta}^{{\mathbb A},{\mathbb B}}$ unless we make some assumptions for
the sequence $(\Delta_i^{{\mathbb A},{\mathbb B}})_{i\in {\mathbb Z}}$ (or $\Delta_1^{{\mathbb A},{\mathbb B}}, \ldots, \Delta_N^{{\mathbb A},{\mathbb B}}$).
The standard assumptions in the DM-test are:
\begin{enumerate}
\item ${\mathbb E}( \Delta_i^{{\mathbb A},{\mathbb B}} ) = \mu_{{\mathbb A},{\mathbb B}} $ for all $i$
\item $\operatorname{{\mathbb C} ov}( \Delta_i^{{\mathbb A},{\mathbb B}}, \Delta_{i-k}^{{\mathbb A},{\mathbb B}} ) = \gamma_{{\mathbb A},{\mathbb B}}(k) $ for all $i$ and $k\geq 0$
\end{enumerate}
Thus, $(\Delta_i^{{\mathbb A},{\mathbb B}})_{i\in {\mathbb Z}}$ is a weakly stationary (or covariance stationary) process.
These assumptions are very restrictive, and can be relaxed substantially.
Still, under these assumption given above it possible to show that
$\ov{\Delta}^{{\mathbb A},{\mathbb B}}$ is asymptotic normal \cite{diebold2015comparing}, i.e. it holds for $N\to \infty$ that
\begin{equation}
\frac{\ov{\Delta}^{{\mathbb A},{\mathbb B}} - \mu_{{\mathbb A},{\mathbb B}} }{ \sigma(\ov{\Delta}^{{\mathbb A},{\mathbb B}}) } \to N(0,1)
\label{eq_null}
\end{equation}
in distribution for where $\sigma(\ov{\Delta}^{{\mathbb A},{\mathbb B}}) = \sqrt{\gamma_{{\mathbb A},{\mathbb B}}(0) }$ is the standard deviation of $\ov{\Delta}^{{\mathbb A},{\mathbb B}}$.
This allows the creation of asymptotic tests, namely the Diebold-Mariano test with the null hypothesis $\text{H}_0: \mu_{{\mathbb A},{\mathbb B}} = 0$. When replacing $\sigma(\ov{\Delta}^{{\mathbb A},{\mathbb B}})$
with a suitable estimator, e.g. the sample standard deviation $\widehat{\sigma}(\ov{\Delta}^{{\mathbb A},{\mathbb B}})$ the same distributional limit is attained.
Note that the DM-test is a \emph{asymptotic} test, as a consequence the out-of-sample window length $N$ should be as large as possible.
In the next section we also study in more detail the impact of the simulation sample size $M$ of the proposed evaluation procedure.
\section{Simulation Studies} \label{sec:studies}
In the subsequent subsections we will carry out some simulation studies to compare the forecasting measure in various forecasting situations.
These experiment are designed with $N$ replications (or instances) indexed by $i$ as described in the reporting section.
For evaluation we will consider always 9 different multivariate measures:
\begin{enumerate}
\item Energy score: $\text{ES}_{i,1}$, estimated by $ \widehat{\text{ES}}^{K\text{band}}_{i,1} $ with $K=1$
\item Variogram score: $\text{VS}_{i,\boldsymbol 1 \boldsymbol 1', 1}$, estimated by $\widehat{\text{VS}}_{i,\boldsymbol 1 \boldsymbol 1', 1}$
\item Dawid-Sebastiani score: $\text{DSS}_{i}$, estimated by $\widehat{\text{DSS}}_i$
\item CRPS-copula energy score: ${(\text{CRPS}(\frac{1}{H}\boldsymbol 1)\text{-}\text{CES}_{1})}_i$, estimated by \eqref{eq_msc_est} with $\widehat{\text{CRPS}}_{i}(\frac{1}{H}\boldsymbol 1)$ and $\widehat{\text{CES}}^{K\text{band}}_{i,1}$ with $K=1$
\item CRPS-copula variogram score: $\text{CRPS}(\frac{1}{H}\boldsymbol 1)\text{-}\text{CVS}_{i,\boldsymbol 1 \boldsymbol 1', 1}$, estimated by \eqref{eq_msc_est} with $\widehat{\text{CRPS}}_{i}(\frac{1}{H}\boldsymbol 1)$ and $\widehat{\text{CVS}}_{i,\boldsymbol 1 \boldsymbol 1', 1}$
\item Continuous ranked probability score: $\text{CRPS}_i$, estimated by $\widehat{\text{CRPS}}_{i}(\frac{1}{H}\boldsymbol 1)$.
\item Copula energy score: $\text{CES}_{i,1}$, estimated by $\widehat{\text{CES}}^{K\text{band}}_{i,1}$ with $K=1$
\item Copula variogram score: $\text{CVS}_{i,\boldsymbol 1 \boldsymbol 1', 1}$, estimated by $\widehat{\text{CVS}}_{i,\boldsymbol 1 \boldsymbol 1,1}$
\item Copula Dawid-Sebastiani score: $\text{CDSS}_{i}$, estimated by $\widehat{\text{CDSS}}_{i}$
\end{enumerate}
The first three are the established multivariate forecasting criterion. The next two are marginal-copula scores with the CRPS as marginal score.
The sixth score is the CRPS itself, which evaluated only the marginals. The latter three ones only evaluate the dependency structure using the copulas.
\subsection{Sensitivity study I}
Here we replicate the study of \cite{pinson2013discrimination} and enlarge the study design to some extend. This study was the reason for \cite{pinson2013discrimination}
to conclude that the energy score has bad discrimination abilities with respect to the dependency behavior.
We consider a bivariate normal distribution with zero mean and variance of $1$ in each component.
So the true model is $\boldsymbol Y \sim {\mathcal{N}}_2( \boldsymbol \mu, \boldsymbol \Sigma(\rho) )$ with $\boldsymbol \mu= (0,0)'$ and $\boldsymbol \Sigma(\rho) = \left( \begin{array}{cc}
1 & \rho \\ \rho & 1
\end{array} \right)$ with correlation $\rho$.
In the simulation setup we choose the true $\rho$ on a grid $\{-1,-0.8, \ldots, 0.8, 1\}$.
For the forecasting model we consider $\boldsymbol Y \sim {\mathcal{N}}_2( \boldsymbol \mu, \boldsymbol \Sigma(\varrho) )$. Here we choose the
forecasted correlation $\varrho$ out of a slightly denser grid $\{-1,-0.9, -0.8 \ldots, 1\}$.
In the experiment we draw $M=2^{14}=16384$ times for a single bivariate forecast. The window length is $N=2^9=512$.
Then we compute the relative change in the score and the DM-test statistic with respect to the true model. Note
that \cite{pinson2013discrimination} evaluated only the relative change.
The results are given in Figures \ref{fig_sim_study_pinson_relch} and \ref{fig_sim_study_pinson_dm}.
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_pinson_relch1}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Variogram score \newline(VS)}
\label{fig_sim_study_pinson_relch2}
\end{subfigure}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_pinson_relch3}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_pinson_relch4}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_pinson_relch5}
\end{subfigure}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_pinson_relch7}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_pinson_relch8}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_pinson_relch9}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_pinson_relch10}
\end{subfigure}
\caption{Relative changes $\text{RelCh}$ (see \eqref{eq_relch})
for all considered scoring rules for the extended simulation study of \cite{pinson2013discrimination}
with an ensemble sample size of $M=2^{14}=16384$ and a rolling window length of $N=2^9=512$.}
\label{fig_sim_study_pinson_relch}
\end{figure}
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_pinson_dm1}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Variogram score \newline(VS)}
\label{fig_sim_study_pinson_dm2}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_pinson_dm3}
\end{subfigure}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_pinson_dm4}
\end{subfigure}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_pinson_dm5}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_pinson_dm7}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_pinson_dm8}
\end{subfigure}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_pinson_dm9}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_pinson_dm10}
\end{subfigure}
\caption{DM-test statistics with corresponding p-value given in squared brackets with respect to the true model for all considered scoring rules for the extended simulation study of \cite{pinson2013discrimination}
with an ensemble sample size of $M=2^{14}=16384$ and a rolling window length of $N=2^9=512$.}
\label{fig_sim_study_pinson_dm}
\end{figure}
In Figures \ref{fig_sim_study_pinson_relch} we observe the same pattern for the energy score as reported in \cite{pinson2013discrimination}.
The relative change in the score is quite small. For instance, if the true correlation is $-0.8$ and the forecasted one is $0.3$, then
the relative change is only 5\% (Fig. \ref{fig_sim_study_pinson_relch1}). In this situation, the variogram score yields a relative change of about 25\% (Fig. \ref{fig_sim_study_pinson_relch2}) and the Dawid-Sebastiani score
even a relative change of approximately 175\% (Fig. \ref{fig_sim_study_pinson_relch3}). However, the latter score is designed for evaluating multivariate normal distributions. Thus, the high sensitivity is not surprising.
Further, the energy score and the Dawid-Sebastiani score exhibit symmetric evaluation behavior for positive and negative correlations. In contrast, the variogram score
reacts more sensitive for positive correlations than for negative ones.
The CRPS-copula energy score (CRPS-CES) was designed to overcome this issue of the energy score. Indeed, the relative changes are substantially larger than
the energy score values. In the mentioned example the relative change is about 12\% (Fig. \ref{fig_sim_study_pinson_relch4}), thus more than twice as sensitive in the relative change.
Similarly, we observe an increase of the relative change of the CRPS-copula variogram score (CRPS-CVS) score when comparing with the variogram score (Fig. \ref{fig_sim_study_pinson_relch5}). However, we want to highlight that the relative change is not a suitable measure to evaluate if a scoring rule can discriminate well a suboptimal forecast and the optimal one. Hence, e.g. the CRPS-CES is not necessarily better than the energy score itself in doing so.
Obviously, the CRPS can not evaluate any change in the correlation structure and has tiny relative changes which are not significantly different from zero.
This is also a reason why, the curve pattern for the energy score look very similar to the CRPS-copula energy score and copula energy score
(Fig. \ref{fig_sim_study_pinson_relch1}, \ref{fig_sim_study_pinson_relch4}, \ref{fig_sim_study_pinson_relch8}), and analogously for the variogram score.
For the copula Dawid-Sebastiani score we see smaller relative changes than for the Dawid-Sebastiani score. This matches again the fact that
Dawid-Sebastiani score should be optimal in the considered situation.
Let us turn to Figure \ref{fig_sim_study_pinson_dm} and DM-test statistic results. We capped the statistics at 10 to improve the interpretability of the graphs.
First, we observe that all scoring rules except the CRPS (Fig. \ref{fig_sim_study_pinson_dm7}) can identify the correct model.
Due to the normal distribution setting it is not surprisingly that the Dawid-Sebastiani score (Fig. \ref{fig_sim_study_pinson_dm3}) yields the
largest test statistics, among all measures. Still, the related copula Dawid-Sebastiani score (Fig. \ref{fig_sim_study_pinson_dm10}) has only slightly
smaller DM-test statistics, despite the observation on the relative changes of Figure \ref{fig_sim_study_pinson_relch}.
When comparing the energy score, the copula energy score and the CRPS-copula energy score (Fig. \ref{fig_sim_study_pinson_dm1}, Fig. \ref{fig_sim_study_pinson_dm4}, Fig. \ref{fig_sim_study_pinson_dm8}) we observe that the largest test statistics for the energy score, followed by the copula energy score, followed by the CRPS-copula
energy score. Thus, among these three scoring rules the energy score seems to be the most sensitive with respect to the DM-test.
For the variogram score this corresponding ordering slightly changes (Fig. \ref{fig_sim_study_pinson_dm2}, Fig. \ref{fig_sim_study_pinson_dm5}, Fig. \ref{fig_sim_study_pinson_dm9}). So the copula variogram score is more sensitive than the variogram score
which is more sensitive than the CRPS-copula variogram score.
Hence, we focus on the comparison between the energy score, the variogram score and the Dawid-Sebastiani score (Fig. \ref{fig_sim_study_pinson_dm1}, Fig. \ref{fig_sim_study_pinson_dm2}, Fig. \ref{fig_sim_study_pinson_dm3}).
Let us consider a true correlation of 0. In this simulation results, the energy score gives DM-statistics approximately 4 or greater
if for the forecasted correlation $\rho$ it holds $\rho \leq -0.5$ or $\rho \geq 0.4$. In contrast, the more sensitive Dawid-Sebastiani score
requires $\rho \leq -0.4$ or $\rho \geq 0.4$ for a DM-statistic above 4. This somewhat surprising result shows, that the energy score is almost as
powerful as the Dawid-Sebastiani score in discriminating the forecasts.
For the variogram score we receive a similar picture to the energy score, we need $\rho \leq -0.5$ or $\rho \geq 0.6$ for a DM-statistic above 4.
Still, the variogram score has a weaker performance than the energy score.
Basically the same results hold for other correlations as well.
However, as the variogram score is more sensitive to larger correlations than to smaller ones, the results improve for the variogram score when
larger correlations are considered.
\subsection{Sensitivity study II}
This study is based on the study from \cite{scheuerer2015variogram} but it is also to some extend adjusted.
Again, we consider the multivariate normal distribution case, more precisely the bivariate normal distribution with zero mean, variance equal to one and
a correlation of $\rho = \sqrt{2}/2 \approx 0.707$.
Now, we bias this underlying distribution and analyze the change in the score. The major difference to the previous study is that we analyze different changes and control the magnitude of the change, whereas in \cite{scheuerer2015variogram} these changes were chosen ad hoc.
With $\boldsymbol \mu= (0,0)'$ and $\boldsymbol \Sigma(\rho) = \left( \begin{array}{cc}
1 & \rho \\ \rho & 1
\end{array} \right)$ we define the following prediction models:
\begin{enumerate}
\item (true setting): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu, \boldsymbol \Sigma)$ with $\rho = \sqrt{2}/2$
\item (symmetric mean bias): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu + a_1\boldsymbol 1, \boldsymbol \Sigma(\rho))$ with $\rho = \sqrt{2}/2$
\item (asymmetric mean bias): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu + (a_2,-a_2)', \boldsymbol \Sigma(\rho))$ with $\rho = \sqrt{2}/2$
\item (smaller variance): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu, a_3\boldsymbol \Sigma(\rho))$ with $a_3<1$ and $\rho = \sqrt{2}/2$
\item (larger variance): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu, a_4\boldsymbol \Sigma(\rho))$ with $a_4>1$ and $\rho = \sqrt{2}/2$
\item (smaller correlation): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu, \boldsymbol \Sigma(a_5))$ with $a_5<\rho$
\item (larger correlation): $\boldsymbol X \sim {\mathcal{N}}_2( \boldsymbol \mu, \boldsymbol \Sigma(a_6))$ with $a_6>\rho$
\end{enumerate}
We determine $a_i$ so that the likelihoods of all settings except the true coincide. As we have one degree of freedom we choose $a_5=0$. This corresponds to a likelihood reduction with respect
to the true model of $\delta = \frac{1}{2}\log(2)$.
Then, the computation of the remaining $a_i$'s is simple, for the first two ones we even have explicit solutions.
They are given by $a_1 = \sqrt{ \frac{\delta}{2-\sqrt{2}} }$, $a_2 = \sqrt{ \frac{\delta}{2+\sqrt{2}} }$, $a_3\approx 0.48124$, $a_4\approx 2.62729$ and $a_6 \approx 0.89032$.
For the simulation study, we consider in total $M=2^{13}=8192$ paths in the simulated ensemble, a rolling window length of $N=2^{8}=256$ and consider
$L=2^6= 64$ replications of the experiment.
Again, we evaluate the relative change in the scores such as the DM-test test statistic.
The results are visualized in Figures \ref{fig_sim_study_sens_relch} and \ref{fig_sim_study_sens_dm}.
\begin{figure}[h!]
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_sens_relch1}
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\caption{Variogram score \newline (VS)}
\label{fig_sim_study_sens_relch2}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_sens_relch3}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_sens_relch4}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_sens_relch5}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_sens_relch7}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_sens_relch8}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_sens_relch9}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_sens_relch10}
\end{subfigure}
\caption{Box plots of relative change in score $\text{RelCh}$ (see \eqref{eq_relch}) for all considered scoring rules among $L=64$ replications
with an ensemble sample size of $M=2^{13}=8192$ and a rolling window length of $N=2^8=265$.}
\label{fig_sim_study_sens_relch}
\end{figure}
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\caption{Variogram score \newline (VS)}
\label{fig_sim_study_sens_dm2}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_sens_dm3}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_sens_dm4}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_sens_dm5}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_sens_dm7}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_sens_dm8}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_sens_dm9}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_sens_dm10}
\end{subfigure}
\caption{Box plots of DM-test statistics with respect to the true model for all considered scoring rules among $L=64$ replications
with an ensemble sample size of $M=2^{13}=8192$ and a rolling window length of $N=2^8=265$.}
\label{fig_sim_study_sens_dm}
\end{figure}
First, we focus on Figures \ref{fig_sim_study_sens_relch} with the relative changes with respect to the true model.
We observe in Figure \ref{fig_sim_study_sens_relch3} that the constructions seems to be correct. So the Dawid-Sebastiani score reacts with
a relative change sensitivity for all distorted setting of about 0.5.
Moreover, we see a similar picture as reported in \cite{scheuerer2015variogram} for the energy score (Fig. \ref{fig_sim_study_sens_relch1}).
For changes in the mean structure we have a relative high relative changes compared to changes in the variance structure which is still
more distinct than changes in the correlation structure. Especially for large correlations, the relative changes are very small, and appears to be
hardly significantly different from zero. The CRPS-copula energy score (CRPS-CES) shows a very similar picture (Fig. \ref{fig_sim_study_sens_relch4}).
For the variogram score (Fig. \ref{fig_sim_study_sens_relch2}) we observe a better behavior in the relative change concerning the correlations. Still,
we see as well that the variogram score can not identify a shift in the mean. The CRPS-copula variogram score (CRPS-CVS) shows a completely different pattern (Fig. \ref{fig_sim_study_sens_relch5}), most
importantly it can identify all settings correctly. Still, the CRPS-CVS is not strictly proper.
Finally, we observe that the CRPS (Fig. \ref{fig_sim_study_sens_relch7}) can not detect changes in the correlation structure. In contrast, all copula scores (Fig. \ref{fig_sim_study_sens_relch8}, \ref{fig_sim_study_sens_relch9}, \ref{fig_sim_study_sens_relch10}) can not identify changes in the structure of
the marginal distribution.
Now, let us turn to the interesting results of Figures \ref{fig_sim_study_sens_dm}, the DM-statistics.
For the Dawid-Sebastiani score (Fig. \ref{fig_sim_study_sens_dm3}), the average DM-test statistics is for all settings between about 5 and 9, where the smallest DM-statistic appear for the \emph{greater correlation} case.
The energy score (Fig. \ref{fig_sim_study_sens_dm1}) has DM-statistics averages are between about 3 and 8 for all settings. Thus, the energy score can identify those models correctly even though the power is noticeable smaller than for the Dawid-Sebastiani score.
For the variogram score (Fig. \ref{fig_sim_study_sens_dm2}) we observe that for changes in mean a DM-statistic that is not significantly different from zero.
Still, for the remaining models we receive DM-statistics between 3 and 9. Thus, similarly to the energy score the remaining settings can be identified.
The CRPS-copula energy score (CRPS-CES) shows similar pattern to the energy score (Fig. \ref{fig_sim_study_sens_dm4}), but all with smaller average DM-statistics in all settings, they vary only between about 2 and 5. Thus, it is clearly less powerful than the energy score in this simulation study setting.
Also the CRPS-copula variogram score (CRPS-CVS) does not perform great (Fig. \ref{fig_sim_study_sens_dm4}).
In two setting (asymmetric mean shift, smaller variance) average DM-statistics are only about 1. Thus, for standard significance levels (e.g. $5\%$) we could not
reject the null hypothesis.
For the marginal settings, the CRPS (Fig. \ref{fig_sim_study_sens_dm7}) has average DM-statistics between about 4 and 7 which seems to be a similar level
as the energy score. Interestingly, the copula energy score (CES) shows similar
average DM-statistics in the correlation settings to the energy score (Fig. \ref{fig_sim_study_sens_dm8}). It seems that
the energy score combines to some extent the marginal discrimination ability of the CRPS with the dependency discrimination ability of the copula energy score.
However, among the three copula scores the copula energy score performs worst with respect to the discrepancy measurement in the correlation structure.
\subsection{Random peak study}
In this section we discuss in detail the application of the energy score and the Diebold-Mariano test to a toy example.
This example is motivated by \cite{haben2014new} with application to smart-meter electricity load data.
It illustrates nicely proper multivariate forecasting evaluation as it is an example where simple measures usually fail.
We assume that $\boldsymbol X_{i}$ is an $H$-dimensional process. It is defined by:
$\boldsymbol X_{i} = \boldsymbol Y_{i} + Q \boldsymbol Z_{i}$ with $\boldsymbol Y_{i} \sim {\mathcal{N}}_H(\boldsymbol 0, \boldsymbol I)$ and
$\boldsymbol Z_{i} \sim {\mathcal{U}}( \{\boldsymbol e_1,\ldots,\boldsymbol e_H\})$ with ${\mathcal{N}}_H$ as $H$-dimensional normal distribution, ${\mathcal{U}}({\mathcal{A}})$ as uniform distribution
on ${\mathcal{A}}$, a constant $Q$ such as unit vectors $\boldsymbol e_1=(1, 0, 0, \ldots)'$, $\ldots$ and $\boldsymbol e_H=(0, \ldots, 0, 0 , 1)'$ and identity matrix $\boldsymbol I$. Each
$\boldsymbol Y_{i}$ and $\boldsymbol Z_{i}$ are independent from each other.
So $\boldsymbol X_{i}$ is a $H$-dimensional standard normal distribution where at one uniformly chosen coordinate we add an additional
constant $Q=5$. This can be interpreted as a peak that occurs always at one of the $H$ dimensions.
An interpretation of this experiment based on \cite{haben2014new} could be that we know that a person will have a shower every morning, say either at
6:00, 7:00 or 8:00. In this $H=3$-dimensional example we expect a peak at exactly one of these hours but non at the others.
Obviously, with standard point forecasting measures MAE (mean absolute error) or RMSE (root mean square error)
the \emph{flat forecast} (having never a shower) outperforms
the forecast \emph{fixed peak forecast} which stats that a person has a shower always at e.g. 7:00. This seems to be counterintuitive as the \emph{fixed peak forecast} seems
to be a better forecast than the \emph{flat forecast} as at least the peak is detected, even though not the correct one.
Back to the simulation study: Next to the perfect model
we will consider seven other models that could serve as (more or less) plausible forecast model.
They are defined by:
\begin{enumerate}
\item (true)
$\boldsymbol X_{i} = \boldsymbol Y_{i} + Q \boldsymbol Z_{i}$ with $\boldsymbol Y_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\boldsymbol 0, \boldsymbol I)$ and
$\boldsymbol Z_{i} \stackrel{\text{iid}}{\sim} {\mathcal{U}}( \{\boldsymbol e_1, \ldots,\boldsymbol e_H\})$
\item (average mean)
$\boldsymbol X_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\mu \boldsymbol 1, \boldsymbol I)$ with $\mu = \frac{Q}{H} $
\item (zero mean)
$\boldsymbol X_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\boldsymbol 0, \boldsymbol I)$
\item (fixed peak)
$\boldsymbol X_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\boldsymbol \mu, \boldsymbol I)$ with $\boldsymbol \mu= (Q,0, \ldots ,0)'$
\item (rolling peak)
$\boldsymbol X_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\boldsymbol \mu_i, \boldsymbol I)$ with
$\boldsymbol \mu_i = \boldsymbol e_{1 + (i-1) \text{mod} H }$ with as unit vector for the $i$'th coordinate.
\item (mixture normal with same marginals)
$\boldsymbol X_{i} = (X_{i,1}, \ldots ,X_{i,H})'$ with \newline
$X_{i,j}\stackrel{\text{iid}}{\sim} \begin{cases}
{\mathcal{N}}_1(0,1)& U_j \leq \frac{H-1}{H} \\
{\mathcal{N}}_1(Q,1) & U_j > \frac{H-1}{H}
\end{cases}$
where $U_j \stackrel{\text{iid}}{\sim} {\mathcal{U}}( [0,1] )$.
\item (shifted mean)
$\boldsymbol X_{i} = \boldsymbol Y_{i} + Q \boldsymbol Z_{i}$ with $\boldsymbol Y_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(Q/H \boldsymbol 1, \boldsymbol I)$ and
$\boldsymbol Z_{i} \stackrel{\text{iid}}{\sim} {\mathcal{U}}( \{\boldsymbol e_1,\ldots,\boldsymbol e_H\})$
\item (normal with true mean and covariance)
$\boldsymbol X_{i} \stackrel{\text{iid}}{\sim} {\mathcal{N}}_H(\boldsymbol \mu , \boldsymbol \Sigma)$ with $\boldsymbol \mu = \frac{Q}{H} \boldsymbol 1$ and $\boldsymbol \Sigma =
(H+ Q^2)/H \boldsymbol I -Q^2/H^2 \boldsymbol 1 \boldsymbol 1' $
\end{enumerate}
The second and third model are two multivariate normal models with a constant mean, around the overall mean $\mu$ for model 2 and around $0$ for model 3.
In the contest of peak or spike forecasting model $3$ is the above mentioned \emph{flat forecast}.
But both model 2 and 3 miss completely the occurring peak.
Model 4 instead, uses a multivariate normal distribution as well and predicts a peak of size $Q$. But this peak is predicted to occur always
in the first coordinate and corresponds to the \emph{fixed peak forecast} example.
Similarly does model 5, it always predicts a peak of size $Q$.
But the time of occurrence is moves with $i$. For $i=1$ it is in the first dimension, with $i=2$ in the second, up to $i=H$ in the $H$'s component
and with $i=H+1$ in the first again. So the peak is predicted to move cyclic (esp. deterministic) within the $H$ dimensions.
The next considered model 6 is a mixture normal model. Every single random variable $X_{i,j}$
is with probability $1/H$ a normal distribution with mean $Q$ (which corresponds to the peak) and with probability $(H-1)/H$ it has a zero mean (no peak).
As a consequence the model can have peaks as well. But the number of peaks in $\boldsymbol X_{t,i}$ can vary between $0$ and $H$.
If $H=3$, there are cases
with no peak (about $30\%$ of all cases), and cases with three peaks (about $4\%$ of all cases). Still, the peaks are randomly spread around the $H$ dimensions.
The seventh model corresponds to the true one with a mean shift. And the latter one is a normal distribution, with exactly the same first and second moment as
the true distribution. Hence, we expect the Dawid-Sebastiani score to fail here.
We conduct the random peak study for the $H=3$-dimensional case with a peak size of $Q=5$.
We carry out $L=2^{6}=64$ experiment replications with an ensemble sample size of $M=2^{14}=16384$ and a rolling window length of $N=2^5=32$.
As we pointed out, that the relative change is not suitable for evaluation, we only report the statistics of the Diebold-Mariano, see Figure \ref{fig_sim_study_peak_dm}.
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_peak_dm1}
\end{subfigure}
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\label{fig_sim_study_peak_dm2}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_peak_dm3}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_peak_dm4}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_peak_dm5}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_peak_dm7}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_peak_dm8}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_peak_dm9}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_peak_dm10}
\end{subfigure}
\caption{Box plots of DM-test statistics with respect to the true model for all considered scoring rules for the random peak study
with dimension $H=3$ and peak size $Q=5$ among $L=2^{6}=64$ replications
with an ensemble sample size of $M=2^{14}=16384$ and a rolling window length of $N=2^5=32$.}
\label{fig_sim_study_peak_dm}
\end{figure}
We observe that the variogram score and Dawid-Sebastiani score (Fig. \ref{fig_sim_study_peak_dm2}, \ref{fig_sim_study_peak_dm3})
can not identify the true model in all settings. This, is not surprising, as they are not strictly proper.
In contrast, the energy score (Fig. \ref{fig_sim_study_peak_dm1}) has an average DM-statistic in all settings between about 4 and 14. Thus, again it can clearly identify the true model.
Also the strictly proper CRPS-Copula energy score (CRPS-CES) can identify all models (Fig. \ref{fig_sim_study_peak_dm4}), even though the average DM-statistics
vary in the settings between 2 and 6. Similarly, the CRPS-Copula variogram score (CRPS-CVS) can identify all settings, even though it is not strictly proper.
Obviously, the CRPS (Fig. \ref{fig_sim_study_peak_dm7}) fails in separating the mixture normal model from the true one, as they have the same marginal distribution.
Interestingly, the copula energy score (Fig. \ref{fig_sim_study_peak_dm8}) is very sensitive to exactly this normal mixture setting. This is at least an indication why
it might be a good idea to combine the copula energy score with the CRPS. Still, overall copula scores show rather small DM-statistics.
\subsection{The role of the ensemble size and forecasting horizon.}
In this simulation study we want to highlight the importance of the ensemble sample size $M$ and the forecasting horizon $H$ when reporting multivariate forecasts. First we focus on the ensemble sample size $M$.
It is clear that for $M\to \infty$ the forecasting distribution is arbitrarily well covered. However, in real applications we face the problem that
can only simulate finite amount of time, and hope that for large $M$ the underlying distribution is approximated sufficiently well. Now, we want to study the impact
of the sample size.
Therefore, we consider again the framework of the peak study of the previous subsection but vary across the ensemble sample size $M$.
This grid is chosen to be ${\mathcal{M}}=\{2^i| i \in \{4,\ldots, 14\}\} = \{2^4, 2^5, \ldots, 2^{14}\}= \{16,32,\ldots, 16384\}$.
We conduct experiments with $L=2^7=128$ replication and a rolling window length of $N=2^4=16$.
As in the previous section we chose the dimension $H=3$ and a peak size of $Q=5$.
The results of the average DM-statistics across the $L$ replications are given in Figure \ref{fig_sim_study_peak_M1}.
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_peak_M1_dm1}
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\caption{Variogram score \newline (VS)}
\label{fig_sim_study_peak_M1_dm2}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_peak_M1_dm3}
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\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_peak_M1_dm4}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_peak_M1_dm5}
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}
\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_peak_M1_dm7}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_peak_M1_dm8}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_peak_M1_dm9}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_peak_M1_dm10}
\end{subfigure}
\caption{Average of DM-test statistics across $L=2^8=128$ experiment replications with respect to the true model for all considered scoring rules for the random peak study
with dimension $H=3$, peak size $Q=5$ and a rolling window length of $N=2^4=16$ on a ensemble size $M$ on the grid ${\mathcal{M}}= \{16,32,\ldots, 16384\}$.}
\label{fig_sim_study_peak_M1}
\end{figure}
There, we observe that small $M$ values are not sufficient to report a stable forecast. When focusing on the energy score (Fig. \ref{fig_sim_study_peak_M1_dm1}) we notice,
that for most settings (all except \emph{fixed peak} and \emph{rolling peak}) an ensemble size of $M=16$ is by far not sufficient to
receive approximately maximal DM-statistics. For three settings (\emph{average mean}, \emph{zero mean} and \emph{shifted mean}) we require
about $2^{10}=1024$ paths in the ensemble to reach stability in the DM-statistic. For the remaining tricky situation s
(\emph{mixture normal} and \emph{normal}) we observe that $M=2^{14}=163846$ might be even not sufficient to reach the theoretical maximum.
For the variogram score (Fig. \ref{fig_sim_study_peak_M1_dm2}) the results seem to be quite robust, even for very small ensemble sizes, like $M=16$ or $M=32$, they work relatively well.
The Dawid-Sebastiani score (Fig. \ref{fig_sim_study_peak_M1_dm3}) requires about $M=2^9=512$ paths in the ensemble seem to be sufficiently close to the optimum
in the given setting. Also for the other scores we observe that larger sample sizes $M$ help to increase the DM-statistics, even for the CRPS which is only based on univariate scoring rules.
However, the copula based scoring rules have relatively small average test statistics for all cases.
In literature the ensemble sample size $M$ is sometimes not reported.
If so, the considered ensemble size is usually surprisingly small, e.g. $M=8$ and $M=19$ in \cite{gneiting2008assessing}, $M=51$ in \cite{pinson2012adaptive} and \cite{junk2014comparison}.
The most common reason is that for multivariate forecasts in the meteorologic applications, only a few meteorologic ensemble forecasts based on a physical/meteorologic weather model are considered.
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\caption{Energy score \newline (ES)}
\label{fig_sim_study_peak_M2_dm1}
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\caption{Variogram score \newline (VS)}
\label{fig_sim_study_peak_M2_dm2}
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\caption{Dawid-Sebastiani score \newline (DSS)}
\label{fig_sim_study_peak_M2_dm3}
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\end{scope}
\end{tikzpicture}
}
\caption{CRPS-Copula Energy score \newline (CRPS-CES)}
\label{fig_sim_study_peak_M2_dm4}
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\caption{CRPS-Copula Variogram score (CRPS-CVS)}
\label{fig_sim_study_peak_M2_dm5}
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\caption{Continuous ranked probability score (CRPS)}
\label{fig_sim_study_peak_M2_dm7}
\end{subfigure}
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\caption{Copula energy score \newline (CES)}
\label{fig_sim_study_peak_M2_dm8}
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\caption{Copula variogram score \newline (CVS)}
\label{fig_sim_study_peak_M2_dm9}
\end{subfigure}
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\caption{Copula Dawid-Sebastiani score (CDSS)}
\label{fig_sim_study_peak_M2_dm10}
\end{subfigure}
\caption{Average of DM-test statistics across $L=2^8=256$ experiment replications with respect to the true model for all considered scoring rules for the random peak study
with dimension $H=9$, peak size $Q=5$ and a rolling window length of $N=2^4=16$ on a ensemble size grid ${\mathcal{M}}= \{16,32,\ldots, 4096\}$.}
\label{fig_sim_study_peak_M2}
\end{figure}
Finally, we want to vary the forecasting horizon. Instead of an $H=3$-dimensional we consider now a problem
of the square size. Thus, we assume $H=9$. The average DM-statistics across the $L$ experiments are given in Figure \ref{fig_sim_study_peak_M2}.
There we see that for almost all scores the DM-statistic descrease compared to the 3-dimensional case \ref{fig_sim_study_peak_M2}. The CRPS for instance is an
exception, here the statistics increase in some cases (Fig. \ref{fig_sim_study_peak_M2_dm7}). Likely, because there are more marginals that can be evaluated.
For larger sample sizes, only the energy score (Fig. \ref{fig_sim_study_peak_M2_dm1}) can identify the true model significantly.
In constrast, the strictly proper CRPS-CES (Fig. \ref{fig_sim_study_peak_M2_dm4}) fails to seperate the true and the mixture normal model, as the copula energy score (CES) does as well.
The reason is likely that the ensemble size is too small such the underlying $9$-dimensional space is explored well. More on this is explained in the next subsection.
\subsection{Discussion}
Let us shortly discuss the simulation results.
As pointed out before, the relative change in a score with respect to the true (or optimal) model is not a good criterion for measuring
the sensitivity of a score. In contrast, the DM-test is a suitable tool for evaluating the sensitivity, as it makes a statement
if a non-optimal model can be identified as non-optimal with respect to the true one.
With respect to the DM-test have seen that in all simulation studies (with multivariate normal design and the peak study) the energy score performs clearly best. It can identify the true model in all studies of article with relatively strong power.
When the authors received this results they somehow wondered as a couple of researchers
(e.g. \cite{scheuerer2015variogram, spath2015time, gamakumara2018probabilisitic}) mentioning to weak discrimination ability of the energy score. But, they are all referring the working paper simulation study of \cite{pinson2013discrimination} which uses the relative change in the score to conclude.
Therefore, it is worth studying again a bit deeper the properties of the energy score.
Let $\boldsymbol Y$ be an $H$-dimensional random variable. The energy score results as the one-sided version of the
energy distance $d_{E}$ (see \cite{szekely2013energy}). The energy distance is defined as
\begin{equation}
d_{E}(\boldsymbol X,\boldsymbol Y) =
{\mathbb E} \|\boldsymbol X-\boldsymbol Y\|_2^{\beta} - \frac{1}{2} {\mathbb E} \|\boldsymbol X-\widetilde{\boldsymbol X}\|_2^{\beta} -\frac{1}{2} {\mathbb E} \|\boldsymbol Y-\widetilde{\boldsymbol Y}\|_2^{\beta}
\label{eq_energy_dist}
\end{equation}
where $\widetilde{\boldsymbol X}$ and $\widetilde{\boldsymbol Y}$ are iid-copies of $\boldsymbol X$ and $\boldsymbol Y$, and $\beta \in (0,1)$. Again, $\beta=1$ gives the standard case which results
in the 1-dimension the Cramer-distance $d_{C}$:
\begin{equation}
d_{C}(X,Y) =
\int_{\mathbb R} | F_{X}(z)-F_{Y}(z) |^2 d \, z =
{\mathbb E}|X-Y| - \frac{1}{2}{\mathbb E}|X-\widetilde{X}| -\frac{1}{2} {\mathbb E}|Y-\widetilde{Y}|
\label{eq_cramer_repr}
\end{equation}
If we replace $Y$ and $\widetilde{Y}$ by a constant $y$ we receive the popular CRPS.
The energy distance allows for many statistical applications, \cite{szekely2013energy, rizzo2016energy}. It allows to construct a
general (non-parametric) test for equality of two multivariate distributions. When replacing one distribution by a certain family of distributions
it allows the derivation of test for various parametric distributions. For instance, the resulting test
for multivariate normality has a better empirical power than standard alternatives \cite{szekely2005new}.
However, another important application of the energy distance is the construction of so called \emph{distance correlations}.
This is a general concept to characterize multivariate dependence, but not just linear dependency as measured by standard Pearson correlation.
Consequently it allows to construct the so called \emph{energy test of independence} which tests for multivariate independence.
To understand the energy distance and its power even better it is useful to highlight a representation of the energy distance in \eqref{eq_energy_dist} using characteristic function.
The energy distance $d_{E}$ can be represented as a weighted $L^2$-distance of characteristic the characteristic functions $\boldsymbol \varphi_{\boldsymbol X}$ and $\boldsymbol \varphi_{\boldsymbol Y}$ of $\boldsymbol X$ and $\boldsymbol Y$
with $\boldsymbol \varphi_{\boldsymbol X}(\boldsymbol z) = {\mathbb E}( e^{i\boldsymbol z'\boldsymbol X})$ and $\boldsymbol \varphi_{\boldsymbol Y}(\boldsymbol z) = {\mathbb E}( e^{i\boldsymbol z'\boldsymbol Y})$:
\begin{equation}
d_{E}(\boldsymbol X,\boldsymbol Y) = \frac{\pi^{\frac{H+1}{2}}}{\Gamma(\frac{H+1}{2})} \int_{{\mathbb R}^H } \frac{|\boldsymbol \varphi_{\boldsymbol X}(\boldsymbol z) - \boldsymbol \varphi_{\boldsymbol Y}(\boldsymbol z)|^2 }{\|\boldsymbol z\|_2^{H+\beta}} \, d \boldsymbol z
\label{eq_energy_dist_characteristic}
\end{equation}
The weight function $\|\boldsymbol z\|_2^{-(H+\beta)}$ is crucial.
It is the only choice of any continuous function $g$ such that
the weighted $L^2$-distance
$ C \int_{{\mathbb R}^H } g(\boldsymbol z) |\boldsymbol \varphi_{\boldsymbol X}(\boldsymbol z) - \boldsymbol \varphi_{\boldsymbol Y}(\boldsymbol z)|^2 \, d \boldsymbol z $
between the characteristic functions $\boldsymbol \varphi_{\boldsymbol X}$ and $\boldsymbol \varphi_{\boldsymbol X}$ is scale equivariant and rotational invariant. Thus, it is suitable for
the tests described above.
Of course, this does not directly explain the high empirical power of the energy score, but it gives an intuition why it works well.
Moreover, we know that for any dimension $H$ $\boldsymbol X$ is uniquely determined by its complex valued
characteristic function $\boldsymbol \varphi_{\boldsymbol X}$. Thus, in \eqref{eq_energy_dist_characteristic} we are measuring distances in the complex plane which
is isomorphic to the 2-dimensional space ${\mathbb R}^2$. Hence, if $H$ is greater than $2$, the energy distance measures de facto the discrepancy in a in a lower dimensional space. Consequently, we should prefer the energy distance especially for long forecasting horizons.
Note that the energy score should not necessarily be the only score to be considered. Of course, any proper scoring rule can be applied. A perfect forecasting model performs well with respect to every proper scoring rule. As any measure is designed for a certain purpose this might help to detect problems in the forecasting model. For instance, if a model scores well in the energy score but the opposite in the Dawid-Sebastiani score we have a
strong indication that either the first or second moment of our forecast is not well captured by the corresponding forecasting model.
Here, it also worth mentioning, that the forecaster should not restrict to $H$-dimensional scores. Instead the forecaster should evaluate scores on lower
marginal distributions as well. The an important special case, this is the evaluation on the $H$ marginal distributions, e.g. by the
CRPS or log-score. This gives an indication in which forecasting horizon which model suffers problems concerning the marginal distribution. This, seems to be quite common practice in some fields
of application, see e.g. \cite{hong2016probabilistic}.
Similarly, we can evaluate any score on low dimensional marginals, especially bivariate distributions. For instance, we can look at the bivariate
copula energy score for the dimension 1 and 2, then 2 and 3, etc.. By doing that, it gives us information about the capturing of the pairwise
dependency structure along the forecasting path. Thus, we can study how the predictive performance concerning the pairwise
dependency structure changes across the forecasting horizon.
The described procedure is also applied in the real data example of the next section.
\section{Illustrative real data example} \label{sec:realdata}
For illustration purpose we consider the popular classic Box and Jenkins airline data, which is denoted \texttt{AirPassengers} in \texttt{R}.
It contains monthly totals of international airline passengers from 1949 to 1960. Thus, there are in total 144 observations.
We apply a rolling window forecasting study with $T=5\times12 = 70$ as in-sample data length. We choose a forecasting horizon on $H=12$ which corresponds to a year
and shift the rolling window always by $4$ which corresponds to a quarter. Hence, we have in total $N=19$ forecasting windows.
The forecasting study design is illustrated in Figure \ref{fig_sample_air}.
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\caption{Illustration of the rolling window forecasting, for rolling window $1$, $2$ and $N=19$ with a small illustrative ensemble forecast
with forecasting horizon $H=12$ and ensemble sample size $M=8$ of the AR(13) model \eqref{item_ar13} shown below.}
\label{fig_sample_air}
\end{figure}
We consider 9 very simple forecasting models, based on 3 AR models with 3 different error terms.
\begin{enumerate}
\item AR(12): $Y_t= \phi_0 + \sum_{k=1}^{12} \phi_k Y_{t-k} + {\varepsilon}_t$ with ${\varepsilon}_t$ iid and ${\mathbb E}({\varepsilon}_t)=0$. \label{item_ar12}
\item AR(13): $Y_t= \phi_0 + \sum_{k=1}^{13} \phi_k Y_{t-k} + {\varepsilon}_t$ with ${\varepsilon}_t$ iid and ${\mathbb E}({\varepsilon}_t)=0$. \label{item_ar13}
\item AR($p$): $Y_t= \phi_0 + \sum_{k=1}^p \phi_k Y_{t-k} + {\varepsilon}_t$ with ${\varepsilon}_t$ iid, ${\mathbb E}({\varepsilon}_t)=0$ and $p\in \{1,\ldots, T/2\}$ such that the corresponding Akaike information criterion (AIC) is minimized. \label{item_arp}
\item AR(12) as in \ref{item_ar12}. but with comonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol M_2$)
\item AR(13) as in \ref{item_ar13}. but with comonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol M_2$)
\item AR($p$) as in \ref{item_arp}. but with comonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol M_2$)
\item AR(12) as in \ref{item_ar12}. but with countermonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol W_2$)
\item AR(13) as in \ref{item_ar13}. but with countermonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol W_2$)
\item AR($p$) as in \ref{item_arp}. but with countermonotone residuals (i.e. $(\widehat{{\varepsilon}}_t,\widehat{{\varepsilon}}_{t+1})$ have the copula $\boldsymbol W_2$)
\end{enumerate}
All models are estimated by solving the Yule-Walker equations which yields stationary estimated process. The ensemble simulations are generated using residual based bootstrap. For the reporting of the forecasts we consider $M=2^{16}=65536$ simulated paths.
Note that the comonotone and countermonotone models have exactly the same predicted marginal distribution, but a
crucially different dependency structure. A small example of simulated paths is given in Figure \ref{fig_sample_monotone} simulated for
the AR(13) in \ref{item_ar13}.
We evaluate all nine evaluation measures as in the simulation studies.
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\caption{Illustration of standard (left), comonotone (center) and countermonotone (right) model simulations for the AR(13) with $M=8$ paths
for the last experiment ($N=19$).}
\label{fig_sample_monotone}
\end{figure}
In Table \ref{tab_air_scores} the sample means (see eqn. \eqref{eq_score_average})
of all nine scores and nine models across all $N=19$ paths are given\footnote{Note that due to collinearity the copula-Dawid-Sebastiani score (CDSS)
can not be computed for the comonotone and countermonotone AR models.}.
DM-statistics and the corresponding p-values for the energy score is given in are Table \ref{tab_DM_ES}. The corresponding tables for
the remaining eight scores are provided in the Appendix. Additionally, to the considered scores for $H$-dimensional forecasts we compute
the selected low-dimensional scores along the forecasting horizon are given in Figure \ref{fig_sample_marg}.
\setlength{\tabcolsep}{2pt}
\begin{table}[ht]
\centering
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\begin{tabular}{rccccccccc}
\hline
\footnotesize Score$\backslash$Model & \footnotesize AR(12) & \footnotesize AR(13) & \footnotesize AR(p) & \footnotesize AR(12)-M & \footnotesize AR(13)-M & \footnotesize AR(p)-M & \footnotesize AR(12)-W & \footnotesize AR(13)-W & \footnotesize AR(p)-W \\
\hline
ES & \cellcolor[rgb]{0.6,1,0.6} 120.8 & \cellcolor[rgb]{1,0.807,0.6} 137.5 & \cellcolor[rgb]{1,0.845,0.6} 134.9 & \cellcolor[rgb]{1,0.522,0.522} 197.3 & \cellcolor[rgb]{1,0.524,0.524} 196.2 & \cellcolor[rgb]{1,0.537,0.537} 190.9 & \cellcolor[rgb]{1,0.508,0.508} 203.2 & \cellcolor[rgb]{1,0.5,0.5} 206.6 & \cellcolor[rgb]{1,0.513,0.513} 201.0 \\
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DSS & \cellcolor[rgb]{0.6,1,0.6} 95.92 & \cellcolor[rgb]{0.6,1,0.6} 90.99 & \cellcolor[rgb]{0.6,1,0.6} 91.78 & \cellcolor[rgb]{1,0.563,0.563} 683635 & \cellcolor[rgb]{1,0.684,0.6} 372563 & \cellcolor[rgb]{1,0.689,0.6} 364726 & \cellcolor[rgb]{1,0.5,0.5} 995252 & \cellcolor[rgb]{1,0.532,0.532} 838370 & \cellcolor[rgb]{1,0.547,0.547} 762531 \\
CRPS-CES & \cellcolor[rgb]{0.6,1,0.6} 3.228 & \cellcolor[rgb]{1,0.937,0.6} 3.887 & \cellcolor[rgb]{1,0.947,0.6} 3.828 & \cellcolor[rgb]{1,0.553,0.553} 8.617 & \cellcolor[rgb]{1,0.508,0.508} 10.240 & \cellcolor[rgb]{1,0.513,0.513} 10.045 & \cellcolor[rgb]{1,0.547,0.547} 8.818 & \cellcolor[rgb]{1,0.5,0.5} 10.537 & \cellcolor[rgb]{1,0.505,0.505} 10.365 \\
CRPS-CVS & \cellcolor[rgb]{0.6,1,0.6} 0.3346 & \cellcolor[rgb]{1,0.806,0.6} 0.4638 & \cellcolor[rgb]{1,0.895,0.6} 0.4166 & \cellcolor[rgb]{1,0.63,0.6} 0.6357 & \cellcolor[rgb]{1,0.568,0.568} 0.7712 & \cellcolor[rgb]{1,0.582,0.582} 0.7248 & \cellcolor[rgb]{1,0.559,0.559} 0.7989 & \cellcolor[rgb]{1,0.5,0.5} 0.9954 & \cellcolor[rgb]{1,0.515,0.515} 0.9464 \\
CRPS & \cellcolor[rgb]{0.6,1,0.6} 28.28 & \cellcolor[rgb]{1,0.5,0.5} 33.94 & \cellcolor[rgb]{1,0.521,0.521} 33.36 & \cellcolor[rgb]{0.67,1,0.6} 28.32 & \cellcolor[rgb]{1,0.501,0.501} 33.90 & \cellcolor[rgb]{1,0.521,0.521} 33.36 & \cellcolor[rgb]{0.623,1,0.6} 28.29 & \cellcolor[rgb]{1,0.501,0.501} 33.92 & \cellcolor[rgb]{1,0.521,0.521} 33.35 \\
CES & \cellcolor[rgb]{0.6,1,0.6} 0.1142 & \cellcolor[rgb]{0.613,1,0.6} 0.1144 & \cellcolor[rgb]{0.622,1,0.6} 0.1146 & \cellcolor[rgb]{1,0.507,0.507} 0.3044 & \cellcolor[rgb]{1,0.51,0.51} 0.3020 & \cellcolor[rgb]{1,0.511,0.511} 0.3010 & \cellcolor[rgb]{1,0.5,0.5} 0.3118 & \cellcolor[rgb]{1,0.501,0.501} 0.3105 & \cellcolor[rgb]{1,0.501,0.501} 0.3106 \\
CVS & \cellcolor[rgb]{0.6,1,0.6} 0.01184 & \cellcolor[rgb]{1,0.919,0.6} 0.01367 & \cellcolor[rgb]{0.96,1,0.6} 0.01247 & \cellcolor[rgb]{1,0.579,0.579} 0.02247 & \cellcolor[rgb]{1,0.575,0.575} 0.02276 & \cellcolor[rgb]{1,0.587,0.587} 0.02171 & \cellcolor[rgb]{1,0.512,0.512} 0.02826 & \cellcolor[rgb]{1,0.5,0.5} 0.02935 & \cellcolor[rgb]{1,0.511,0.511} 0.02836 \\
CDSS & \cellcolor[rgb]{0.6,1,0.6} -26.06 & \cellcolor[rgb]{1,0.567,0.567} -24.54 & \cellcolor[rgb]{1,0.5,0.5} -23.78 & \cellcolor[rgb]{.9,.9,.9} - & \cellcolor[rgb]{.9,.9,.9} - & \cellcolor[rgb]{.9,.9,.9} - & \cellcolor[rgb]{.9,.9,.9} - & \cellcolor[rgb]{.9,.9,.9} - & \cellcolor[rgb]{.9,.9,.9} - \\
\hline
\end{tabular}}
\caption{Score averages $\ov{\text{SC}}$ across the $N=19$ out-of-sample windows for the considered scores and models.
-M indecates models with comonotone residuals, -W for countermonotone residuals.}
\label{tab_air_scores}
\end{table}
\begin{table}[ht]
\centering
\begin{tabular}{rlllllllll}
\hline
& \footnotesize AR(12) & \footnotesize AR(13) & \footnotesize AR(p) & \footnotesize AR(12)-M & \footnotesize AR(13)-M & \footnotesize AR(p)-M & \footnotesize AR(12)-W & \footnotesize AR(13)-W & \footnotesize AR(p)-W \\
\hline
AR(12) & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{0.601,1,0.6} $\underset{[<0.001]}{ -4.04}$ & \cellcolor[rgb]{0.73,1,0.6} $\underset{[0.005]}{ -2.57}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -26.53}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -19.57}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -13.39}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -30.02}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -17.90}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -13.02}$ \\
AR(13) & \cellcolor[rgb]{1,0.601,0.6} $\underset{[>0.999]}{ 4.04}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{1,0.923,0.6} $\underset{[0.766]}{ 0.73}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -11.15}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -31.58}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -13.67}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -13.57}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -35.14}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -14.51}$ \\
AR(p) & \cellcolor[rgb]{1,0.73,0.6} $\underset{[0.995]}{ 2.57}$ & \cellcolor[rgb]{0.923,1,0.6} $\underset{[0.234]}{ -0.73}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -9.12}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -14.06}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -29.95}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -10.74}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -16.87}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -31.22}$ \\
AR(12)-M & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 26.53}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 11.15}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 9.12}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{1,0.971,0.6} $\underset{[0.602]}{ 0.26}$ & \cellcolor[rgb]{1,0.896,0.6} $\underset{[0.860]}{ 1.08}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -7.18}$ & \cellcolor[rgb]{0.812,1,0.6} $\underset{[0.031]}{ -1.86}$ & \cellcolor[rgb]{0.94,1,0.6} $\underset{[0.292]}{ -0.55}$ \\
AR(13)-M & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 19.57}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 31.58}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 14.06}$ & \cellcolor[rgb]{0.971,1,0.6} $\underset{[0.398]}{ -0.26}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{1,0.879,0.6} $\underset{[0.918]}{ 1.39}$ & \cellcolor[rgb]{0.788,1,0.6} $\underset{[0.024]}{ -1.98}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -9.32}$ & \cellcolor[rgb]{0.896,1,0.6} $\underset{[0.138]}{ -1.09}$ \\
AR(p)-M & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 13.39}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 13.67}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 29.95}$ & \cellcolor[rgb]{0.896,1,0.6} $\underset{[0.140]}{ -1.08}$ & \cellcolor[rgb]{0.879,1,0.6} $\underset{[0.082]}{ -1.39}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{0.749,1,0.6} $\underset{[0.011]}{ -2.28}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -4.18}$ & \cellcolor[rgb]{0.6,1,0.6} $\underset{[<0.001]}{ -9.51}$ \\
AR(12)-W & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 30.02}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 13.57}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 10.74}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 7.18}$ & \cellcolor[rgb]{1,0.788,0.6} $\underset{[0.976]}{ 1.98}$ & \cellcolor[rgb]{1,0.749,0.6} $\underset{[0.989]}{ 2.28}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{0.919,1,0.6} $\underset{[0.219]}{ -0.78}$ & \cellcolor[rgb]{1,0.959,0.6} $\underset{[0.642]}{ 0.36}$ \\
AR(13)-W & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 17.90}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 35.14}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 16.87}$ & \cellcolor[rgb]{1,0.812,0.6} $\underset{[0.969]}{ 1.86}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 9.32}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 4.18}$ & \cellcolor[rgb]{1,0.919,0.6} $\underset{[0.781]}{ 0.78}$ & \cellcolor[rgb]{0.6,0.6,0.6} & \cellcolor[rgb]{1,0.879,0.6} $\underset{[0.918]}{ 1.39}$ \\
AR(p)-W & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 13.02}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 14.51}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 31.22}$ & \cellcolor[rgb]{1,0.94,0.6} $\underset{[0.708]}{ 0.55}$ & \cellcolor[rgb]{1,0.896,0.6} $\underset{[0.862]}{ 1.09}$ & \cellcolor[rgb]{1,0.6,0.6} $\underset{[>0.999]}{ 9.51}$ & \cellcolor[rgb]{0.959,1,0.6} $\underset{[0.358]}{ -0.36}$ & \cellcolor[rgb]{0.879,1,0.6} $\underset{[0.082]}{ -1.39}$ & \cellcolor[rgb]{0.6,0.6,0.6} \\
\hline
\end{tabular}
\caption{DM-test statistics with corresponding p-value given in squared brackets for the energy score (ES).}
\label{tab_DM_ES}
\end{table}
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(205.76,308.55) --
(202.09,312.22) --
(198.43,308.55) --
cycle;
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}
\caption{1-dim. energy score \newline (CRPS)}
\label{fig_sample_marg_dim1}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{2-dim. copula energy score \newline (CES)}
\label{fig_sample_marg_dim2}
\end{subfigure}
\begin{subfigure}[b]{0.32\textwidth}
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\caption{3-dim. copula energy score \newline (CES)}
\label{fig_sample_marg_dim3}
\end{subfigure}
\caption{Low-dimensional scores across the forecasting horizon for all considered models.}
\label{fig_sample_marg}
\end{figure}
In Table \ref{tab_air_scores} we observe that the energy score is the smallest for the AR(12).
Even though for the AR(13) and AR(p) the energy score has a magnitude their performance is significantly worse than the AR(12) with
respect to the energy score, see Table \ref{tab_DM_VS}.
Moreover, we see that the three AR models with comonotone residuals have similar scores, as well as the three models with countermonotone residuals but with larger scores. It shows us that the energy score takes the dependency structure into account and favors a comonotone
to a countermonotone. This makes intuitively sense as the time series data tends to be positively correlated.
However, when looking at the variogram and Dawid-Sebastiani score we get a similar picture.
As both scores are proper and
the corresponding DM-statistics (Tab.
\ref{tab_DM_VS}
and \ref{tab_DM_DSS}) are clearly greater than 2 this indicated that the AR(12) model can not be optimal. It is likely that
the correlation structure of the AR(13) and AR(p) is better captured than for the AR(12). But the AR(12) seems to have a better predictive
performance concerning the marginal distributions as CRPS is significantly smaller (Tab. \ref{tab_air_scores} and \ref{tab_DM_CRPS}).
This illustrative study, also shows that any proper scoring rule should never be used alone without considering a strictly proper one.
For instance, variogram score (Tab. \ref{tab_DM_VS}) states that the
co- and countermonotone AR(p) provide significantly better forecasts than the AR(12), which is hard do agree if you just eyeball the corresponding forecasts.
Given the simulation studies conducted before, it is not surprising that
the CRPS-copula energy score (Tab. \ref{tab_DM_CRPS-CES}) gives similar results to the energy score. In some cases it even more sensitive than
the energy score, e.g. the AR(12)-M has significantly worse forecasting accuracy than the AR(p), but the DM-statistic is about 9
for the energy score but about 15 for the CRPS-CES. The reason is the strong sensitivity of the copula energy score (CES) (Tab. \ref{tab_DM_CES}). It detects easily that the comonotone and countermonotone AR processes do not match the data behaviour. All
test statistics are extremely large, from at least 28 to about 158 which is amazing having in mind that we have only $N=19$ rolling window experiments.
Indeed, the copula energy score is the most sensitive score concerning the detection on mis-predicted dependency structures.
However, it is followed by energy score which is still quite sensitive for dependency misspecifications.
Now, we take a closer look at the lower dimensional marginal evaluation (Fig. \ref{fig_sample_marg}).
Obviously, we observe that the 1-dimensional energy score is essentially the same for the standard, co- and countermonotone AR models of the same type
(Fig. \eqref{fig_sample_marg_dim1}). But, we see that AR(12) has a larger CRPS than
the AR(13) and the similar performing AR(p) for the first three time steps. For larger forecasting horizons the
AR(12) is clearly best. Thus, it might be worth to combine both model aspects into a better one.
When looking at the pairwise evaluation based on the copula energy score (Fig. \eqref{fig_sample_marg_dim2}) we observe that the co- and countermonotone
models can be well separated from the more plausible AR models. Interestingly the 3-dimensional copula energy score (Fig. \eqref{fig_sample_marg_dim3}) gives roughly the same picture as
the bivariate one. Which allows the conclusion that the main differences in the dependency structure is likely due to pairwise dependencies.
\section{Summary and Conclusion} \label{sec:conclusion}
We analyzed in detail established and newly introduced scoring rules for multivariate forecasting evaluation.
The introduced \textit{marginal-copula scoring rules} can excellently evaluate the dependency structure in forecasts, this
holds especially for the \textit{copula energy score}. Additionally, we also discuss reporting and forecasting study design.
Thus, our clear suggestion is to apply \textit{ensemble forecasting} in a rolling window forecasting study framework. This is to report multivariate forecast by providing a large ensemble from the forecasting model, which can be realized by Monte-Carlo simulation for statistical models.
Across several simulation studies, the strictly proper \textit{energy score} is the only evaluation measure which can clearly separate the true model from
all alternatives. This contradicts the conclusion from \cite{pinson2013discrimination} that the energy score is not suitable for evaluating differences in the dependency structure,
especially correlations. A crucial element is that the \textit{Diebold-Mariono-test} is considered for checking the discrimination ability.
Our simulation results allow the conclusion that the energy score should be preferred evaluation measure for multivariate predictions. Still, it should be combined in practice with
other measures, esp. the \textit{CRPS and bivariate copula energy scores}, the first one for checking the marginal distribution fit, the latter one for the pairwise dependency fit.
Summarizing, our guideline for forecasters is:
\begin{itemize}
\item[(1)] Do \textbf{ensemble forecasts}
with a \textbf{large} Monte-Carlo / ensemble \textbf{sample size}
\item[(2)] Evaluate on the full-dimensional \textbf{energy score}
\item[(3)] Apply the \textbf{Diebold-Mariano-test for significance} evaluation
\item[(4)] (\textit{optional but recommended}) back it up by other scores, esp. \newline \textbf{CRPS and bivariate copula energy score} across the forecasting horizon
\end{itemize}
From the practical point of view it is nice to remember that there is nothing beyond the multivariate forecasting, as the every characteristic can be derived from
a large ensemble. Thus, now further forecasting concept has to be developed. Still, there are a couple of open questions.
Concerning the marginal copula scores, this is for example to
look for better isotonic functions which combine marginal scores and copula scores more efficiently than the considered multiplicative structure.
As it seems that the copula energy score is suitable for analyzing the dependency structure of the forecasts it might be worth to study more analytical properties,
esp. better or even sharp bounds.
As the energy score seems to be the most promising known evaluation criterion so far, it might be worth to study it and the energy distance in more detail as well. For instance,
the energy score definition \eqref{eq_energy_score}, depends on $\beta$ which we fixed essentially to $1$ in this paper, even though
$\beta\in(0,2)$ could be analyzed.
As pointed out by \cite{szekely2013energy}, the energy distance also works for more general function than just $x\mapsto |x|^{\beta}$ in its definition, such as $x\mapsto \log(1+|x|^2)$.
In fact, every function which is the negative logarithm of an infinitely divisible characteristic function works out and can be studied in this context.
Finally, we want to remark that there might exists other scoring rules suitable for forecasting evaluation that were not studied in literature so far. Promising candidates
could be derived from probability distances, as e.g. the Wasserstein distance or the Hellinger distance.