Función de reacción del banco Central de Chile

Originally introduced by Sklar (1959), copulas play an important part in probability and statistics whenever modeling of stochastic dependence is required. Essentially, a copula is a multivariate CDF with standard uniform margins. A precise mathematical definition is given in the following (Nelsen, 2006).

Definition 3.1. A function C : [0, 1]L→ [0, 1] is an L-dimensional copula or L-copula if it satisfies the following conditions.

(C1) C is grounded, in that C(u1, . . . , uL) = 0 if u = 0 for at least one ∈ {1, . . . , L}.

(C3) C is L-increasing, in that

ΔbL

aL· · · Δ

b1

a1C(u1, . . . , uL)≥ 0

for all a_, b_ ∈ [0, 1] such that a_≤ b_ for all ∈ {1, . . . , L}, where Δb

aC(u1, . . . , uL) := C(u1, . . . , u−1, b, u+1, . . . , uL) −C(u1, . . . , u−1, a, u+1, . . . , uL).

Definition 3.2. Let J1, . . . , JL be subsets of [0, 1] containing at least the points 0 and 1. A function C∗ : J1× · · · × JL → [0, 1] is an L-dimensional subcopula or L-subcopula if it analogously satisfies the conditions (C1), (C2) and (C3) in Definition 3.1, but is defined on

J1× · · · × JL rather than on the entire L-cube [0, 1]L.

The field of copulas has been developing rapidly over the last decades, and copulas have been applied to a wide range of problems in various areas, such as climatology, meteorology and hydrology (Möller et al., 2013; Genest and Favre, 2007; Schoelzel and Friederichs, 2008; Zhang et al., 2012) or econometrics, insurance and mathematical finance (Cherubini et al., 2004; Embrechts et al., 2003; Pfeifer and Nešlehová, 2003; Genest et al., 2009), with Mikosch (2006) providing a critical review on their use. However, copulas are also of immense the- oretical interest, due to their appealing mathematical properties. For a general overview of the mathematical theory of copulas, we refer to the textbooks by Joe (1997) and Nelsen (2006), as well as to the survey paper by Sempi (2011).

The relevance of copulas in multivariate dependence modeling is based on the famous theo- rem of Sklar (1959), which states that any multivariate CDF can be linked to its univariate marginal CDFs via a copula function.

For the purposes of this thesis, let us suppose that we have a predictive CDF F_ for each univariate weather variable Y_ for ∈ {1, . . . , L}, where the multi-index  := (i, j, k) refers to weather quantity i ∈ {1, . . . , I}, location j ∈ {1, . . . , J} and prediction horizon

k ∈ {1, . . . , K}, with L := I × J × K. Our goal is then to provide a physically realistic

multivariate joint predictive CDF F with margins F1, . . . , FL. With this setting in mind, Sklar’s theorem can be formulated as follows.

Theorem 3.3. (Sklar) (Sklar, 1959; Nelsen, 2006)

1. For any multivariate CDF H with marginal CDFs F1, . . . , FL, there exists a copula C such that

H(y1, . . . , yL) = C(F1(y1), . . . , FL(yL)) (3.7) for y1, . . . , yL∈ R := R ∪ {−∞, ∞}. Moreover, C is uniquely determined if F1, . . . , FL are continuous; otherwise, C is uniquely determined on Ran(F1)× · · · × Ran(FL). 2. Conversely, if C is a copula and F1, . . . , FLare univariate CDFs, then the function H

as defined in (3.7) is a multivariate CDF with margins F1, . . . , FL.

Due to Sklar’s theorem, the desired multivariate distribution can thus be constructed by combining margins obtained by univariate postprocessing and a multivariate dependence structure contained in a suitable copula function. Hence, the main challenge now is to choose and fit the copula C appropriately.

Various different types of copulas are available, including but not limited to Gaussian, elliptical, Archimedean, vine or pair, extremal and discrete or empirical copulas.

The most common type used in parametric approaches is that of a Gaussian copula (Em- brechts et al., 2003), under which the joint multivariate CDF H is given by

H(y1, . . . , yL|R) := ΦL(Φ−1(F1(y1)), . . . , Φ−1(FL(yL))|R), (3.8) where Φ_L(·|R) denotes the CDF of an L-dimensional normal distribution with mean vector 0 := (0, . . . , 0) and correlation matrix R, and Φ−1 _{the inverse of the CDF of the univariate} standard normal distribution. A major advantage of Gaussian copula models is that only the margins F1, . . . , FL and the correlation matrix R are required. Some examples for the use of Gaussian copulas in multivariate postprocessing are discussed in the next subsection, while parametric or semi-parametric alternatives comprise elliptical (Demarta and McNeil, 2005), Archimedean (Nelsen, 2006; McNeil and Nešlehová, 2009), extremal (Davison et al., 2012) and vine or pair copulas (Aas et al., 2009; Erhardt et al., 2014), for instance. All these types of copula are typically employed if the dimension L is rather small, or if a specific structure can be exploited.

By contrast, for large L, and if no specific structure can be utilized, it is reasonable to use non-parametric approaches, which rely on discrete or empirical copulas (Rüschendorf, 1976; Kolesárová et al., 2006; Rüschendorf, 2009), which are also known under the term “empirical dependence functions” (Deheuvels, 1979). Letting {x1, . . . ,xM} with xm := (x1

m, . . . , xLm) ∈ RL for m ∈ {1, . . . , M} be a data set of size M with values in RL and assuming for simplicity that there are no ties, the corresponding empirical copula E_M is given by E_{M i} 1 M, . . . , i_L M  := 1 M M  m=1 1_{rank(x1

m)≤i1,...,rank(xLm)≤iL} (3.9)

for integers i1, . . . , iL∈ {0, . . . , M}, where rank(xm) denotes the rank of xm in{x1, . . . , xM} for ∈ {1, . . . , L} and m ∈ {1, . . . , M}. Detailed theoretical aspects of discrete and empirical copulas, respectively, are presented in Chapter 6. Examples of multivariate postprocessing approaches based on empirical copulas include the Schaake shuffle (Clark et al., 2004), which is discussed in the next subsection, and ensemble copula coupling (Schefzik et al., 2013), which forms the main subject of this thesis and is introduced in Chapter 4.