Copula (probability theory)

Consider a random vector ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$. Suppose its marginals are continuous, i.e. the marginal CDFs ${\displaystyle F_{i}(x)=\Pr[X_{i}\leq x]}$ are continuous functions. By applying the probability integral transform to each component, the random vector

${\displaystyle (U_{1},U_{2},\dots ,U_{d})=\left(F_{1}(X_{1}),F_{2}(X_{2}),\dots ,F_{d}(X_{d})\right)}$

has marginals that are uniformly distributed on the interval [0, 1].

The copula of ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$ is defined as the joint cumulative distribution function of ${\displaystyle (U_{1},U_{2},\dots ,U_{d})}$:

${\displaystyle C(u_{1},u_{2},\dots ,u_{d})=\Pr[U_{1}\leq u_{1},U_{2}\leq u_{2},\dots ,U_{d}\leq u_{d}].}$

The copula C contains all information on the dependence structure between the components of ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$ whereas the marginal cumulative distribution functions ${\displaystyle F_{i}}$ contain all information on the marginal distributions of ${\displaystyle X_{i}}$.

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 ${\displaystyle (U_{1},U_{2},\dots ,U_{d})}$ from the copula function, the required sample can be constructed as

${\displaystyle (X_{1},X_{2},\dots ,X_{d})=\left(F_{1}^{-1}(U_{1}),F_{2}^{-1}(U_{2}),\dots ,F_{d}^{-1}(U_{d})\right).}$

The inverses ${\displaystyle F_{i}^{-1}}$ are unproblematic almost surely, since the ${\displaystyle F_{i}}$ were assumed to be continuous. Furthermore, the above formula for the copula function can be rewritten as:

${\displaystyle C(u_{1},u_{2},\dots ,u_{d})=\Pr[X_{1}\leq F_{1}^{-1}(u_{1}),X_{2}\leq F_{2}^{-1}(u_{2}),\dots ,X_{d}\leq F_{d}^{-1}(u_{d})].}$

In probabilistic terms, ${\displaystyle C:[0,1]^{d}\rightarrow [0,1]}$ is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube ${\displaystyle [0,1]^{d}}$ with uniform marginals.[4]

In analytic terms, ${\displaystyle C:[0,1]^{d}\rightarrow [0,1]}$ is a d-dimensional copula if

• ${\displaystyle C(u_{1},\dots ,u_{i-1},0,u_{i+1},\dots ,u_{d})=0}$, the copula is zero if any one of the arguments is zero,
• ${\displaystyle C(1,\dots ,1,u,1,\dots ,1)=u}$, 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 ${\displaystyle B=\prod _{i=1}^{d}[x_{i},y_{i}]\subseteq [0,1]^{d}}$ the C-volume of B is non-negative:

  ${\displaystyle \int _{B}\mathrm {d} C(u)=\sum _{\mathbf {z} \in \prod _{i=1}^{d}\{x_{i},y_{i}\}}(-1)^{N(\mathbf {z} )}C(\mathbf {z} )\geq 0,}$

where the ${\displaystyle N(\mathbf {z} )=\#\{k:z_{k}=x_{k}\}}$.

For instance, in the bivariate case, ${\displaystyle C:[0,1]\times [0,1]\rightarrow [0,1]}$ is a bivariate copula if ${\displaystyle C(0,u)=C(u,0)=0}$, ${\displaystyle C(1,u)=C(u,1)=u}$ and ${\displaystyle C(u_{2},v_{2})-C(u_{2},v_{1})-C(u_{1},v_{2})+C(u_{1},v_{1})\geq 0}$ for all ${\displaystyle 0\leq u_{1}\leq u_{2}\leq 1}$ and ${\displaystyle 0\leq v_{1}\leq v_{2}\leq 1}$.

Density and contour plot of a Bivariate Gaussian Distribution

Density and contour plot of two Normal marginals joint with a Gumbel copula

Sklar's theorem, named after Abe Sklar, provides the theoretical foundation for the application of copulas.[5][6] Sklar's theorem states that every multivariate cumulative distribution function

${\displaystyle H(x_{1},\dots ,x_{d})=\Pr[X_{1}\leq x_{1},\dots ,X_{d}\leq x_{d}]}$

of a random vector ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$ can be expressed in terms of its marginals ${\displaystyle F_{i}(x_{i})=\Pr[X_{i}\leq x_{i}]}$ and a copula ${\displaystyle C}$. Indeed:

${\displaystyle H(x_{1},\dots ,x_{d})=C\left(F_{1}(x_{1}),\dots ,F_{d}(x_{d})\right).}$

In case that the multivariate distribution has a density ${\displaystyle h}$, and if this is available, it holds further that

${\displaystyle h(x_{1},\dots ,x_{d})=c(F_{1}(x_{1}),\dots ,F_{d}(x_{d}))\cdot f_{1}(x_{1})\cdot \dots \cdot f_{d}(x_{d}),}$

where ${\displaystyle c}$ is the density of the copula.

The theorem also states that, given ${\displaystyle H}$, the copula is unique on ${\displaystyle \operatorname {Ran} (F_{1})\times \cdots \times \operatorname {Ran} (F_{d})}$, which is the cartesian product of the ranges of the marginal cdf's. This implies that the copula is unique if the marginals ${\displaystyle F_{i}}$ are continuous.

The converse is also true: given a copula ${\displaystyle C:[0,1]^{d}\rightarrow [0,1]}$ and marginals ${\displaystyle F_{i}(x)}$ then ${\displaystyle C\left(F_{1}(x_{1}),\dots ,F_{d}(x_{d})\right)}$ defines a d-dimensional cumulative distribution function with marginal distributions ${\displaystyle F_{i}(x)}$.

Copulas mainly work when time series are stationary[7] and continuous.[8] 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 existence dependence between sets of variables and result in incorrect Copula dependence structure.[9]

Graphs of the bivariate Fréchet–Hoeffding copula limits and of the independence copula (in the middle).

The Fréchet–Hoeffding Theorem (after Maurice René Fréchet and Wassily Hoeffding[10]) states that for any Copula ${\displaystyle C:[0,1]^{d}\rightarrow [0,1]}$ and any ${\displaystyle (u_{1},\dots ,u_{d})\in [0,1]^{d}}$ the following bounds hold:

${\displaystyle W(u_{1},\dots ,u_{d})\leq C(u_{1},\dots ,u_{d})\leq M(u_{1},\dots ,u_{d}).}$

The function W is called lower Fréchet–Hoeffding bound and is defined as

${\displaystyle W(u_{1},\ldots ,u_{d})=\max \left\{1-d+\sum \limits _{i=1}^{d}{u_{i}},\,0\right\}.}$

The function M is called upper Fréchet–Hoeffding bound and is defined as

${\displaystyle M(u_{1},\ldots ,u_{d})=\min\{u_{1},\dots ,u_{d}\}.}$

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 ${\displaystyle {\tilde {C}}}$ such that ${\displaystyle {\tilde {C}}(u)=W(u)}$. 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

${\displaystyle \max\{u+v-1,\,0\}\leq C(u,v)\leq \min\{u,v\}}$.

Several families of copulas have been described.

Gaussian copula

Cumulative and density distribution of Gaussian copula with ρ = 0.4

The Gaussian copula is a distribution over the unit hypercube ${\displaystyle [0,1]^{d}}$. It is constructed from a multivariate normal distribution over ${\displaystyle \mathbb {R} ^{d}}$ by using the probability integral transform.

For a given correlation matrix ${\displaystyle R\in [-1,1]^{d\times d}}$, the Gaussian copula with parameter matrix ${\displaystyle R}$ can be written as

${\displaystyle C_{R}^{\text{Gauss}}(u)=\Phi _{R}\left(\Phi ^{-1}(u_{1}),\dots ,\Phi ^{-1}(u_{d})\right),}$

where ${\displaystyle \Phi ^{-1}}$ is the inverse cumulative distribution function of a standard normal and ${\displaystyle \Phi _{R}}$ is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix equal to the correlation matrix ${\displaystyle R}$. While there is no simple analytical formula for the copula function, ${\displaystyle C_{R}^{\text{Gauss}}(u)}$, it can be upper or lower bounded, and approximated using numerical integration.[11][12] The density can be written as[13]

${\displaystyle c_{R}^{\text{Gauss}}(u)={\frac {1}{\sqrt {\det {R}}}}\exp \left(-{\frac {1}{2}}{\begin{pmatrix}\Phi ^{-1}(u_{1})\\\vdots \\\Phi ^{-1}(u_{d})\end{pmatrix}}^{T}\cdot \left(R^{-1}-I\right)\cdot {\begin{pmatrix}\Phi ^{-1}(u_{1})\\\vdots \\\Phi ^{-1}(u_{d})\end{pmatrix}}\right),}$

where ${\displaystyle \mathbf {I} }$ is the identity matrix.

Most important Archimedean copulas

The following tables highlight the most prominent bivariate Archimedean copulas, with their corresponding generator. Not all of them are completely monotone, i.e. d-monotone for all ${\displaystyle d\in \mathbb {N} }$ or d-monotone for certain ${\displaystyle \theta \in \Theta }$ only.

Table with the most important Archimedean copulas[14]

Name of copula	Bivariate copula ${\displaystyle \;C_{\theta }(u,v)}$	parameter ${\displaystyle \,\theta }$	generator ${\displaystyle \,\psi _{\theta }(t)}$	generator inverse ${\displaystyle \,\psi _{\theta }^{-1}(t)}$
Ali–Mikhail–Haq[16]	${\displaystyle {\frac {uv}{1-\theta (1-u)(1-v)}}}$	${\displaystyle \theta \in [-1,1]}$	${\displaystyle \log \!\left[{\frac {1-\theta (1-t)}{t}}\right]}$	${\displaystyle {\frac {1-\theta }{\exp(t)-\theta }}}$
Clayton[17]	${\displaystyle \left[\max \left\{u^{-\theta }+v^{-\theta }-1;0\right\}\right]^{-1/\theta }}$	${\displaystyle \theta \in [-1,\infty )\backslash \{0\}}$	${\displaystyle {\frac {1}{\theta }}\,(t^{-\theta }-1)}$	${\displaystyle \left(1+\theta t\right)^{-1/\theta }}$
Frank	${\displaystyle -{\frac {1}{\theta }}\log \!\left[1+{\frac {(\exp(-\theta u)-1)(\exp(-\theta v)-1)}{\exp(-\theta )-1}}\right]}$	${\displaystyle \theta \in \mathbb {R} \backslash \{0\}}$	${\textstyle -\log \!\left({\frac {\exp(-\theta t)-1}{\exp(-\theta )-1}}\right)}$	${\displaystyle -{\frac {1}{\theta }}\,\log(1+\exp(-t)(\exp(-\theta )-1))}$
Gumbel	${\textstyle \exp \!\left[-\left((-\log(u))^{\theta }+(-\log(v))^{\theta }\right)^{1/\theta }\right]}$	${\displaystyle \theta \in [1,\infty )}$	${\displaystyle \left(-\log(t)\right)^{\theta }}$	${\displaystyle \exp \!\left(-t^{1/\theta }\right)}$
Independence	${\textstyle uv}$		${\displaystyle -\log(t)}$	${\displaystyle \exp(-t)}$
Joe	${\textstyle {1-\left[(1-u)^{\theta }+(1-v)^{\theta }-(1-u)^{\theta }(1-v)^{\theta }\right]^{1/\theta }}}$	${\displaystyle \theta \in [1,\infty )}$	${\displaystyle -\log \!\left(1-(1-t)^{\theta }\right)}$	${\displaystyle 1-\left(1-\exp(-t)\right)^{1/\theta }}$

In statistical applications, many problems can be formulated in the following way. One is interested in the expectation of a response function ${\displaystyle g:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ applied to some random vector ${\displaystyle (X_{1},\dots ,X_{d})}$.[18] If we denote the cdf of this random vector with ${\displaystyle H}$, the quantity of interest can thus be written as

${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]=\int _{\mathbb {R} ^{d}}g(x_{1},\dots ,x_{d})\,\mathrm {d} H(x_{1},\dots ,x_{d}).}$

If ${\displaystyle H}$ is given by a copula model, i.e.,

${\displaystyle H(x_{1},\dots ,x_{d})=C(F_{1}(x_{1}),\dots ,F_{d}(x_{d}))}$

this expectation can be rewritten as

${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]=\int _{[0,1]^{d}}g(F_{1}^{-1}(u_{1}),\dots ,F_{d}^{-1}(u_{d}))\,\mathrm {d} C(u_{1},\dots ,u_{d}).}$

In case the copula C is absolutely continuous, i.e. C has a density c, this equation can be written as

${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]=\int _{[0,1]^{d}}g(F_{1}^{-1}(u_{1}),\dots ,F_{d}^{-1}(u_{d}))\cdot c(u_{1},\dots ,u_{d})\,du_{1}\cdots \mathrm {d} u_{d},}$

and if each marginal distribution has the density ${\displaystyle f_{i}}$ it holds further that

${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]=\int _{\mathbb {R} ^{d}}g(x_{1},\dots x_{d})\cdot c(F_{1}(x_{1}),\dots ,F_{d}(x_{d}))\cdot f_{1}(x_{1})\cdots f_{d}(x_{d})\,\mathrm {d} x_{1}\cdots \mathrm {d} x_{d}.}$

If copula and marginals are known (or if they have been estimated), this expectation can be approximated through the following Monte Carlo algorithm:

1. Draw a sample ${\displaystyle (U_{1}^{k},\dots ,U_{d}^{k})\sim C\;\;(k=1,\dots ,n)}$ of size n from the copula C
2. By applying the inverse marginal cdf's, produce a sample of ${\displaystyle (X_{1},\dots ,X_{d})}$ by setting ${\displaystyle (X_{1}^{k},\dots ,X_{d}^{k})=(F_{1}^{-1}(U_{1}^{k}),\dots ,F_{d}^{-1}(U_{d}^{k}))\sim H\;\;(k=1,\dots ,n)}$
3. Approximate ${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]}$ by its empirical value:

${\displaystyle \operatorname {E} \left[g(X_{1},\dots ,X_{d})\right]\approx {\frac {1}{n}}\sum _{k=1}^{n}g(X_{1}^{k},\dots ,X_{d}^{k})}$

When studying multivariate data, one might want to investigate the underlying copula. Suppose we have observations

${\displaystyle (X_{1}^{i},X_{2}^{i},\dots ,X_{d}^{i}),\,i=1,\dots ,n}$

from a random vector ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$ with continuous marginals. The corresponding “true” copula observations would be

${\displaystyle (U_{1}^{i},U_{2}^{i},\dots ,U_{d}^{i})=\left(F_{1}(X_{1}^{i}),F_{2}(X_{2}^{i}),\dots ,F_{d}(X_{d}^{i})\right),\,i=1,\dots ,n.}$

However, the marginal distribution functions ${\displaystyle F_{i}}$ are usually not known. Therefore, one can construct pseudo copula observations by using the empirical distribution functions

${\displaystyle F_{k}^{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} (X_{k}^{i}\leq x)}$

instead. Then, the pseudo copula observations are defined as

${\displaystyle ({\tilde {U}}_{1}^{i},{\tilde {U}}_{2}^{i},\dots ,{\tilde {U}}_{d}^{i})=\left(F_{1}^{n}(X_{1}^{i}),F_{2}^{n}(X_{2}^{i}),\dots ,F_{d}^{n}(X_{d}^{i})\right),\,i=1,\dots ,n.}$

The corresponding empirical copula is then defined as

${\displaystyle C^{n}(u_{1},\dots ,u_{d})={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} \left({\tilde {U}}_{1}^{i}\leq u_{1},\dots ,{\tilde {U}}_{d}^{i}\leq u_{d}\right).}$

The components of the pseudo copula samples can also be written as ${\displaystyle {\tilde {U}}_{k}^{i}=R_{k}^{i}/n}$, where ${\displaystyle R_{k}^{i}}$ is the rank of the observation ${\displaystyle X_{k}^{i}}$:

${\displaystyle R_{k}^{i}=\sum _{j=1}^{n}\mathbf {1} (X_{k}^{j}\leq X_{k}^{i})}$

Therefore, the empirical copula can be seen as the empirical distribution of the rank transformed data.

The sample version of Spearman's rho: [19]

${\displaystyle r={\frac {12}{n^{2}-1}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left[C^{n}\left({\frac {i}{n}},{\frac {j}{n}}\right)-{\frac {i}{n}}\cdot {\frac {j}{n}}\right]}$

For any two random variables X and Y, the continuous joint probability distribution function can be written as

${\displaystyle F_{XY}(x,y)=\Pr {\begin{Bmatrix}X\leq {x},Y\leq {y}\end{Bmatrix}},}$

where ${\textstyle F_{X}(x)=\Pr {\begin{Bmatrix}X\leq {x}\end{Bmatrix}}}$ and ${\textstyle F_{Y}(y)=\Pr {\begin{Bmatrix}Y\leq {y}\end{Bmatrix}}}$ are the marginal cumulative distribution functions of the random variables X and Y, respectively.

then the copula distribution function ${\displaystyle C(u,v)}$ can be defined using Sklar's theorem[72][73] as:

${\displaystyle F_{XY}(x,y)=C(F_{X}(x),F_{Y}(y))\triangleq C(u,v)}$,

where ${\displaystyle u=F_{X}(x)}$ and ${\displaystyle v=F_{Y}(y)}$ are marginal distribution functions, ${\displaystyle F_{XY}(x,y)}$ joint and ${\displaystyle u,v\in (0,1)}$.

We start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.

${\displaystyle {\begin{alignedat}{6}f_{XY}(x,y)={}&{\partial ^{2}F_{XY}(x,y) \over \partial x\,\partial y}\\\vdots \\f_{XY}(x,y)={}&{\partial ^{2}C(F_{X}(x),F_{Y}(y)) \over \partial x\,\partial y}\\\vdots \\f_{XY}(x,y)={}&{\partial ^{2}C(u,v) \over \partial u\,\partial v}\cdot {\partial F_{X}(x) \over \partial x}\cdot {\partial F_{Y}(y) \over \partial y}\\\vdots \\f_{XY}(x,y)={}&c(u,v)f_{X}(x)f_{Y}(y)\\\vdots \\{\frac {f_{XY}(x,y)}{f_{X}(x)f_{Y}(y)}}={}&c(u,v)\end{alignedat}}}$

where ${\displaystyle c(u,v)}$ is the copula density function, ${\displaystyle f_{X}(x)}$ and ${\displaystyle f_{Y}(y)}$ are the marginal probability density functions of X and Y, respectively. It is important to understand that there are four elements in this equation, and if any three elements are known, the fourth element can be calculated. For example, it may be used,

• when joint probability density function between two random variables is known, the copula density function is known, and one of the two marginal functions are known, then, the other marginal function can be calculated, or
• when the two marginal functions and the copula density function are known, then the joint probability density function between the two random variables can be calculated, or
• when the two marginal functions and the joint probability density function between the two random variables are known, then the copula density function can be calculated.

List of copula density functions and applications

Various bivariate copula density functions are important in the area of signal processing. ${\displaystyle u=F_{X}(x)}$ and ${\displaystyle v=F_{Y}(y)}$ are marginal distributions functions and ${\displaystyle f_{X}(x)}$ and ${\displaystyle f_{Y}(y)}$ are marginal density functions. Extension and generalization of copulas for statistical signal processing have been shown to construct new bivariate copulas for exponential, Weibull, and Rician distributions.[74] Zeng et al.[75] presented algorithms, simulation, optimal selection, and practical applications of these copulas in signal processing.

	Copula density: c(u, v)	Use
Gaussian	${\displaystyle {\begin{aligned}={}&{\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {(a^{2}+b^{2})\rho ^{2}-2ab\rho }{2(1-\rho ^{2})}}\right)\\&{\text{where }}\rho \in (-1,1)\\&{\text{where }}a={\sqrt {2}}\operatorname {erf} ^{-1}({2u-1})\\&{\text{where }}b={\sqrt {2}}\operatorname {erf} ^{-1}({2v-1})\\&{\text{where }}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int \limits _{0}^{z}\exp(-t^{2})\,dt\end{aligned}}}$	supervised classification of synthetic aperture radar (SAR) images,[76] validating biometric authentication,[77] modeling stochastic dependence in large-scale integration of wind power,[78] unsupervised classification of radar signals[79]
Exponential	${\displaystyle {\begin{aligned}={}&{\frac {1}{1-\rho }}\exp \left({\frac {\rho (\ln(1-u)+\ln(1-v))}{1-\rho }}\right)\cdot I_{0}\left({\frac {2{\sqrt {\rho \ln(1-u)\ln(1-v)}}}{1-\rho }}\right)\\&{\text{where }}x=F_{X}^{-1}(u)=-\ln(1-u)/\lambda \\&{\text{where }}y=F_{Y}^{-1}(v)=-\ln(1-v)/\mu \end{aligned}}}$	queuing system with infinitely many servers[80]
Rayleigh	bivariate exponential, Rayleigh, and Weibull copulas have been proved to be equivalent[81][82][83]	change detection from SAR images[84]
Weibull	bivariate exponential, Rayleigh, and Weibull copulas have been proved to be equivalent[81][82][83]	digital communication over fading channels[85]
Log-normal	bivariate log-normal copula and Gaussian copula are equivalent[83][82]	shadow fading along with multipath effect in wireless channel[86][87]
Farlie–Gumbel–Morgenstern (FGM)	${\displaystyle {\begin{aligned}={}&1+\theta (1-2u)(1-2v)\\&{\text{where }}\theta \in [-1,1]\end{aligned}}}$	information processing of uncertainty in knowledge-based systems[88]
Clayton	${\displaystyle {\begin{aligned}={}&(1+\theta )(uv)^{(-1-\theta )}(-1+u^{-\theta }+v^{-\theta })^{(-2-1/\theta )}\\&{\text{where }}\theta \in (-1,\infty ),\theta \neq 0\end{aligned}}}$	location estimation of random signal source and hypothesis testing using heterogeneous data[89][90]
Frank	${\displaystyle {\begin{aligned}={}&{\frac {\theta e^{\theta (u+v)}(e^{\theta }-1)}{(e^{\theta }-e^{\theta u}-e^{\theta v}+e^{\theta (u+v)})^{2}}}\\&{\text{where }}\theta \in (-\infty ,+\infty ),\theta \neq 0\end{aligned}}}$	change detection in remote sensing applications[91]
Student's t	${\displaystyle {\begin{aligned}={}&{\frac {\Gamma (0.5v)\Gamma (0.5v+1)(1+(t_{v}^{-2}(u)+t_{v}^{-2}(v)-2\rho t_{v}^{-1}(u)t_{v}^{-1}(v))/(v(1-\rho ^{2})))^{-0.5(v+2)})}{{\sqrt {1-\rho ^{2}}}\cdot \Gamma (0.5(v+1))^{2}(1+t_{v}^{-2}(u)/v)^{-0.5(v+1)}(1+t_{v}^{-2}(v)/v)^{-0.5(v+1)}}}\\&{\text{where }}\rho \in (-1,1)\\&{\text{where }}\phi (z)={\frac {1}{\sqrt {2\pi }}}\int \limits _{-\infty }^{z}\exp \left({\frac {-t^{2}}{2}}\right)\,dt\\&{\text{where }}t_{v}(x\mid v)=\int \limits _{-\infty }^{x}{\frac {\Gamma {(0.5(v+1))}}{{\sqrt {v\pi }}(\Gamma {0.5v})(1+v^{-1}t^{2})^{0.5(v+1)}}}dt\\&{\text{where }}v={\text{degrees of freedom}}\\&{\text{where }}\Gamma {\text{ is the Gamma function}}\end{aligned}}}$	supervised SAR image classification,[84] fusion of correlated sensor decisions[92]
Nakagami-m
Rician

• Coupling (probability)

• The standard reference for an introduction to copulas. Covers all fundamental aspects, summarizes the most popular copula classes, and provides proofs for the important theorems related to copulas

Roger B. Nelsen (1999), "An Introduction to Copulas", Springer. ISBN 978-0-387-98623-4

• A book covering current topics in mathematical research on copulas:

Piotr Jaworski, Fabrizio Durante, Wolfgang Karl Härdle, Tomasz Rychlik (Editors): (2010): "Copula Theory and Its Applications" Lecture Notes in Statistics, Springer. ISBN 978-3-642-12464-8

• A reference for sampling applications and stochastic models related to copulas is

Jan-Frederik Mai, Matthias Scherer (2012): Simulating Copulas (Stochastic Models, Sampling Algorithms and Applications). World Scientific. ISBN 978-1-84816-874-9

• A paper covering the historic development of copula theory, by the person associated with the "invention" of copulas, Abe Sklar.

Abe Sklar (1997): "Random variables, distribution functions, and copulas – a personal look backward and forward" in Rüschendorf, L., Schweizer, B. und Taylor, M. (eds) Distributions With Fixed Marginals & Related Topics (Lecture Notes – Monograph Series Number 28). ISBN 978-0-940600-40-9

• The standard reference for multivariate models and copula theory in the context of financial and insurance models

Alexander J. McNeil, Rudiger Frey and Paul Embrechts (2005) "Quantitative Risk Management: Concepts, Techniques, and Tools", Princeton Series in Finance. ISBN 978-0-691-12255-7

• "Copula", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Copula Wiki: community portal for researchers with interest in copulas
• A collection of Copula simulation and estimation codes
• Copulas & Correlation using Excel Simulation Articles
• Chapter 1 of Jan-Frederik Mai, Matthias Scherer (2012) "Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications"