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
:
where
and
are the marginal cumulative distribution functions of the random variables ''X'' and ''Y'', respectively.
then the copula distribution function
can be defined using Sklar's theorem
as:
,
where
and
are marginal distribution functions,
joint and
.
We start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.
:
:(Equation 1)
where
is the copula density function,
and
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,
and
are marginal distributions functions and
and
are marginal density functions
TABLE 1: Copula density function of a family of copulas used in signal processing.
References
{{Reflist
Signal processing