Copula (statistics)

Consider a random vector $(X\_1,X\_2,\backslash dots,X\_d)$. Suppose its marginals are continuous, i.e. the marginal CDFs $F\_i(x)\; =\; \backslash Pr[X\_i\backslash leq\; x]$ are continuous functions. By applying the probability integral transform to each component, the random vector

:$(U\_1,U\_2,\backslash dots,U\_d)=\backslash left(F\_1(X\_1),F\_2(X\_2),\backslash dots,F\_d(X\_d)\backslash right)$

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

The copula of $(X\_1,X\_2,\backslash dots,X\_d)$ is defined as the joint cumulative distribution function of $(U\_1,U\_2,\backslash dots,U\_d)$:

:$C(u\_1,u\_2,\backslash dots,u\_d)=\backslash Pr[U\_1\backslash leq\; u\_1,U\_2\backslash leq\; u\_2,\backslash dots,U\_d\backslash leq\; u\_d].$

The copula C contains all information on the dependence structure between the components of $(X\_1,X\_2,\backslash dots,X\_d)$ whereas the marginal cumulative distribution functions $F\_i$ contain all information on the marginal distributions of $X\_i$.

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 $(U\_1,U\_2,\backslash dots,U\_d)$ from the copula function, the required sample can be constructed as

:$(X\_1,X\_2,\backslash dots,X\_d)\; =\; \backslash left(F\_1^\{-1\}(U\_1),F\_2^\{-1\}(U\_2),\backslash dots,F\_d^\{-1\}(U\_d)\backslash right).$

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

:$C(u\_1,u\_2,\backslash dots,u\_d)=\backslash Pr[X\_1\backslash leq\; F\_1^\{-1\}(u\_1),X\_2\backslash leq\; F\_2^\{-1\}(u\_2),\backslash dots,X\_d\backslash leq\; F\_d^\{-1\}(u\_d)]\; .$

In probabilistic terms, $C:[0,1]^d\backslash rightarrow\; [0,1]$ is a d-dimensional copula if C is a joint cumulative distribution function of a d-dimensional random vector on the unit cube $[0,1]^d$ with uniform marginals.{{Citation |first=Roger B. |last=Nelsen |year=1999 |title=An Introduction to Copulas |location=New York |publisher=Springer |isbn=978-0-387-98623-4 }}

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

:* $C(u\_1,\backslash dots,u\_\{i-1\},0,u\_\{i+1\},\backslash dots,u\_d)=0$, the copula is zero if any one of the arguments is zero,

:* $C(1,\backslash dots,1,u,1,\backslash dots,1)=u$, 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 $B=\backslash prod\_\{i=1\}^\{d\}[x\_i,y\_i]\backslash subseteq\; [0,1]^d$ the C-volume of B is non-negative:

:*:$\backslash int\_B\; \backslash mathrm\{d\}\; C(u)\; =\backslash sum\_\{\backslash mathbf\; z\backslash in\; \backslash prod\_\{i=1\}^\{d\}\backslash \{x\_i,y\_i\backslash \}\}\; (-1)^\{N(\backslash mathbf\; z)\}\; C(\backslash mathbf\; z)\backslash ge\; 0,$

::where the $N(\backslash mathbf\; z)=\backslash \#\backslash \{k\; :\; z\_k=x\_k\backslash \}$.

For instance, in the bivariate case, $C:[0,1]\; \backslash times[0,1]\backslash rightarrow\; [0,1]$ is a bivariate copula if $C(0,u)\; =\; C(u,0)\; =\; 0$, $C(1,u)\; =\; C(u,1)\; =\; u$ and $C(u\_2,v\_2)-C(u\_2,v\_1)-C(u\_1,v\_2)+C(u\_1,v\_1)\; \backslash geq\; 0$ for all $0\; \backslash leq\; u\_1\; \backslash leq\; u\_2\; \backslash leq\; 1$ and $0\; \backslash leq\; v\_1\; \backslash leq\; v\_2\; \backslash leq\; 1$.

File:Gaussian copula gaussian marginals.png

File:Biv gumbel dist.png

Sklar's theorem, named after Abe Sklar, provides the theoretical foundation for the application of copulas.{{citation | last=Sklar | first=A. | author-link = Abe Sklar | title=Fonctions de répartition à n dimensions et leurs marges | journal=Publ. Inst. Statist. Univ. Paris | year=1959 | volume=8 | pages=229–231 }}{{citation |first1=Fabrizio |last1=Durante |first2=Juan |last2=Fernández-Sánchez |first3=Carlo |last3=Sempi |title=A Topological Proof of Sklar's Theorem |journal=Applied Mathematics Letters |volume=26 |issue=9 |year=2013 |pages=945–948 |doi=10.1016/j.aml.2013.04.005 |doi-access=free }} Sklar's theorem states that every multivariate cumulative distribution function

:$H(x\_1,\backslash dots,x\_d)=\backslash Pr[X\_1\backslash leq\; x\_1,\backslash dots,X\_d\backslash leq\; x\_d]$

of a random vector $(X\_1,X\_2,\backslash dots,X\_d)$ can be expressed in terms of its marginals $F\_i(x\_i)\; =\; \backslash Pr[X\_i\backslash leq\; x\_i]$ and

a copula $C$. Indeed:

:$H(x\_1,\backslash dots,x\_d)\; =\; C\backslash left(F\_1(x\_1),\backslash dots,F\_d(x\_d)\; \backslash right).$

If the multivariate distribution has a density $h$, and if this density is available, it also holds that

:$h(x\_1,\backslash dots,x\_d)=\; c(F\_1(x\_1),\backslash dots,F\_d(x\_d))\backslash cdot\; f\_1(x\_1)\backslash cdot\backslash dots\backslash cdot\; f\_d(x\_d),$

where $c$ is the density of the copula.

The theorem also states that, given $H$, the copula is unique on $\backslash operatorname\{Ran\}(F\_1)\backslash times\backslash cdots\backslash times\; \backslash operatorname\{Ran\}(F\_d)$ which is the cartesian product of the ranges of the marginal cdf's. This implies that the copula is unique if the marginals $F\_i$ are continuous.

The converse is also true: given a copula $C:[0,1]^d\backslash rightarrow\; [0,1]$ and marginals $F\_i(x)$ then $C\backslash left(F\_1(x\_1),\backslash dots,F\_d(x\_d)\; \backslash right)$ defines a d-dimensional cumulative distribution function with marginal distributions $F\_i(x)$.

Copulas mainly work when time series are stationary{{Cite journal|last1=Sadegh|first1=Mojtaba|last2=Ragno|first2=Elisa|last3=AghaKouchak|first3=Amir|date=2017|title=Multivariate Copula Analysis Toolbox (MvCAT): Describing dependence and underlying uncertainty using a Bayesian framework|journal=Water Resources Research|language=en|volume=53|issue=6|pages=5166–5183|doi=10.1002/2016WR020242|issn=1944-7973|bibcode=2017WRR....53.5166S|url=https://scholarworks.boisestate.edu/civileng_facpubs/92|doi-access=free}} and continuous.{{Cite journal|last1=AghaKouchak|first1=Amir|last2=Bárdossy|first2=András|last3=Habib|first3=Emad|date=2010|title=Copula-based uncertainty modelling: application to multisensor precipitation estimates|journal=Hydrological Processes|language=en|volume=24|issue=15|pages=2111–2124|doi=10.1002/hyp.7632|bibcode=2010HyPr...24.2111A |s2cid=12283329 |issn=1099-1085}} 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 existing dependence between sets of variables and result in incorrect copula dependence structure.{{Cite journal|last1=Tootoonchi|first1=Faranak|last2=Haerter|first2=Jan Olaf|last3=Räty|first3=Olle|last4=Grabs|first4=Thomas|last5=Sadegh|first5=Mojtaba|last6=Teutschbein|first6=Claudia|date=2020-07-21|title=Copulas for hydroclimatic applications – A practical note on common misconceptions and pitfalls|url=https://hess.copernicus.org/preprints/hess-2020-306/|journal=Hydrology and Earth System Sciences Discussions|language=en|pages=1–31|doi=10.5194/hess-2020-306|s2cid=224352645 |issn=1027-5606 |doi-access=free }}

File:copule ord.svg

The Fréchet–Hoeffding theorem (after Maurice René Fréchet and Wassily Hoeffding{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Hoeffding.html |title=Biography of Wassily Hoeffding |author=J. J. O'Connor and E. F. Robertson |date= March 2011 |publisher= School of Mathematics and Statistics, University of St Andrews, Scotland |access-date=14 February 2019}}) states that for any copula $C:[0,1]^d\backslash rightarrow\; [0,1]$ and any $(u\_1,\backslash dots,u\_d)\backslash in[0,1]^d$ the following bounds hold:

: $W(u\_1,\backslash dots,u\_d)\; \backslash leq\; C(u\_1,\backslash dots,u\_d)\; \backslash leq\; M(u\_1,\backslash dots,u\_d).$

The function {{mvar|W}} is called lower Fréchet–Hoeffding bound and is defined as

:$W(u\_1,\backslash ldots,u\_d)\; =\; \backslash max\backslash left\backslash \{1-d+\backslash sum\backslash limits\_\{i=1\}^d\; \{u\_i\}\; ,\backslash ,\; 0\; \backslash right\backslash \}.$

The function {{mvar|M}} is called upper Fréchet–Hoeffding bound and is defined as

:$M(u\_1,\backslash ldots,u\_d)\; =\; \backslash min\; \backslash \{u\_1,\backslash dots,u\_d\backslash \}.$

The upper bound is sharp: {{mvar|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 $\backslash tilde\{C\}$ such that $\backslash tilde\{C\}(u)\; =\; W(u)$. However, {{mvar|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

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

Several families of copulas have been described.

File:Copula gaussian.svg

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

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

:$C\_R^\{\backslash text\{Gauss\}\}(u)\; =\; \backslash Phi\_R\backslash left(\backslash Phi^\{-1\}(u\_1),\backslash dots,\; \backslash Phi^\{-1\}(u\_d)\; \backslash right),$

where $\backslash Phi^\{-1\}$ is the inverse cumulative distribution function of a standard normal and $\backslash Phi\_R$ is the joint cumulative distribution function of a multivariate normal distribution with mean vector zero and covariance matrix equal to the correlation matrix $R$. While there is no simple analytical formula for the copula function, $C\_R^\{\backslash text\{Gauss\}\}(u)$, it can be upper or lower bounded, and approximated using numerical integration.{{cite journal|last1=Botev|first1=Z. I.|title=The normal law under linear restrictions: simulation and estimation via minimax tilting|journal=Journal of the Royal Statistical Society, Series B|volume=79|pages=125–148|date=2016|doi=10.1111/rssb.12162|arxiv=1603.04166|bibcode=2016arXiv160304166B|s2cid=88515228}}{{cite web|url=https://cran.r-project.org/package=TruncatedNormal|title=TruncatedNormal: Truncated Multivariate Normal|first=Zdravko I.|last=Botev|date=10 November 2015|via=R-Packages}} The density can be written as{{cite journal |first=Philipp |last=Arbenz |year=2013 |title=Bayesian Copulae Distributions, with Application to Operational Risk Management—Some Comments |journal=Methodology and Computing in Applied Probability |volume=15 |issue=1 |pages=105–108 |doi=10.1007/s11009-011-9224-0 |hdl=20.500.11850/64244 |s2cid=121861059 |hdl-access=free }}

:$c\_R^\{\backslash text\{Gauss\}\}(u)$

= \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 $I$ is the identity matrix.

Archimedean copulas are an associative class of copulas. Most common Archimedean copulas admit an explicit formula, something not possible for instance for the Gaussian copula.

In practice, Archimedean copulas are popular because they allow modeling dependence in arbitrarily high dimensions with only one parameter, governing the strength of dependence.

A copula C is called Archimedean if it admits the representation{{cite book |last=Nelsen |first=R. B. |year=2006 |title=An Introduction to Copulas |edition=Second |publisher=Springer |location=New York |isbn=978-1-4419-2109-3 }}

:$C(u\_1,\backslash dots,u\_d;\backslash theta)\; =\; \backslash psi^\{-1\}\backslash left(\backslash psi(u\_1;\backslash theta)+\backslash cdots+\backslash psi(u\_d;\backslash theta);\backslash theta\backslash right)$

where $\backslash psi\backslash !:[0,1]\backslash times\backslash Theta\; \backslash rightarrow\; [0,\backslash infty)$ is a continuous, strictly decreasing and convex function such that $\backslash psi(1;\backslash theta)=0$, $\backslash theta$ is a parameter within some parameter space $\backslash Theta$, and $\backslash psi$ is the so-called generator function and $\backslash psi^\{-1\}$ is its pseudo-inverse defined by

:

Moreover, the above formula for C yields a copula for $\backslash psi^\{-1\}$ if and only if $\backslash psi^\{-1\}$ is d-monotone on $[0,\backslash infty)$.{{cite journal |last1=McNeil |first1=A. J. |last2=Nešlehová |first2=J.|author2-link= Johanna G. Nešlehová |year=2009 |title=Multivariate Archimedean copulas, d-monotone functions and $\backslash mathit\{l\}$1-norm symmetric distributions |journal=Annals of Statistics |volume=37 |issue=5b |pages=3059–3097 |doi=10.1214/07-AOS556 |arxiv=0908.3750 |s2cid=9858856 }}

That is, if it is $d-2$ times differentiable and the derivatives satisfy

:$(-1)^k\backslash psi^\{-1,(k)\}(t;\backslash theta)\; \backslash geq\; 0$

for all $t\backslash geq\; 0$ and $k=0,1,\backslash dots,d-2$ and $(-1)^\{d-2\}\backslash psi^\{-1,(d-2)\}(t;\backslash theta)$ is nonincreasing and convex.

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 $d\backslash in\backslash mathbb\{N\}$ or d-monotone for certain $\backslash theta\; \backslash in\; \backslash Theta$ only.

In statistical applications, many problems can be formulated in the following way. One is interested in the expectation of a response function $g:\backslash mathbb\{R\}^d\backslash rightarrow\backslash mathbb\{R\}$ applied to some random vector $(X\_1,\backslash dots,X\_d)$.Alexander J. McNeil, Rudiger Frey and Paul Embrechts (2005) "Quantitative Risk Management: Concepts, Techniques, and Tools", Princeton Series in Finance If we denote the CDF of this random vector with $H$, the quantity of interest can thus be written as

: $\backslash operatorname\{E\}\backslash left[\; g(X\_1,\backslash dots,X\_d)\; \backslash right]\; =\; \backslash int\_\{\backslash mathbb\{R\}^d\}\; g(x\_1,\backslash dots,x\_d)\; \backslash ,\; \backslash mathrm\{d\}H(x\_1,\backslash dots,x\_d).$

If $H$ is given by a copula model, i.e.,

:$H(x\_1,\backslash dots,x\_d)=C(F\_1(x\_1),\backslash dots,F\_d(x\_d))$

this expectation can be rewritten as

:$\backslash operatorname\{E\}\backslash left[g(X\_1,\backslash dots,X\_d)\backslash right]=\backslash int\_\{[0,1]^d\}g(F\_1^\{-1\}(u\_1),\backslash dots,F\_d^\{-1\}(u\_d))\; \backslash ,\; \backslash mathrm\{d\}C(u\_1,\backslash dots,u\_d).$

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

:$\backslash operatorname\{E\}\backslash left[g(X\_1,\backslash dots,X\_d)\backslash right]=\backslash int\_\{[0,1]^d\}g(F\_1^\{-1\}(u\_1),\backslash dots,F\_d^\{-1\}(u\_d))\backslash cdot\; c(u\_1,\backslash dots,u\_d)\; \backslash ,\; du\_1\backslash cdots\; \backslash mathrm\{d\}u\_d,$

and if each marginal distribution has the density $f\_i$ it holds further that

:$\backslash operatorname\{E\}\backslash left[g(X\_1,\backslash dots,X\_d)\backslash right]=\backslash int\_\{\backslash mathbb\{R\}^d\}g(x\_1,\backslash dots\; x\_d)\backslash cdot\; c(F\_1(x\_1),\backslash dots,F\_d(x\_d))\backslash cdot\; f\_1(x\_1)\backslash cdots\; f\_d(x\_d)\; \backslash ,\; \backslash mathrm\{d\}x\_1\backslash cdots\; \backslash mathrm\{d\}x\_d.$

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 $(U\_1^k,\backslash dots,U\_d^k)\backslash sim\; C\backslash ;\backslash ;(k=1,\backslash dots,n)$ of size n from the copula C
2. By applying the inverse marginal cdf's, produce a sample of $(X\_1,\backslash dots,X\_d)$ by setting $(X\_1^k,\backslash dots,X\_d^k)=(F\_1^\{-1\}(U\_1^k),\backslash dots,F\_d^\{-1\}(U\_d^k))\backslash sim\; H\backslash ;\backslash ;(k=1,\backslash dots,n)$
3. Approximate $\backslash operatorname\{E\}\backslash left[g(X\_1,\backslash dots,X\_d)\backslash right]$ by its empirical value:

:::$\backslash operatorname\{E\}\backslash left[g(X\_1,\backslash dots,X\_d)\backslash right]\backslash approx\; \backslash frac\{1\}\{n\}\backslash sum\_\{k=1\}^n\; g(X\_1^k,\backslash dots,X\_d^k)$

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

:$(X\_1^i,X\_2^i,\backslash dots,X\_d^i),\; \backslash ,\; i=1,\backslash dots,n$

from a random vector $(X\_1,X\_2,\backslash dots,X\_d)$ with continuous marginals. The corresponding “true” copula observations would be

:$(U\_1^i,U\_2^i,\backslash dots,U\_d^i)=\backslash left(F\_1(X\_1^i),F\_2(X\_2^i),\backslash dots,F\_d(X\_d^i)\backslash right),\; \backslash ,\; i=1,\backslash dots,n.$

However, the marginal distribution functions $F\_i$ are usually not known. Therefore, one can construct pseudo copula observations by using the empirical distribution functions

:$F\_k^n(x)=\backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\{1\}(X\_k^i\backslash leq\; x)$

instead. Then, the pseudo copula observations are defined as

:$(\backslash tilde\{U\}\_1^i,\backslash tilde\{U\}\_2^i,\backslash dots,\backslash tilde\{U\}\_d^i)=\backslash left(F\_1^n(X\_1^i),F\_2^n(X\_2^i),\backslash dots,F\_d^n(X\_d^i)\backslash right),\; \backslash ,\; i=1,\backslash dots,n.$

The corresponding empirical copula is then defined as

:$C^n(u\_1,\backslash dots,u\_d)\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\{1\}\backslash left(\backslash tilde\{U\}\_1^i\backslash leq\; u\_1,\backslash dots,\backslash tilde\{U\}\_d^i\backslash leq\; u\_d\backslash right).$

The components of the pseudo copula samples can also be written as $\backslash tilde\{U\}\_k^i=R\_k^i/n$, where $R\_k^i$ is the rank of the observation $X\_k^i$:

:$R\_k^i=\backslash sum\_\{j=1\}^n\; \backslash mathbf\{1\}(X\_k^j\backslash leq\; X\_k^i)$

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

The sample version of Spearman's rho:{{cite book |last1=Nelsen |first1=Roger B. |title=An introduction to copulas |date=2006 |publisher=Springer |location=New York |isbn=978-0-387-28678-5 |page=220 |edition=2nd}}

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

File:Four_Correlations.svg

In quantitative finance copulas are applied to risk management, to portfolio management and optimization, and to derivatives pricing.

For the former, copulas are used to perform stress-tests and robustness checks that are especially important during "downside/crisis/panic regimes" where extreme downside events may occur (e.g., the 2008 financial crisis). The formula was also adapted for financial markets and was used to estimate the probability distribution of losses on pools of loans or bonds.

During a downside regime, a large number of investors who have held positions in riskier assets such as equities or real estate may seek refuge in 'safer' investments such as cash or bonds. This is also known as a flight-to-quality effect and investors tend to exit their positions in riskier assets in large numbers in a short period of time. As a result, during downside regimes, correlations across equities are greater on the downside as opposed to the upside and this may have disastrous effects on the economy.{{Citation

| last1 = Longin |first1= F |last2= Solnik |first2= B| year = 2001

| title = Extreme correlation of international equity markets

| journal = Journal of Finance

| volume= 56

| issue= 2

| pages=649–676

| doi=10.1111/0022-1082.00340

|citeseerx= 10.1.1.321.4899 |s2cid= 6143150 }}{{Citation

| last1 = Ang |first1= A |last2= Chen |first2= J

| year = 2002

| title = Asymmetric correlations of equity portfolios

| journal = Journal of Financial Economics

| volume=63

| issue=3

| pages=443–494

| doi=10.1016/s0304-405x(02)00068-5

}} For example, anecdotally, we often read financial news headlines reporting the loss of hundreds of millions of dollars on the stock exchange in a single day; however, we rarely read reports of positive stock market gains of the same magnitude and in the same short time frame.

Copulas aid in analyzing the effects of downside regimes by allowing the modelling of the marginals and dependence structure of a multivariate probability model separately. For example, consider the stock exchange as a market consisting of a large number of traders each operating with his/her own strategies to maximize profits. The individualistic behaviour of each trader can be described by modelling the marginals. However, as all traders operate on the same exchange, each trader's actions have an interaction effect with other traders'. This interaction effect can be described by modelling the dependence structure. Therefore, copulas allow us to analyse the interaction effects which are of particular interest during downside regimes as investors tend to herd their trading behaviour and decisions. (See also agent-based computational economics, where price is treated as an emergent phenomenon, resulting from the interaction of the various market participants, or agents.)

The users of the formula have been criticized for creating "evaluation cultures" that continued to use simple copulæ despite the simple versions being acknowledged as inadequate for that purpose.{{Cite magazine |last=Salmon |first=Felix |title=Recipe for Disaster: The Formula That Killed Wall Street |language=en-US |magazine=Wired |url=https://www.wired.com/2009/02/wp-quant/ |access-date=2023-08-11 |issn=1059-1028}}{{Cite journal |last1=MacKenzie |first1=Donald |last2=Spears |first2=Taylor |date=2014 |title='The formula that killed Wall Street': The Gaussian copula and modelling practices in investment banking |url=https://www.jstor.org/stable/43284238 |journal=Social Studies of Science |volume=44 |issue=3 |pages=393–417 |doi=10.1177/0306312713517157 |jstor=43284238 |pmid=25051588 |hdl=20.500.11820/3095760a-6d7c-4829-b327-98c9c28c1db6 |s2cid=15907952 |issn=0306-3127|hdl-access=free }} Thus, previously, scalable copula models for large dimensions only allowed the modelling of elliptical dependence structures (i.e., Gaussian and Student-t copulas) that do not allow for correlation asymmetries where correlations differ on the upside or downside regimes. However, the development of vine copulas{{cite book|last1=Cooke|first1=R.M.|last2=Joe|first2=H.|last3=Aas|first3=K.|title=Dependence Modeling Vine Copula Handbook|date= January 2011|publisher=World Scientific|isbn= 978-981-4299-87-9|pages=37–72|url=http://rogermcooke.net/rogermcooke_files/Vines%20Arise%20Handbook%20VCM.pdf|editor1-last=Kurowicka|editor1-first=D.|editor2-last=Joe|editor2-first=H.}} (also known as pair copulas) enables the flexible modelling of the dependence structure for portfolios of large dimensions.{{Citation

| last1 = Aas |first1= K |last2= Czado |first2= C|author2-link= Claudia Czado |last3= Bakken|first3= H | year = 2009

| title = Pair-copula constructions of multiple dependence

| journal = Insurance: Mathematics and Economics

| volume=44

| issue=2

| pages=182–198

| doi=10.1016/j.insmatheco.2007.02.001

|citeseerx= 10.1.1.61.3984 |s2cid= 18320750 }}

The Clayton canonical vine copula allows for the occurrence of extreme downside events and has been successfully applied in portfolio optimization and risk management applications. The model is able to reduce the effects of extreme downside correlations and produces improved statistical and economic performance compared to scalable elliptical dependence copulas such as the Gaussian and Student-t copula.{{Citation

| last1 = Low |first1= R |last2= Alcock |first2= J|last3= Brailsford|first3= T |last4= Faff|first4= R

| year = 2013

| title = Canonical vine copulas in the context of modern portfolio management: Are they worth it?

| journal = Journal of Banking and Finance

| volume=37

| issue=8

| pages=3085–3099

| doi=10.1016/j.jbankfin.2013.02.036

|s2cid= 154138333 }}

Other models developed for risk management applications are panic copulas that are glued with market estimates of the marginal distributions to analyze the effects of panic regimes on the portfolio profit and loss distribution. Panic copulas are created by Monte Carlo simulation, mixed with a re-weighting of the probability of each scenario.{{Citation

| last = Meucci

| first = Attilio

| year = 2011

| title = A New Breed of Copulas for Risk and Portfolio Management

| journal = Risk

| volume=24

| issue=9

| pages=122–126

| url = http://ssrn.com/abstract=1752702

}}

As regards derivatives pricing, dependence modelling with copula functions is widely used in applications of financial risk assessment and actuarial analysis – for example in the pricing of collateralized debt obligations (CDOs).{{Citation|last1=Meneguzzo|first1=David|title=Copula sensitivity in collateralized debt obligations and basket default swaps|date=Nov 2003|journal=Journal of Futures Markets|volume=24|issue=1|pages=37–70|doi=10.1002/fut.10110|last2=Vecchiato|first2=Walter}} Some believe the methodology of applying the Gaussian copula to credit derivatives to be one of the causes of the 2008 financial crisis;[https://www.wired.com/techbiz/it/magazine/17-03/wp_quant?currentPage=all Recipe for Disaster: The Formula That Killed Wall Street] Wired, 2/23/2009{{Citation

| last = MacKenzie

| first = Donald

| publication-date = 2008-05-08

| year = 2008

| title = End-of-the-World Trade

| pages = 24–26

| periodical = London Review of Books

| url = http://www.lrb.co.uk/v30/n09/mack01_.html

| access-date = 2009-07-27

}}{{Citation |title=The formula that felled Wall St |url=http://www.ft.com/cms/s/2/912d85e8-2d75-11de-9eba-00144feabdc0.html |archive-url=https://ghostarchive.org/archive/20221211181245/https://www.ft.com/content/912d85e8-2d75-11de-9eba-00144feabdc0 |archive-date=2022-12-11 |url-access=subscription |url-status=live |first=Sam |last=Jones |newspaper=Financial Times |date=April 24, 2009 |access-date=2010-05-05 }} see {{slink|David X. Li #CDOs and Gaussian copula}}.

Despite this perception, there are documented attempts within the financial industry, occurring before the crisis, to address the limitations of the Gaussian copula and of copula functions more generally, specifically the lack of dependence dynamics. The Gaussian copula is lacking as it only allows for an elliptical dependence structure, as dependence is only modeled using the variance-covariance matrix. This methodology is limited such that it does not allow for dependence to evolve as the financial markets exhibit asymmetric dependence, whereby correlations across assets significantly increase during downturns compared to upturns. Therefore, modeling approaches using the Gaussian copula exhibit a poor representation of extreme events.{{cite book| first1=Alexander|last1= Lipton |first2= Andrew |last2=Rennie |title=Credit Correlation: Life After Copulas|publisher= World Scientific|isbn= 978-981-270-949-3|year= 2008 }} There have been attempts to propose models rectifying some of the copula limitations.{{Cite journal

| last1 = Donnelly|first1= C|last2= Embrechts|first2= P

| year = 2010

| title = The devil is in the tails: actuarial mathematics and the subprime mortgage crisis

| journal = ASTIN Bulletin

|volume = 40

|issue = 1

|pages=1–33

|doi= 10.2143/AST.40.1.2049222|hdl= 20.500.11850/20517|hdl-access= free

}}{{Cite book

| last1 = Brigo |first1=D |last2= Pallavicini |first2= A|last3= Torresetti |first3= R

| year = 2010

| title = Credit Models and the Crisis: A Journey into CDOs, Copulas, Correlations and dynamic Models

| publisher = Wiley and Sons

}}

Additional to CDOs, copulas have been applied to other asset classes as a flexible tool in analyzing multi-asset derivative products. The first such application outside credit was to use a copula to construct a basket implied volatility surface,{{Cite journal

| last = Qu, Dong

| year = 2001

| title = Basket Implied Volatility Surface

| journal= Derivatives Week|issue=4 June

}} taking into account the volatility smile of basket components. Copulas have since gained popularity in pricing and risk management{{Cite journal

| last = Qu, Dong

| year = 2005

| title = Pricing Basket Options With Skew

| journal = Wilmott Magazine |issue= July

}} of options on multi-assets in the presence of a volatility smile, in equity-, foreign exchange- and fixed income derivatives.

Recently, copula functions have been successfully applied to the database formulation for the reliability analysis of highway bridges, and to various multivariate simulation studies in civil engineering,{{Citation

| last1 = Thompson| first1 = David

| last2 = Kilgore| first2 = Roger

| publication-date = 2011| year = 2011

| title = Estimating Joint Flow Probabilities at Stream Confluences using Copulas

| journal = Transportation Research Record

| volume=2262

| pages=200–206

| url = http://trb.metapress.com/content/m3146tg612k80771/?p=d6b0d7200af148b8a4e18e592ca1e269&pi=3

| access-date = 2012-02-21

| doi=10.3141/2262-20

| s2cid = 17179491

| url-access = subscription

}} reliability of wind and earthquake engineering,{{Cite journal|last1=Yang|first1=S.C.|last2=Liu|first2=T.J.|last3=Hong|first3=H.P.|date=2017|title=Reliability of Tower and Tower-Line Systems under Spatiotemporally Varying Wind or Earthquake Loads|journal=Journal of Structural Engineering|volume=143|issue=10|pages=04017137|doi=10.1061/(ASCE)ST.1943-541X.0001835}} and mechanical & offshore engineering.{{Cite journal|last1=Zhang|first1=Yi|last2=Beer|first2=Michael|last3=Quek|first3=Ser Tong|date=2015-07-01|title=Long-term performance assessment and design of offshore structures|journal=Computers & Structures|volume=154|pages=101–115|doi=10.1016/j.compstruc.2015.02.029}} Researchers are also trying these functions in the field of transportation to understand the interaction between behaviors of individual drivers which, in totality, shapes traffic flow.

Copulas are being used for reliability analysis of complex systems of machine components with competing failure modes.

{{citation | last = Pham | first = Hong | title = Handbook of Reliability Engineering | pages = 150–151 | publisher = Springer | year = 2003}}

Copulas are being used for warranty data analysis in which the tail dependence is analysed.{{Citation | year = 2014 |last1=Wu |first1 = S. |title= Construction of asymmetric copulas and its application in two-dimensional reliability modelling | journal = European Journal of Operational Research |doi=10.1016/j.ejor.2014.03.016 |volume=238 |issue=2 |pages=476–485

|s2cid=22916401 |url=http://kar.kent.ac.uk/38763/1/ShaominWu.pdf }}

Copulas are used in modelling turbulent partially premixed combustion, which is common in practical combustors.{{Citation

| year = 2014

| last1= Ruan

| first1 = S.

| last2 = Swaminathan

| first2 = N

| last3 = Darbyshire

| first3 = O

| title = Modelling of turbulent lifted jet flames using flamelets: a priori assessment and a posteriori validation

| volume = 18 | issue = 2

| pages = 295–329

| journal = Combustion Theory and Modelling

| doi=10.1080/13647830.2014.898409

| bibcode= 2014CTM....18..295R

| s2cid= 53641133

}}{{Citation

| year = 2012

| last1=Darbyshire

| first1 = O.R.

| last2 = Swaminathan

| first2 = N

| title = A presumed joint pdf model for turbulent combustion with varying equivalence ratio

| volume = 184 | issue = 12

| pages = 2036–2067

| journal = Combustion Science and Technology

| doi=10.1080/00102202.2012.696566

| s2cid=98096093

}}

Copulæ have many applications in the area of medicine, for example,

1. Copulæ have been used in the field of magnetic resonance imaging (MRI), for example, to segment images,{{Cite journal|last1=Lapuyade-Lahorgue|first1=Jerome|last2=Xue|first2=Jing-Hao|last3=Ruan|first3=Su|date=July 2017|title=Segmenting Multi-Source Images Using Hidden Markov Fields With Copula-Based Multivariate Statistical Distributions|journal=IEEE Transactions on Image Processing|volume=26|issue=7|pages=3187–3195|doi=10.1109/tip.2017.2685345|pmid=28333631|bibcode=2017ITIP...26.3187L|s2cid=11762408|issn=1057-7149|url=https://discovery.ucl.ac.uk/id/eprint/1557403/}} to fill a vacancy of graphical models in imaging genetics in a study on schizophrenia,{{Cite book|last1=Zhang|first1=Aiying|last2=Fang|first2=Jian|last3=Calhoun|first3=Vince D.|last4=Wang|first4=Yu-ping|title=2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018) |chapter=High dimensional latent Gaussian copula model for mixed data in imaging genetics |date=April 2018|pages=105–109|publisher=IEEE|doi=10.1109/isbi.2018.8363533|isbn=978-1-5386-3636-7|s2cid=44114562}} and to distinguish between normal and Alzheimer patients.{{Cite book|last1=Bahrami|first1=Mohsen|last2=Hossein-Zadeh|first2=Gholam-Ali|title=2015 23rd Iranian Conference on Electrical Engineering |chapter=Assortativity changes in Alzheimer's diesease: A resting-state FMRI study |date=May 2015|pages=141–144|publisher=IEEE|doi=10.1109/iraniancee.2015.7146198|isbn=978-1-4799-1972-7|s2cid=20649428}}
2. Copulæ have been in the area of brain research based on EEG signals, for example, to detect drowsiness during daytime nap,{{Cite journal|last1=Qian|first1=Dong|last2=Wang|first2=Bei|last3=Qing|first3=Xiangyun|last4=Zhang|first4=Tao|last5=Zhang|first5=Yu|last6=Wang|first6=Xingyu|last7=Nakamura|first7=Masatoshi|date=April 2017|title=Drowsiness Detection by Bayesian-Copula Discriminant Classifier Based on EEG Signals During Daytime Short Nap|journal=IEEE Transactions on Biomedical Engineering|volume=64|issue=4|pages=743–754|doi=10.1109/tbme.2016.2574812|pmid=27254855|s2cid=24244444|issn=0018-9294}} to track changes in instantaneous equivalent bandwidths (IEBWs),{{Cite book|last1=Yoshida|first1=Hisashi|last2=Kuramoto|first2=Haruka|last3=Sunada|first3=Yusuke|last4=Kikkawa|first4=Sho|title=2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society |chapter=EEG Analysis in Wakefulness Maintenance State against Sleepiness by Instantaneous Equivalent Bandwidths |date=August 2007|volume=2007|pages=19–22|publisher=IEEE|doi=10.1109/iembs.2007.4352212|pmid=18001878|isbn=978-1-4244-0787-3|s2cid=29527332}} to derive synchrony for early diagnosis of Alzheimer's disease,{{Cite book|last1=Iyengar|first1=Satish G.|last2=Dauwels|first2=Justin|last3=Varshney|first3=Pramod K.|author-link3= Pramod Varshney|last4=Cichocki|first4=Andrzej|title=2010 IEEE International Conference on Acoustics, Speech and Signal Processing |chapter=Quantifying EEG synchrony using copulas |date=2010|pages=505–508|publisher=IEEE|doi=10.1109/icassp.2010.5495664|isbn=978-1-4244-4295-9|s2cid=16476449}} to characterize dependence in oscillatory activity between EEG channels,{{Cite book|last1=Gao|first1=Xu|last2=Shen|first2=Weining|last3=Ting|first3=Chee-Ming|last4=Cramer|first4=Steven C.|last5=Srinivasan|first5=Ramesh|last6=Ombao|first6=Hernando|title=2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019) |chapter=Estimating Brain Connectivity Using Copula Gaussian Graphical Models |date=April 2019|pages=108–112|publisher=IEEE|doi=10.1109/isbi.2019.8759538|isbn=978-1-5386-3641-1|s2cid=195881851|url=https://escholarship.org/uc/item/8tk041wr }} and to assess the reliability of using methods to capture dependence between pairs of EEG channels using their time-varying envelopes.{{Cite book|last1=Fadlallah|first1=B. H.|last2=Brockmeier|first2=A. J.|last3=Seth|first3=S.|last4=Lin Li|last5=Keil|first5=A.|last6=Principe|first6=J. C.|title=2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society |chapter=An Association Framework to Analyze Dependence Structure in Time Series |date=August 2012|volume=2012|pages=6176–6179|publisher=IEEE|doi=10.1109/embc.2012.6347404|pmid=23367339|isbn=978-1-4577-1787-1|s2cid=9061806}} Copula functions have been successfully applied to the analysis of neuronal dependencies{{Citation|last1=Eban|first1=E|title=Dynamic Copula Networks for Modeling Real-valued Time Series|url=http://jmlr.org/proceedings/papers/v31/eban13a.pdf|journal=Journal of Machine Learning Research|volume=31|year=2013|editor1-last=Carvalho|editor1-first=C|last2=Rothschild|first2=R|last3=Mizrahi|first3=A|last4=Nelken|first4=I|last5=Elidan|first5=G|editor2-last=Ravikumar|editor2-first=P}} and spike counts in neuroscience .{{Citation|last1=Onken|first1=A|title=Analyzing Short-Term Noise Dependencies of Spike-Counts in Macaque Prefrontal Cortex Using Copulas and the Flashlight Transformation|journal=PLOS Computational Biology|volume=5|issue=11|pages=e1000577|year=2009|editor1-last=Aertsen|editor1-first=Ad|bibcode=2009PLSCB...5E0577O|doi=10.1371/journal.pcbi.1000577|pmc=2776173|pmid=19956759|last2=Grünewälder|first2=S|last3=Munk|first3=MH|last4=Obermayer|first4=K|doi-access=free}}
3. A copula model has been developed in the field of oncology, for example, to jointly model genotypes, phenotypes, and pathways to reconstruct a cellular network to identify interactions between specific phenotype and multiple molecular features (e.g. mutations and gene expression change). Bao et al.{{Cite book|last1=Bao|first1=Le|last2=Zhu|first2=Zhou|last3=Ye|first3=Jingjing|title=2009 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology |chapter=Modeling oncology gene pathways network with multiple genotypes and phenotypes via a copula method |date=March 2009|pages=237–246|publisher=IEEE|doi=10.1109/cibcb.2009.4925734|isbn=978-1-4244-2756-7|s2cid=16779505}} used NCI60 cancer cell line data to identify several subsets of molecular features that jointly perform as the predictors of clinical phenotypes. The proposed copula may have an impact on biomedical research, ranging from cancer treatment to disease prevention. Copula has also been used to predict the histological diagnosis of colorectal lesions from colonoscopy images,{{Cite book|last1=Kwitt|first1=Roland|last2=Uhl|first2=Andreas|last3=Hafner|first3=Michael|last4=Gangl|first4=Alfred|last5=Wrba|first5=Friedrich|last6=Vecsei|first6=Andreas|title=2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Workshops |chapter=Predicting the histology of colorectal lesions in a probabilistic framework |date=June 2010|pages=103–110|publisher=IEEE|doi=10.1109/cvprw.2010.5543146|isbn=978-1-4244-7029-7|s2cid=14841548}} and to classify cancer subtypes.{{Cite book|last1=Kon|first1=M. A.|last2=Nikolaev|first2=N.|title=2011 10th International Conference on Machine Learning and Applications and Workshops |chapter=Empirical Normalization for Quadratic Discriminant Analysis and Classifying Cancer Subtypes |date=December 2011|pages=374–379|publisher=IEEE|doi=10.1109/icmla.2011.160|arxiv=1203.6345 |isbn=978-1-4577-2134-2|hdl=2144/38445|s2cid=346934|hdl-access=free}}
4. A copula-based analysis model has been developed in the field of heart and cardiovascular disease, for example, to predict heart rate (HR) variation. Heart rate (HR) is one of the most critical health indicators for monitoring exercise intensity and load degree because it is closely related to heart rate. Therefore, an accurate short-term HR prediction technique can deliver efficient early warning for human health and decrease harmful events. Namazi (2022){{Cite journal|last1=Namazi|first1=Asieh|date=December 2022|title=On the improvement of heart rate prediction using the combination of singular spectrum analysis and copula-based analysis approach|journal=PeerJ|volume=10 |pages=e14601|doi=10.7717/peerj.14601|pmid=36570014 |pmc=9774013 |issn=2167-8359 |doi-access=free }} used a novel hybrid algorithm to predict HR.

The combination of SSA and copula-based methods have been applied for the first time as a novel stochastic tool for Earth Orientation Parameters prediction.{{Cite journal | last1 = Modiri | first1 = S. | last2 = Belda | first2 = S. | last3 = Heinkelmann | first3 = R. | last4 = Hoseini | first4 = M. | last5 = Ferrándiz | first5 = J.M. | last6 = Schuh | first6 = H. |doi = 10.1186/s40623-018-0888-3 | title = Polar motion prediction using the combination of SSA and Copula-based analysis | journal = Earth, Planets and Space | volume = 70 | issue = 70 | pages = 115 | year = 2018 | pmid = 30996648| pmc =6434970 | bibcode =2018EP&S...70..115M | doi-access = free }}{{Cite journal | last1 = Modiri | first1 = S. | last2 = Belda | first2 = S. | last3 = Hoseini | first3 = M.| last4 = Heinkelmann | first4 = R. | last5 = Ferrándiz | first5 = J.M. | last6 = Schuh | first6 = H. |doi = 10.1007/s00190-020-01354-y | title = A new hybrid method to improve the ultra-short-term prediction of LOD | journal = Journal of Geodesy | volume = 94 | issue = 23 | year = 2020 | page = 23 | pmid = 32109976 | pmc = 7004433 | bibcode = 2020JGeod..94...23M | doi-access = free }}

Copulas have been used in both theoretical and applied analyses of hydroclimatic data. Theoretical studies adopted the copula-based methodology for instance to gain a better understanding of the dependence structures of temperature and precipitation, in different parts of the world.{{Cite journal|last1=Lazoglou|first1=Georgia|last2=Anagnostopoulou|first2=Christina|date=February 2019|title=Joint distribution of temperature and precipitation in the Mediterranean, using the Copula method|journal=Theoretical and Applied Climatology|language=en|volume=135|issue=3–4|pages=1399–1411|doi=10.1007/s00704-018-2447-z|bibcode=2019ThApC.135.1399L|s2cid=125268690|issn=0177-798X}}{{Cite journal|last1=Cong|first1=Rong-Gang|last2=Brady|first2=Mark|date=2012|title=The Interdependence between Rainfall and Temperature: Copula Analyses|journal=The Scientific World Journal|language=en|volume=2012|page=405675|doi=10.1100/2012/405675|issn=1537-744X|pmc=3504421|pmid=23213286 |doi-access=free }} Applied studies adopted the copula-based methodology to examine e.g., agricultural droughts{{Cite journal|last1=Wang|first1=Long|last2=Yu|first2=Hang|last3=Yang|first3=Maoling|last4=Yang|first4=Rui|last5=Gao|first5=Rui|last6=Wang|first6=Ying|date=April 2019|title=A drought index: The standardized precipitation evapotranspiration runoff index|journal=Journal of Hydrology|language=en|volume=571|pages=651–668|doi=10.1016/j.jhydrol.2019.02.023|bibcode=2019JHyd..571..651W|s2cid=134409125 }} or joint effects of temperature and precipitation extremes on vegetation growth.{{Cite journal|last1=Alidoost|first1=Fakhereh|last2=Su|first2=Zhongbo|last3=Stein|first3=Alfred|date=December 2019|title=Evaluating the effects of climate extremes on crop yield, production and price using multivariate distributions: A new copula application|journal=Weather and Climate Extremes|language=en|volume=26|pages=100227|doi=10.1016/j.wace.2019.100227|bibcode=2019WCE....2600227A |doi-access=free}}

Copulas have been extensively used in climate- and weather-related research.{{Cite journal | last1 = Schölzel | first1 = C. | last2 = Friederichs | first2 = P. | doi = 10.5194/npg-15-761-2008 | title = Multivariate non-normally distributed random variables in climate research – introduction to the copula approach | journal = Nonlinear Processes in Geophysics | volume = 15 | issue = 5 | pages = 761–772 | year = 2008 | bibcode = 2008NPGeo..15..761S | doi-access = free }}{{Cite journal | last1 = Laux | first1 = P. | first2 = S. |last2 = Vogl | first3 = W. | last3 = Qiu | first4 = H.R. | last4 = Knoche | first5 = H. | last5 = Kunstmann | doi = 10.5194/hess-15-2401-2011 | title = Copula-based statistical refinement of precipitation in RCM simulations over complex terrain | journal = Hydrol. Earth Syst. Sci. | volume = 15 | issue = 7 | pages = 2401–2419 | year = 2011 | bibcode = 2011HESS...15.2401L | doi-access = free }}

Copulas have been used to estimate the solar irradiance variability in spatial networks and temporally for single locations.{{cite journal|last1=Munkhammar|first1=J.|last2=Widén|first2=J.|title=A copula method for simulating correlated instantaneous solar irradiance in spatial networks|journal=Solar Energy |volume=143|pages=10–21|year=2017|doi=10.1016/j.solener.2016.12.022|bibcode=2017SoEn..143...10M}}{{cite journal|last1=Munkhammar|first1=J.|last2=Widén|first2=J.|title=An autocorrelation-based copula model for generating realistic clear-sky index time-series|journal=Solar Energy |volume=158|pages=9–19|year=2017|doi=10.1016/j.solener.2017.09.028|bibcode=2017SoEn..158....9M}}

Large synthetic traces of vectors and stationary time series can be generated using empirical copula while preserving the entire dependence structure of small datasets.{{cite conference|last=Strelen|first=Johann Christoph|title=Tools for Dependent Simulation Input with Copulas|conference=2nd International ICST Conference on Simulation Tools and Techniques|year=2009|doi=10.4108/icst.simutools2009.5596|doi-access=free}} Such empirical traces are useful in various simulation-based performance studies.{{cite book|last1=Bandara|first1=H. M. N. D.|last2=Jayasumana |first2=A. P. |title=2011 IEEE Global Telecommunications Conference - GLOBECOM 2011 |chapter=On Characteristics and Modeling of P2P Resources with Correlated Static and Dynamic Attributes |publisher=IEEE Globecom|date=Dec 2011|pages=1–6|doi=10.1109/GLOCOM.2011.6134288|isbn=978-1-4244-9268-8|citeseerx=10.1.1.309.3975|s2cid=7135860}}

Copulas have been used for quality ranking in the manufacturing of electronically commutated motors.{{Cite journal|last1=Mileva Boshkoska|first1=Biljana|last2=Bohanec|first2=Marko|last3=Boškoski|first3=Pavle|last4=Juričić|first4=Ðani|date=2015-04-01|title=Copula-based decision support system for quality ranking in the manufacturing of electronically commutated motors|journal=Journal of Intelligent Manufacturing|language=en|volume=26|issue=2|pages=281–293|doi=10.1007/s10845-013-0781-7|s2cid=982081|issn=1572-8145}}

Copulas are important because they represent a dependence structure without using marginal distributions. Copulas have been widely used in the field of finance, but their use in signal processing is relatively new. Copulas have been employed in the field of wireless communication for classifying radar signals, change detection in remote sensing applications, and EEG signal processing in medicine. In this section, a short mathematical derivation to obtain copula density function followed by a table providing a list of copula density functions with the relevant signal processing applications are presented.

Copulas have been used for determining the core radio luminosity function of Active galactic Nuclei (AGNs),{{cite journal |last1=Zunli |first1=Yuan |last2=Jiancheng |first2=Wang | last3=Diana |first3=Worrall | last4=Bin-Bin |first4=Zhang | last5=Jirong |first5=Mao |title=Determining the Core Radio Luminosity Function of Radio AGNs via Copula |journal=The Astrophysical Journal Supplement Series |date=2018 |volume=239 |issue=2 |page=33 |doi=10.3847/1538-4365/aaed3b |arxiv=1810.12713 |bibcode=2018ApJS..239...33Y |s2cid=59330508 |doi-access=free }} while this cannot be realized using traditional methods due to the difficulties in sample completeness.

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

: $F\_\{XY\}(x,y)\; =\; \backslash Pr\; \backslash begin\{Bmatrix\}\; X\; \backslash leq\{x\},Y\backslash leq\{y\}\; \backslash end\{Bmatrix\},$

where $F\_X(x)\; =\; \backslash Pr\; \backslash begin\{Bmatrix\}\; X\; \backslash leq\{x\}\; \backslash end\{Bmatrix\}$ and

$F\_Y(y)\; =\; \backslash Pr\; \backslash begin\{Bmatrix\}\; Y\; \backslash leq\{y\}\; \backslash end\{Bmatrix\}$ are the marginal cumulative distribution functions of the random variables X and Y, respectively.

then the copula distribution function $C(u,\; v)$ can be defined using Sklar's theorem{{Cite book|last1=Appell|first1=Paul|title=Théorie des fonctions algébriques et de leurs intégrales étude des fonctions analytiques sur une surface de Riemann / par Paul Appell, Édouard Goursat.|last2=Goursat|first2=Edouard|date=1895|publisher=Gauthier-Villars|location=Paris|doi=10.5962/bhl.title.18731|url=https://www.biodiversitylibrary.org/item/58385}} as:

: $F\_\{XY\}(x,y)\; =\; C(\; F\_X\; (x)\; ,\; F\_Y\; (y)\; )\; \backslash triangleq\; C(\; u,\; v\; ),$

where $u\; =\; F\_X(x)$ and $v\; =\; F\_Y(y)$ are marginal distribution functions, $F\_\{XY\}(x,y)$ joint and $u,\; v\; \backslash in\; (0,1)$.

Assuming $F\_\{XY\}(\backslash cdot,\backslash cdot)$ is a.e. twice differentiable, we start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.

:$\backslash begin\{alignat\}\{6\}$

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{alignat}

where $c(u,v)$ is the copula density function, $f\_X(x)$ and $f\_Y(y)$ are the marginal probability density functions of X and Y, respectively. 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.

Various bivariate copula density functions are important in the area of signal processing. $u=F\_X(x)$ and $v=F\_Y(y)$ are marginal distributions functions and $f\_X(x)$ and $f\_Y(y)$ 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. Zeng et al.{{Cite journal|last1=Zeng|first1=Xuexing|last2=Ren|first2=Jinchang|last3=Sun|first3=Meijun|last4=Marshall|first4=Stephen|last5=Durrani|first5=Tariq|date=January 2014|title=Copulas for statistical signal processing (Part II): Simulation, optimal selection and practical applications|journal=Signal Processing|volume=94|pages=681–690|doi=10.1016/j.sigpro.2013.07.006|bibcode=2014SigPr..94..681Z |issn=0165-1684|url=https://strathprints.strath.ac.uk/48371/1/Copulas_Part2s_v2_5_2.pdf}} presented algorithms, simulation, optimal selection, and practical applications of these copulas in signal processing.

• Coupling (probability)

{{Reflist|30em}}

• 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}}

• {{springer|title=Copula|id=p/c110410}}
• [http://sites.google.com/site/copulawiki/ Copula Wiki: community portal for researchers with interest in copulas] {{Webarchive|url=https://web.archive.org/web/20160911110231/https://sites.google.com/site/copulawiki/ |date=2016-09-11 }}
• [https://web.archive.org/web/20160416142558/http://www.mathfinance.cn/tags/copula/ A collection of Copula simulation and estimation codes]
• [http://www.crystalballservices.com/Resources/ConsultantsCornerBlog/tagid/21/Correlation.aspx Copulas & Correlation using Excel Simulation Articles]
• [http://www.worldscientific.com/doi/suppl/10.1142/p842/suppl_file/p842_chap01.pdf Chapter 1 of Jan-Frederik Mai, Matthias Scherer (2012) "Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications"]

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