A copula is a mathematical function that provides a relationship between marginal distributions of random variables and their joint distributions. Copulas are important because it represents 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 article, a short introduction to copulas is presented, followed by a mathematical derivation to obtain copula density functions, and then a section with a list of copula density functions with applications in signal processing.
Introduction.
Using Sklar's theorem, a copula can be described as a cumulative distribution function (CDF) on a unit-space with uniform marginal distributions on the interval (0, 1). The CDF of a random variable "X" is the probability that "X" will take a value less than or equal to "x" when evaluated at "x" itself. A copula can represent a dependence structure without using marginal distributions. Therefore, it is simple to transform the uniformly distributed variables of copula ("u", "v", and so on) into the marginal variables ("x", "y", and so on) by the inverse marginal cumulative distribution function. Using the chain rule, copula distribution function can be partially differentiated with respect to the uniformly distributed variables of copula, and it is possible to express the multivariate probability density function (PDF) as a product of a multivariate copula density function and marginal PDF"s. The mathematics for converting a copula distribution function into a copula density function is shown for a bivariate case, and a family of copulas used in signal processing are listed in a TABLE 1.
Mathematical derivation.
For any two random variables "X" and "Y", the continuous joint probability distribution function can be written as
 formula_1
where formula_2 and
formula_3 are the marginal cumulative distribution functions of the random variables "X" and "Y", respectively.
then the copula distribution function formula_4 can be defined using Sklar's theorem as:
formula_5,
where formula_6 and formula_7 are marginal distribution functions, formula_8 joint and formula_9.
We start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.
formula_10
(Equation 1)
where formula_11 is the copula density function, formula_12 and formula_13 are the marginal probability density functions of "X" and "Y", respectively. It is important understand that there are four elements in the equation 1, and if three of the four are know, the fourth element can me calculated. For example, equation 1 may be used
Summary table.
The use of copula in signal processing is fairly new compared to finance. Here, a family of new bivariate copula density functions are listed with importance in the area of signal processing. Here, formula_14 and formula_15 are marginal distributions functions and formula_16 and formula_17 are marginal density functions
TABLE 1: Copula density function of a family of copulas used in signal processing.
References.
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