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.. _copula:
Copulas
-------
Let :math:`F` be a multivariate distribution function of dimension
:math:`\inputDim` whose marginal distribution functions are
:math:`F_1,\dots,F_{\inputDim}`. There exists a copula
:math:`C: [0,1]^{\inputDim} \mapsto [0,1]` of dimension :math:`\inputDim` such that for :math:`\vect{x}\in \Rset^{\inputDim}`, we have:
.. math::
\begin{aligned}
F(\vect{x}) = C \left( F_1(x_1),\cdots,F_{\inputDim}(x_{\inputDim}) \right)
\end{aligned}
where :math:`F_i` is the cumulative distribution function of the margin :math:`X_i`.
In the case of continuous marginal distributions, for all :math:`\vect{u}\in[0,1]^{\inputDim}`, the copula is uniquely defined by:
.. math::
\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 :math:`U_i = F_i(X_i)` is a random variable following the uniform distribution on :math:`[0,1]`.
A copula of dimension :math:`\inputDim` is the restriction to the unit cube :math:`[0,1]^{\inputDim}` of a
multivariate distribution function with uniform univariate marginals on :math:`[0,1]`.
It has the following properties:
- :math:`\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 :math:`\vect{u}` with at least one component equal to 0, :math:`C(\vect{u})=0`,
- :math:`C` is :math:`\inputDim`-increasing which means that:
.. math::
\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 :math:`x_{j1}=a_j` and :math:`x_{j2}=b_j` for all :math:`j \in \{1,\dots,\inputDim\}` and
:math:`\vect{a}`, :math:`\vect{b}\in[0,1]^{\inputDim}`, :math:`\vect{a}\leq \vect{b}`,
- :math:`\vect{u}` with all its components equal to 1 except :math:`u_k`,
:math:`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.
.. topic:: API:
- See the list of available :ref:`copulas <copulas>`.
.. topic:: Examples:
- See :doc:`/auto_probabilistic_modeling/copulas/plot_create_copula`
- See :doc:`/auto_probabilistic_modeling/copulas/plot_composed_copula`
- See :doc:`/auto_probabilistic_modeling/copulas/plot_extract_copula`
.. topic:: References:
- Nelsen, *Introduction to Copulas*
- Embrechts P., Lindskog F., Mc Neil A., *Modelling dependence with copulas and application to Risk Management*, ETZH 2001.
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