In this part, we will define the concept of copula.
To define the joined probability density function of the random input vector \uX by composition, one needs:
  • the specification of the copula of interest C with its parameters,

  • the specification of the {n_X} marginal laws of interest F_{X_i} of the {n_X} input variables X_i.

The joined cumulative density function is therefore defined by:

    \Prob{X^1 \leq x^1, X^2 \leq x^2, \cdots, X^{n_X} \leq x^{n_X}}       = C\left(F_{X^1}(x^1),F_{X^2}(x^2),\cdots,F_{X^{n_X}}(x^{n_X}) \right)

Copulas allow one to represent the part of the joined cumulative density function which is not described by the marginal laws. It enables to represent the dependency structure of the input variables. A copula is a special cumulative density function defined on [0,1]^{n_X} whose marginal distributions are uniform on [0,1]. The choice of the dependence structure is disconnected from the choice of the marginal distributions.
A copula, restricted to [0,1]^{n_X} is a n_U-dimensional cumulative density function with uniform marginals.
  • C(\vect{u}) \geq 0, \forall \vect{u} \in [0,1]^{n_U}

  • C(\vect{u}) = u_i, \forall \vect{u}=(1,\ldots,1,u_i,1,\ldots,1)

  • For all N-box \cB = [a_1,b_1] \times \cdots \times [a_{n_U},b_{n_U}] \in [0,1]^{n_U}, we have \cV_C(\cB) \geq 0, where:

    • \cV_C(\cB) = \sum_{i=1,\cdots, 2^{n_U}} \sgn(\vect{v}_i) \times C(\vect{v}_i), the summation being made over the 2^{n_U} vertices \vect{v}_i of \cB.

    • \sgn(\vect{v}_i)= +1 if v_i^k = a_k for an even number of k's, \sgn(\vect{v}_i)= -1 otherwise.