In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe / model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but only metaphorically related to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulas separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below. Two-dimensional copulas are known in some other areas of mathematics under the name permutons and doubly-stochastic measures.
Mathematical motivation Consider a random vector ( X 1 , X 2 , … , X d ) . {\displaystyle \ {\bigl (}X_{1},X_{2},\dots ,X_{d}{\bigr )}.} Suppose its marginals are continuous, i.e. the marginal CDFs F i ( x ) = Pr [ X i ≤ x ] {\displaystyle \ F_{i}(x)=\Pr {\bigl [}\ X_{i}\leq x\ {\bigr ]}\ } are continuous functions. By applying the probability integral transform to each component, the random vector
( U 1 , U 2 , … , U d ) = ( F 1 ( X 1 ) , F 2 ( X 2 ) , … , F d ( X d ) ) {\displaystyle {\bigl (}U_{1},U_{2},\dots ,U_{d}{\bigr )}={\Bigl (}\ F_{1}(X_{1}),F_{2}(X_{2}),\dots ,F_{d}(X_{d})\ {\Bigr )}}
has marginals that are uniformly distributed on the interval [0, 1]. The copula of ( X 1 , X 2 , … , X d ) {\displaystyle \ {\bigl (}\ X_{1},X_{2},\dots ,X_{d}\ {\bigr )}\ } is defined as the joint cumulative distribution function of ( U 1 , U 2 , … , U d ) : {\displaystyle \ {\bigl (}\ U_{1},U_{2},\dots ,U_{d}\ {\bigr )}\ :}
C ( u 1 , u 2 , … , u d ) = Pr [ U 1 ≤ u 1 , U 2 ≤ u 2 , … , U d ≤ u d ] . {\displaystyle C\!\left(\ u_{1},u_{2},\dots ,u_{d}\ \right)=\Pr {\Bigl [}\ U_{1}\leq u_{1},\ U_{2}\leq u_{2},\ \dots ,\ U_{d}\leq u_{d}\ {\Bigr ]}~.}
The copula C contains all information on the dependence structure between the components of ( X 1 , X 2 , … , X d ) {\displaystyle \ {\bigl (}\ X_{1},X_{2},\dots ,X_{d}\ {\bigr )}\ } whereas the marginal cumulative distribution functions F i {\displaystyle F_{i}} contain all information on the marginal distributions of X i . {\displaystyle \ X_{i}~.}
The reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample ( U 1 , U 2 , … , U d ) {\displaystyle (U_{1},U_{2},\dots ,U_{d})} from the copula function, the required sample can be constructed as
( X 1 , X 2 , … , X d ) = ( F 1 − 1 ( U 1 ) , F 2 − 1 ( U 2 ) , … , F d − 1 ( U d ) ) . {\displaystyle {\bigl (}\ X_{1},X_{2},\dots ,X_{d}\ {\bigr )}={\Bigl (}\ F_{1}^{-1}(U_{1}),\ F_{2}^{-1}(U_{2}),\ \dots ,\ F_{d}^{-1}(U_{d})\ {\Bigr )}~.}
The generalized inverses F i − 1 {\displaystyle \ F_{i}^{-1}\ } are unproblematic almost surely, since the F i {\displaystyle \ F_{i}\ } were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:
C ( u 1 , u 2 , … , u d ) = Pr [ X 1 ≤ F 1 − 1 ( u 1 ) , X 2 ≤ F 2 − 1 ( u 2 ) , … , X d ≤ F d − 1 ( u d ) ] . {\displaystyle C\!\left(\ u_{1},u_{2},\dots ,u_{d}\ \right)=\Pr {\Bigl [}\ X_{1}\leq F_{1}^{-1}(u_{1}),\ X_{2}\leq F_{2}^{-1}(u_{2}),\ \dots ,\ X_{d}\leq F_{d}^{-1}(u_{d})\ {\Bigr ]}~.}
Definition In probabilistic terms, C : [ 0 , 1 ] d → [ 0 , 1 ] {\displaystyle C:[0,1]^{d}\rightarrow [0,1]} is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube [ 0 , 1 ] d {\displaystyle [0,1]^{d}} with uniform marginals. In analytic terms, C : [ 0 , 1 ] d → [ 0 , 1 ] {\displaystyle C:[0,1]^{d}\rightarrow [0,1]} is a d-dimensional copula if
C ( u 1 , … , u i − 1 , 0 , u i + 1 , … , u d ) = 0 {\displaystyle C(u_{1},\dots ,u_{i-1},0,u_{i+1},\dots ,u_{d})=0} , the copula is zero if any one of the arguments is zero,
C ( 1 , … , 1 , u , 1 , … , 1 ) = u {\displaystyle C(1,\dots ,1,u,1,\dots ,1)=u} , the copula is equal to u if one argument is u and all others 1, C is d-non-decreasing, i.e., for each hyperrectangle B = ∏ i = 1 d [ x i , y i ] ⊆ [ 0 , 1 ] d {\displaystyle B=\prod _{i=1}^{d}[x_{i},y_{i}]\subseteq [0,1]^{d}} the C-volume of B is non-negative:
∫ B d C ( u ) = ∑ z ∈ ∏ i = 1 d { x i , y i } ( − 1 ) N ( z ) C ( z ) ≥ 0 , {\displaystyle \int _{B}\mathrm {d} C(u)=\sum _{\mathbf {z} \in \prod _{i=1}^{d}\{x_{i},y_{i}\}}(-1)^{N(\mathbf {z} )}C(\mathbf {z} )\geq 0,}
where the N ( z ) = # { k : z k = x k } {\displaystyle N(\mathbf {z} )=\#\{k:z_{k}=x_{k}\}} . For instance, in the bivariate case, C : [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle C:[0,1]\times [0,1]\rightarrow [0,1]} is a bivariate copula if C ( 0 , u ) = C ( u , 0 ) = 0 {\displaystyle C(0,u)=C(u,0)=0} , C ( 1 , u ) = C ( u , 1 ) = u {\displaystyle C(1,u)=C(u,1)=u} and C ( u 2 , v 2 ) − C ( u 2 , v 1 ) − C ( u 1 , v 2 ) + C ( u 1 , v 1 ) ≥ 0 {\displaystyle C(u_{2},v_{2})-C(u_{2},v_{1})-C(u_{1},v_{2})+C(u_{1},v_{1})\geq 0} for all 0 ≤ u 1 ≤ u 2 ≤ 1 {\displaystyle 0\leq u_{1}\leq u_{2}\leq 1} and 0 ≤ v 1 ≤ v 2 ≤ 1 {\displaystyle 0\leq v_{1}\leq v_{2}\leq 1} .
Sklar's theorem
Sklar's theorem, named after Abe Sklar, provides the theoretical foundation for the application of copulas. Sklar's theorem states that every multivariate cumulative distribution function
H ( x 1 , … , x d ) = Pr [ X 1 ≤ x 1 , … , X d ≤ x d ] {\displaystyle H(x_{1},\dots ,x_{d})=\Pr[X_{1}\leq x_{1},\dots ,X_{d}\leq x_{d}]}
of a random vector ( X 1 , X 2 , … , X d ) {\displaystyle (X_{1},X_{2},\dots ,X_{d})} can be expressed in terms of its marginals F i ( x i ) = Pr [ X i ≤ x i ] {\displaystyle F_{i}(x_{i})=\Pr[X_{i}\leq x_{i}]} and a copula C {\displaystyle C} . Indeed:
H ( x 1 , … , x d ) = C ( F 1 ( x 1 ) , … , F d ( x d ) ) . {\displaystyle H(x_{1},\dots ,x_{d})=C\left(F_{1}(x_{1}),\dots ,F_{d}(x_{d})\right).}
If the multivariate distribution has a density h {\displaystyle h} , and if this density is available, it also holds that
h ( x 1 , … , x d ) = c ( F 1 ( x 1 ) , … , F d ( x d ) ) ⋅ f 1 ( x 1 ) ⋅ ⋯ ⋅ f d ( x d ) , {\displaystyle h(x_{1},\dots ,x_{d})=c(F_{1}(x_{1}),\dots ,F_{d}(x_{d}))\cdot f_{1}(x_{1})\cdot \dots \cdot f_{d}(x_{d}),}
where c {\displaystyle c} is the density of the copula. The theorem also states that, given H {\displaystyle H} , the copula is unique on Ran ( F 1 ) × ⋯ × Ran ( F d ) {\displaystyle \operatorname {Ran} (F_{1})\times \cdots \times \operatorname {Ran} (F_{d})} which is the cartesian product of the ranges of the marginal cdf's. This implies that the copula is unique if the marginals F i {\displaystyle F_{i}} are continuous. The converse is also true: given a copula C : [ 0 , 1 ] d → [ 0 , 1 ] {\displaystyle C:[0,1]^{d}\rightarrow [0,1]} and marginals F i ( x ) {\displaystyle F_{i}(x)} then C ( F 1 ( x 1 ) , … , F d ( x d ) ) {\displaystyle C\left(F_{1}(x_{1}),\dots ,F_{d}(x_{d})\right)} defines a d-dimensional cumulative distribution function with marginal distributions F i ( x ) {\displaystyle F_{i}(x)} .
Stationarity condition Copulas mainly work when time series are stationary and continuous. Thus, a very important pre-processing step is to check for the auto-correlation, trend and seasonality within time series. When time series are auto-correlated, they may generate a non existing dependence between sets of variables and result in incorrect copula dependence structure.
Fréchet–Hoeffding copula bounds
The Fréchet–Hoeffding theorem (after Maurice René Fréchet and Wassily Hoeffding) states that for any copula C : [ 0 , 1 ] d → [ 0 , 1 ] {\displaystyle C:[0,1]^{d}\rightarrow [0,1]} and any ( u 1 , … , u d ) ∈ [ 0 , 1 ] d {\displaystyle (u_{1},\dots ,u_{d})\in [0,1]^{d}} the following bounds hold:
W ( u 1 , … , u d ) ≤ C ( u 1 , … , u d ) ≤ M ( u 1 , … , u d ) . {\displaystyle W(u_{1},\dots ,u_{d})\leq C(u_{1},\dots ,u_{d})\leq M(u_{1},\dots ,u_{d}).}
The function W is called lower Fréchet–Hoeffding bound and is defined as
W ( u 1 , … , u d ) = max { 1 − d + ∑ i = 1 d u i , 0 } . {\displaystyle W(u_{1},\ldots ,u_{d})=\max \left\{1-d+\sum \limits _{i=1}^{d}{u_{i}},\,0\right\}.}
The function M is called upper Fréchet–Hoeffding bound and is defined as
M ( u 1 , … , u d ) = min { u 1 , … , u d } . {\displaystyle M(u_{1},\ldots ,u_{d})=\min\{u_{1},\dots ,u_{d}\}.}
The upper bound is sharp: M is always a copula, it corresponds to comonotone random variables. The lower bound is point-wise sharp, in the sense that for fixed u, there is a copula C ~ {\displaystyle {\tilde {C}}} such that C ~ ( u ) = W ( u ) . {\displaystyle {\tilde {C}}(u)=W(u)~.} However, W is a copula only in two dimensions, in which case it corresponds to countermonotonic random variables. In two dimensions, i.e. the bivariate case, the Fréchet–Hoeffding theorem states
max { u + v − 1 , 0 } ≤ C ( u , v ) ≤ min { u , v } . {\displaystyle \max\{u+v-1,\,0\}\leq C(u,v)\leq \min\{u,v\}.}
Families of copulas Several families of copulas have been described.
Gaussian copula
The Gaussian copula is a distribution over the unit hypercube [ 0 , 1 ] d {\displaystyle [0,1]^{d}} . It is constructed from a multivariate normal distribution over R d {\displaystyle \mathbb {R} ^{d}} by using the probability integral transform. For a given correlation matrix R ∈ [ − 1 , 1 ] d × d {\displaystyle R\in [-1,1]^{d\times d}} , the Gaussian copula with parameter matrix R {\displaystyle R} can be written as
C R Gauss ( u ) = Φ R ( Φ − 1 ( u 1 ) , … , Φ − 1 ( u d ) ) , {\displaystyle C_{R}^{\text{Gauss}}(u)=\Phi _{R}\left(\Phi ^{-1}(u_{1}),\dots ,\Phi ^{-1}(u_{d})\right),}
where Φ − 1 {\displaystyle \Phi ^{-1}} is the inverse cumulative distribution function of a standard normal and Φ R {\displaystyle \Phi _{R}} is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix equal to the correlation matrix R {\displaystyle R} . While there is no simple analytical formula for the copula function, C R Gauss ( u ) {\displaystyle C_{R}^{\text{Gauss}}(u)} , it can be upper or lower bounded, and approximated using numerical integration. The density can be written as
c R Gauss ( u ) = 1 det R exp ( − 1 2 [ Φ − 1 ( u 1 ) ⋮ Φ − 1 ( u d ) ] T ( R − 1 − I ) [ Φ − 1 ( u 1 ) ⋮ Φ − 1 ( u d ) ] ) , {\displaystyle c_{R}^{\text{Gauss}}(u)={\frac {1}{\ {\sqrt {\det {R}\ }}\ }}\ \exp \left(-{\frac {1}{2}}{\begin{bmatrix}\Phi ^{-1}(u_{1})\\\vdots \\\Phi ^{-1}(u_{d})\end{bmatrix}}^{\mathsf {T}}\,\left(R^{-1}-I\right)\,{\begin{bmatrix}\Phi ^{-1}(u_{1})\\\vdots \\\Phi ^{-1}(u_{d})\end{bmatrix}}\right)\ ,}
where I {\displaystyle I} is the identity matrix.
Archimedean copulas Archimedean copulas are an associative class of copulas. Most common Archimedean copulas admit an explicit formula, something not possible for instance for the Gaussian copula. In practice, Archimedean copulas are popular because they allow modeling dependence in arbitrarily high dimensions with only one parameter, governing the strength of dependence. A copula C is called Archimedean if it admits the representation
C ( u 1 , … , u d ; θ ) = ψ − 1 ( ψ ( u 1 ; θ ) + ⋯ + ψ ( u d ; θ ) ; θ ) {\displaystyle C(\ u_{1},\dots ,u_{d}\ ;\ \theta \ )=\psi ^{-1}\!{\bigl (}\ \psi (u_{1};\theta )\ +\ \cdots \ +\ \psi (u_{d};\theta )\ ;\ \theta \ {\bigr )}}
where ψ : [ 0 , 1 ] × Θ → [ 0 , ∞ ) {\displaystyle \ \psi \!:[0,1]\times \Theta \rightarrow [0,\infty )\ } is a continuous, strictly decreasing and convex function such that ψ ( 1 ; θ ) = 0 , {\displaystyle \ \psi (1;\theta )=0\ ,} θ {\displaystyle \ \theta \ } is a parameter within some parameter space Θ , {\displaystyle \ \Theta \ ,} and ψ {\displaystyle \ \psi \ } is the so-called generator function and ψ − 1 {\displaystyle \ \psi ^{-1}\ } is its pseudo-inverse, defined by
ψ − 1 ( t ; θ ) = { ψ − 1 ( t ; θ ) if 0 ≤ t ≤ ψ ( 0 ; θ ) 0 if ψ ( 0 ; θ ) ≤ t ≤ ∞ . {\displaystyle \psi ^{-1}(t;\theta )=\left\{{\begin{array}{ll}\psi ^{-1}(t;\theta )&{\mbox{ if }}\qquad ~0~\quad \leq \ t\ \leq \ \psi (0;\theta )\\0&{\mbox{ if }}\quad \psi (0;\theta )\ \leq \ t\ \leq \ \infty ~.\end{array}}\right.}
Moreover, the above formula for C yields a copula for ψ − 1 {\displaystyle \ \psi ^{-1}\ } if and only if ψ − 1 {\displaystyle \psi ^{-1}} is d-monotone on [ 0 , ∞ ) . {\displaystyle \ [0,\infty )~.}
That is, if it is differentiable d − 2 {\displaystyle \ d\ -\ 2\ } times, and those derivatives satisfy