CopulasΒΆ

Let F be a multivariate distribution function of dimension \inputDim whose marginal distribution functions are F_1,\dots,F_{\inputDim}. There exists a copula C: [0,1]^{\inputDim} \mapsto [0,1] of dimension \inputDim such that for \vect{x}\in \Rset^{\inputDim}, we have:

\begin{aligned} F(\vect{x}) = C \left( F_1(x_1),\cdots,F_{\inputDim}(x_{\inputDim}) \right) \end{aligned}

where F_i is the cumulative distribution function of the margin X_i.

In the case of continuous marginal distributions, for all \vect{u}\in[0,1]^{\inputDim}, the copula is uniquely defined by:

\begin{aligned} C(\vect{u}) & = F(F_1^{-1}(u_1),\hdots,F_{\inputDim}^{-1}(u_{\inputDim}))\\ & = \Prob{U_1 \leq u_1, \hdots, U_{\inputDim} \leq u_{\inputDim}} \end{aligned}

where U_i = F_i(X_i) is a random variable following the uniform distribution on [0,1].

A copula of dimension \inputDim is the restriction to the unit cube [0,1]^{\inputDim} of a multivariate distribution function with uniform univariate marginals on [0,1]. It has the following properties:

  • \forall \vect{u},\vect{v}\in[0,1]^{\inputDim}, |C(\vect{u})-C(\vect{v})|\leq \sum_{i=1}^{\inputDim} |u_i-v_i|,

  • for all \vect{u} with at least one component equal to 0, C(\vect{u})=0,

  • C is \inputDim-increasing which means that:

    \sum_{i_1=1}^2 \dots \sum_{i_{\inputDim}=1}^2 (-1)^{i_1 + \dots + i_{\inputDim}} C(x_{1i_1}, \dots, x_{\inputDim i_{\inputDim}})\geq 0

where x_{j1}=a_j and x_{j2}=b_j for all j \in \{1,\dots,\inputDim\} and \vect{a}, \vect{b}\in[0,1]^{\inputDim}, \vect{a}\leq \vect{b},

  • \vect{u} with all its components equal to 1 except u_k, C(\vect{u})=u_k.

The copula represents the part of the joint cumulative density function which is not described by the marginal distributions. It models the dependence structure of the input variables.

Note that a multivariate distribution is characterized by its marginal distributions and its copula. Therefore, a multivariate distribution can be built by choosing the marginals and the copula independently.