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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 distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
s. Copulas have been widely used in the field of finance, but their use in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
is relatively new. Copulas have been employed in the field of
wireless Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The mos ...
communication Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inqu ...
for classifying
radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
signals, change detection in
remote sensing Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object, in contrast to in situ or on-site observation. The term is applied especially to acquiring information about Ear ...
applications, and
EEG Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocortex ...
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
in
medicine Medicine is the science and Praxis (process), practice of caring for a patient, managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, and Health promotion ...
. 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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
(CDF) on a unit-space with uniform
marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
s 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 In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
, 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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(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 : F_(x,y) = \Pr \begin X \leq,Y\leq \end, where F_X(x) = \Pr \begin X \leq \end and F_Y(y) = \Pr \begin Y \leq \end 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 as: F_(x,y) = C( F_X (x) , F_Y (y) ) \triangleq C( u, v ) , where u = F_X(x) and v = F_Y(y) are marginal distribution functions, F_(x,y) joint and 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. :\begin f_(x,y) = & \\ \vdots \\ f_(x,y) = & \\ \vdots \\ f_(x,y) = & \cdot \cdot \\ \vdots \\ f_(x,y) = & c(u,v) f_X(x) f_Y(y) \\ \vdots \\ \frac = & c(u,v) \end :(Equation 1) 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. 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 * 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.


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, 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 TABLE 1: Copula density function of a family of copulas used in signal processing.


References

{{Reflist Signal processing