Overview
What is a Copula?
A (\(d\)-dimensional) copula, \(C: [0,1]^d \rightarrow [0,1]\), is simply a (joint) cumulative distribution function (CDF) with standard uniform marginal distributions (defined between \(0\) and \(1\)). Alternatively, we say a function \(C: [0,1]^d \rightarrow [0,1]\) is a copula if and only if it fulfills all of the following properties:
- \(C(u_1, \dots, u_d)\) is non-decreasing in each \(u_i\), \(i \in \{1,\dots, d\}\)
- \[C(1,1, \dots, u_i,\dots,1,1) = u_i\]
- \(\forall a_i\leq b_i, \mathbb{P} (U_1 \in [a_1, b_1], \dots , U_d \in [a_d, b_d]) \leq 0\) (implies rectangle inequality)
Given a marginal CDF, \(F_X: x\mapsto F_X(x)=u\), its generalised inverse, \(F_X^{(-1)}\) is defined as \(F_X^{(-1)}(u) := \inf\{x: F_X(x) \geq u \}\). Then for \(U\sim U[0,1]\), we have
\[\begin{align} \mathbb{P} (F_X^{(-1)} (U) \leq x ) = F_X(x) \end{align}\]When \(F_X\) is continuous, we also have
\[\begin{align} F_X(X) \sim U[0,1]. \end{align}\]We now extend the above into the multivariate scenario where \(\mathbf{X} = (X_1, X_2, \dots, X_d)\) is a multivariate random vector with joint CDF, \(H\), with continuous and increasing marginal CDFs, \(F_{X_1} \dots, F_{X_d}\). We have
\[\begin{align} H(x_1, \dots, x_d) = \mathbb{P} ( X_1 \leq x_1, \dots, X_d \leq x_d ) \end{align}\]Since \(F_{X_i}(X_i) \sim U[0,1], \forall i \in \{1,\dots, d\}\), the joint distribution of \(F_{X_1}(X_1) \dots F_{X_d}(X_d)\) fulfils the definition of a copula. Using the definition of joint distributions, we have
\[\begin{align*} C(u_1, \dots, u_d) &= \mathbb{P} ( F_{X_1}(X_1) \leq u_1, \dots, F_{X_d}(X_d) \leq u_d )\\ &= \mathbb{P} ( X_1 \leq F^{(-1)}_{X_1}(u_1), \dots, X_d \leq F^{(-1)}_{X_d}(u_d) )\\ &= H(F^{(-1)}_{X_1}(u_1), \dots, F^{(-1)}_{X_d}(u_d)) \end{align*}\]Let \(u_i = F_{X_i}(x_i)\). Then we have
\[\begin{align} C(F_{X_1}(x_1), \dots, F_{X_d}(x_d)) = H(x_1, \dots, x_d), \htmlId{eq:copula_function}{\tag{4}} \end{align}\]Sklar’s Theorem and Building Joint Distributions
This gives us first half of the Sklar’s Theorem: Given a (\(d\)-dimensional) CDF, \(H\), with marginals, \(F_{X_1} \dots, F_{X_d}\), there exists a copula, \(C\), such that
\[\begin{align*} C(F_{X_1}(x_1), \dots, F_{X_d}(x_d)) = H(x_1, \dots, x_d). \end{align*}\]When \(F_{X_i}\) is continuous for all \(i \in \{1,\dots, d\}\), \(C\) is unique; otherwise \(C\) is uniquely determined only on \(\text{Ran}(F_{X_1}) \times \dots \times \text{Ran}(F_{X_d})\) (\(\text{Ran}(F_{X_i})\) is the range of \(F_{X_i}\) ).
Unfortunately, we seldom, if ever, know the joint CDF of a multivariate dataset. To generate synthetic data, we often use the second half the Sklar’s Theorem: Given some copula, \(C\), and univariate CDFs, \(F_{X_1} \dots, F_{X_d}\), we can find a unique joint CDF, \(H\), as defined in equation \(\href{#eq:copula_function}{(4)}\), with marginals \(F_{X_1} \dots, F_{X_d}\).
Density of a Copula
If the multivariate distribution has a density, \(h(x_1, \dots, x_d)\), we can then get a closed formed formula for copula density \(c(F_{X_1}(x_1), \dots, F_{X_d}(x_d))\). Using the definition of \(h(x_1, \dots, x_d)\), we have
\[\begin{align*} h(x_1, \dots, x_d) &= \frac{\partial^d H(x_1, \dots, x_d)}{\partial x_1 \dots \partial x_d} \\ &= \frac{\partial^d C(F_{X_1}(x_1), \dots, F_{X_d}(x_d))}{\partial x_1 \dots \partial x_d} \\ &= \frac{\partial^d C(F_{X_1}(x_1), \dots, F_{X_d}(x_d))}{\partial F_{X_1}(x_1) \dots \partial F_{X_d}(x_d)} \prod^d_{i=1} \frac{\partial F_{X_i}(x_i)}{\partial x_i} \\ &= c(F_{X_1}(x_1), \dots, F_{X_d}(x_d)) \prod^d_{i=1} f_{X_i}(x_i)\\ \end{align*}\]Conditional Copula
Given a three dimensional vector \((X_1, X_2, Y)\), we study the dependence structure of \((X_1, X_2)\) for a given value of \((Y=y)\).
Let
\[\begin{equation} H_y(x_1, x_2) := \mathbb{P} (X_1 \leq x_1, X_2 \leq x_2 \vert Y=y) % \htmlId{eq:test}{\tag{1}} \end{equation}\]According to the Sklar’s Theorem, there exists a conditional copula function \(C_y\) such that
\[\begin{equation} H_y(x_1, x_2) = C_y(F_{1y}(x_1), F_{2y}(x_2)), % \htmlId{eq:conditional_copula_function}{\tag{2}} \end{equation}\]where \(F_{1y}\) and \(F_{2y}\) are the corresponding conditional marginals of \(X_1\) and \(X_2\) respectively.
We can derive this from the known identities of conditional CDFs:
\[\begin{align*} \frac{\partial H(X_1, X_2, Y)}{\partial y} &= H_y(x_1, x_2) f_Y(y) \\ \frac{\partial C(F_{X_1}(x_1), F_{X_2}(x_2), F_Y(y))}{\partial y} &= H_y(x_1, x_2) f_Y(y) \\ \frac{\partial C(F_{X_1}(x_1), F_{X_2}(x_2), F_Y(y))}{\partial F_Y(y)} f_Y(y) &= H_y(x_1, x_2) f_Y(y) \\ \frac{\partial C(F_{X_1}(x_1), F_{X_2}(x_2), F_Y(y))}{\partial F_Y(y)} &= H_y(x_1, x_2)\\ C_y(F_{1y}(x_1), F_{2y}(x_2)) \cdot \mathbb{P}(F_Y(Y)=F_Y(y))&= H_y(x_1, x_2)\\ C_y(F_{1y}(x_1), F_{2y}(x_2)) &= H_y(x_1, x_2)\\ \end{align*}\]Partial Copula
Let \(U_1:= F_{1y}(X_1)\), \(U_2:= F_{2y}(X_2)\). The partial copula function is defined as
\[\begin{align*} \bar{C}(u_1, u_2) &:= \mathbb{P} (U_1 \leq u_1, U_2 \leq u_2) \\ &= \int \mathbb{P} (U_1 \leq u_1, U_2 \leq u_2 \vert Y=y) f_Y(y) dy \\ &= \int C_x(u_1, u_2) f_Y(y) dy % \htmlId{eq:partial_copula_function}{\tag{3}} \end{align*}\]