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.
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.
[1] 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.
For any two random variables X and Y, the continuous joint probability distribution function can be written as where
are the marginal cumulative distribution functions of the random variables X and Y, respectively.
We start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.
are the marginal probability density functions of X and Y, respectively.
For example, equation 1 may be used 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.
are marginal density functions validating biometric authentication,[6] modeling stochastic dependence in large-scale integration of wind power,[7] unsupervised classification of radar signals[8] fusion of correlated sensor decisions[21] TABLE 1: Copula density function of a family of copulas used in signal processing.