Copulas in signal processing

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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 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.[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.[2] 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

[math]\displaystyle{ F_{XY}(x,y) = \Pr \begin{Bmatrix} X \leq{x},Y\leq{y} \end{Bmatrix}, }[/math]

where [math]\displaystyle{ F_X(x) = \Pr \begin{Bmatrix} X \leq{x} \end{Bmatrix} }[/math] and [math]\displaystyle{ F_Y(y) = \Pr \begin{Bmatrix} Y \leq{y} \end{Bmatrix} }[/math] are the marginal cumulative distribution functions of the random variables X and Y, respectively.

then the copula distribution function [math]\displaystyle{ C(u, v) }[/math] can be defined using Sklar's theorem[3][4] as:

[math]\displaystyle{ F_{XY}(x,y) = C( F_X (x) , F_Y (y) ) \triangleq C( u, v ) }[/math],

where [math]\displaystyle{ u = F_X(x) }[/math] and [math]\displaystyle{ v = F_Y(y) }[/math] are marginal distribution functions, [math]\displaystyle{ F_{XY}(x,y) }[/math] joint and [math]\displaystyle{ u, v \in (0,1) }[/math].

We start by using the relationship between joint probability density function (PDF) and joint cumulative distribution function (CDF) and its partial derivatives.

[math]\displaystyle{ \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} }[/math]
(Equation 1)

where [math]\displaystyle{ c(u,v) }[/math]is the copula density function, [math]\displaystyle{ f_X(x) }[/math] and [math]\displaystyle{ f_Y(y) }[/math] 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, [math]\displaystyle{ u=F_X(x) }[/math] and [math]\displaystyle{ v=F_Y(y) }[/math] are marginal distributions functions and [math]\displaystyle{ f_X(x) }[/math] and [math]\displaystyle{ f_Y(y) }[/math] are marginal density functions

Coupla density: c(u, v) Use
Gaussian [math]\displaystyle{ \begin{align} = {} & \frac{1}{\sqrt{1-\rho^2}} \exp\left (-\frac{(a^2+b^2)\rho^2-2 ab\rho}{ 2(1-\rho^2) } \right ) \\ & \text{where } \rho\in (-1,1)\\ & \text{where } a=\sqrt{2} \operatorname{erf}^{-1}({2u-1}) \\ & \text{where } b =\sqrt{2}\operatorname{erf}^{-1}({2v-1}) \\ & \text{where } \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int\limits_0^z \exp (-t^2) \, dt \end{align} }[/math] supervised classification of synthetic aperture radar (SAR) images,[5]

validating biometric authentication,[6] modeling stochastic dependence in large-scale integration of wind power,[7] unsupervised classification of radar signals[8]

Exponential [math]\displaystyle{ \begin{align} = {} & \frac{1}{1-\rho} \exp\left ( \frac{\rho(\ln(1-u)+\ln(1-v))}{1-\rho} \right ) \cdot I_0\left ( \frac{2\sqrt{\rho \ln(1-u)\ln(1-v)}}{1-\rho} \right )\\ & \text{where } x=F_X^{-1}(u)=-\ln(1-u)/\lambda \\ & \text{where } y=F_Y^{-1}(v)=-\ln(1-v)/\mu \end{align} }[/math] queuing system with infinitely servers[9]
Rayleigh bivariate exponential, Rayleigh, and Weibull copulas have been proved to be equivalent[10][11][12] change detection from SAR images[13]
Weibull bivariate exponential, Rayleigh, and Weibull copulas have been proved to be equivalent[10][11][12] digital communication over fading channels[14]
Log-normal bivariate log-normal copula and Gaussian copula are equivalent[12][11] shadow fading along with multipath effect in wireless channel[15][16]
Farlie–Gumbel–Morgenstern (FGM) [math]\displaystyle{ \begin{align} = {} & 1+\theta(1-2u)(1-2v) \\ & \text{where } \theta \in[-1,1] \end{align} }[/math]
Clayton [math]\displaystyle{ \begin{align} = {} & (1+\theta)u^{(-1-\theta)}(-1 +u^{-\theta} + v^{-\theta})^{(-2-1/\theta)} \\ & \text{where } \theta \in(-1,\infty), \theta\neq0 \end{align} }[/math] location estimation of random signal source and hypothesis testing using heterogeneous data[17][18]
Frank [math]\displaystyle{ \begin{align} = {} & \frac {\theta e^{\theta(1+u+v)(e^{\theta}-1)}} {(e^\theta-e^{\theta(1+u)}-e^{\theta(1+v)}+e^{\theta(u+v)})^2}\\ & \text{where } \theta \in(-\infty,+\infty), \theta\neq0 \end{align} }[/math] change detection in remote sensing applications[19]
Student's t [math]\displaystyle{ \begin{align} = {} & \frac{\Gamma(0.5v)\Gamma(0.5v+1)( 1+(t_v^{-2}(u)+t_v^{-2}(v) -2 \rho t_v^{-1}(u) t_v^{-1}(v))/(v(1-\rho^2)))^{-0.5(v+2)} )} {\sqrt{1-\rho^2} \cdot \Gamma(0.5(v+1))^2 (1+ t_v^{-2}(u)/v)^{-0.5(v+1)} (1+ t_v^{-2}(v)/v)^{-0.5(v+1)} } \\ & \text{where } \rho\in (-1,1)\\ & \text{where } \phi(z)= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^z \exp \left(\frac{-t^2}{2}\right) \, dt \\ & \text{where } t_v(x\mid v)= \int\limits_{-\infty}^x \frac{\Gamma{(0.5(v+1))}}{\sqrt{v\pi}(\Gamma{0.5v})(1+v^{-1}t^2)^{0.5(v+1)}} dt\\ & \text{where } v=\text{degrees of freedom} \\ & \text{where } \Gamma \text{ is the Gamma function} \end{align} }[/math] supervised SAR image classification,[13]

fusion of correlated sensor decisions[20]

Nakagami-m
Rician

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

References

  1. ↑ Bellmann, K. (1978). "BATSCHELET, E.: Introduction to Mathematics for Life Scientists. 2nd Ed. Springer-Verlag, Berlin-Heidelberg-New York 1975. 643 S., 227 Abb., DM 38,-.". Biometrical Journal 20 (5): 531. doi:10.1002/bimj.4710200510. ISSN 0323-3847. http://dx.doi.org/10.1002/bimj.4710200510. 
  2. ↑ Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula Methods in Finance. doi:10.1002/9781118673331. ISBN 9781118673331. http://dx.doi.org/10.1002/9781118673331. 
  3. ↑ Appell, Paul; Goursat, Edouard (1895). Théorie des fonctions algébriques et de leurs intégrales étude des fonctions analytiques sur une surface de Riemann / par Paul Appell, Édouard Goursat.. Paris: Gauthier-Villars. doi:10.5962/bhl.title.18731. http://dx.doi.org/10.5962/bhl.title.18731. 
  4. ↑ Durante, Fabrizio; Fernández-Sánchez, Juan; Sempi, Carlo (2013). "A topological proof of Sklar's theorem". Applied Mathematics Letters 26 (9): 945–948. doi:10.1016/j.aml.2013.04.005. ISSN 0893-9659. 
  5. ↑ Storvik, B.; Storvik, G.; Fjortoft, R. (2009). "On the Combination of Multisensor Data Using Meta-Gaussian Distributions". IEEE Transactions on Geoscience and Remote Sensing 47 (7): 2372–2379. doi:10.1109/tgrs.2009.2012699. ISSN 0196-2892. Bibcode: 2009ITGRS..47.2372S. http://dx.doi.org/10.1109/tgrs.2009.2012699. 
  6. ↑ Dass, S.C.; Yongfang Zhu; Jain, A.K. (2006). "Validating a Biometric Authentication System: Sample Size Requirements". IEEE Transactions on Pattern Analysis and Machine Intelligence 28 (12): 1902–1319. doi:10.1109/tpami.2006.255. ISSN 0162-8828. PMID 17108366. http://dx.doi.org/10.1109/tpami.2006.255. 
  7. ↑ Papaefthymiou, G.; Kurowicka, D. (2009). "Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis". IEEE Transactions on Power Systems 24 (1): 40–49. doi:10.1109/tpwrs.2008.2004728. ISSN 0885-8950. Bibcode: 2009ITPSy..24...40P. http://dx.doi.org/10.1109/tpwrs.2008.2004728. 
  8. ↑ Brunel, N.J.-B.; Lapuyade-Lahorgue, J.; Pieczynski, W. (2010). "Modeling and Unsupervised Classification of Multivariate Hidden Markov Chains With Copulas". IEEE Transactions on Automatic Control 55 (2): 338–349. doi:10.1109/tac.2009.2034929. ISSN 0018-9286. http://dx.doi.org/10.1109/tac.2009.2034929. 
  9. ↑ Lai, Chin Diew; Balakrishnan, N. (2009). Continuous Bivariate Distributions. doi:10.1007/b101765. ISBN 978-0-387-09613-1. http://dx.doi.org/10.1007/b101765. 
  10. ↑ 10.0 10.1 Durrani, T.S.; Zeng, X. (2007). "Copulas for bivariate probability distributions". Electronics Letters 43 (4): 248. doi:10.1049/el:20073737. ISSN 0013-5194. Bibcode: 2007ElL....43..248D. http://dx.doi.org/10.1049/el:20073737. 
  11. ↑ 11.0 11.1 11.2 Liu, X. (2010). "Copulas of bivariate Rayleigh and log-normal distributions". Electronics Letters 46 (25): 1669. doi:10.1049/el.2010.2777. ISSN 0013-5194. Bibcode: 2010ElL....46.1669L. http://dx.doi.org/10.1049/el.2010.2777. 
  12. ↑ 12.0 12.1 12.2 Zeng, Xuexing; Ren, Jinchang; Wang, Zheng; Marshall, Stephen; Durrani, Tariq (2014). "Copulas for statistical signal processing (Part I): Extensions and generalization". Signal Processing 94: 691–702. doi:10.1016/j.sigpro.2013.07.009. ISSN 0165-1684. http://dx.doi.org/10.1016/j.sigpro.2013.07.009. 
  13. ↑ 13.0 13.1 Hachicha, S.; Chaabene, F. (2010). Frouin, Robert J; Yoo, Hong Rhyong; Won, Joong-Sun et al.. eds. "SAR change detection using Rayleigh copula". Remote Sensing of the Coastal Ocean, Land, and Atmosphere Environment (SPIE) 7858: 78581F. doi:10.1117/12.870023. Bibcode: 2010SPIE.7858E..1FH. http://dx.doi.org/10.1117/12.870023. 
  14. ↑ "Coded Communication over Fading Channels", Digital Communication over Fading Channels (John Wiley & Sons, Inc.): pp. 758–795, 2005, doi:10.1002/0471715220.ch13, ISBN 978-0-471-71522-1, http://dx.doi.org/10.1002/0471715220.ch13, retrieved 2020-04-06 
  15. ↑ Das, Saikat; Bhattacharya, Amitabha (2020). "Application of the Mixture of Lognormal Distribution to Represent the First-Order Statistics of Wireless Channels". IEEE Systems Journal 14 (3): 4394–4401. doi:10.1109/JSYST.2020.2968409. ISSN 1932-8184. Bibcode: 2020ISysJ..14.4394D. https://ieeexplore.ieee.org/document/8993782. 
  16. ↑ Alouini, M.-S.; Simon, M.K. (2002). "Dual diversity over correlated log-normal fading channels" (in en). IEEE Transactions on Communications 50 (12): 1946–1959. doi:10.1109/TCOMM.2002.806552. ISSN 0090-6778. https://ieeexplore.ieee.org/document/1175472. 
  17. ↑ Sundaresan, Ashok; Varshney, Pramod K. (2011). "Location Estimation of a Random Signal Source Based on Correlated Sensor Observations". IEEE Transactions on Signal Processing 59 (2): 787–799. doi:10.1109/tsp.2010.2084084. ISSN 1053-587X. Bibcode: 2011ITSP...59..787S. http://dx.doi.org/10.1109/tsp.2010.2084084. 
  18. ↑ Iyengar, Satish G.; Varshney, Pramod K.; Damarla, Thyagaraju (2011). "A Parametric Copula-Based Framework for Hypothesis Testing Using Heterogeneous Data". IEEE Transactions on Signal Processing 59 (5): 2308–2319. doi:10.1109/tsp.2011.2105483. ISSN 1053-587X. Bibcode: 2011ITSP...59.2308I. http://dx.doi.org/10.1109/tsp.2011.2105483. 
  19. ↑ Mercier, G.; Moser, G.; Serpico, S.B. (2008). "Conditional Copulas for Change Detection in Heterogeneous Remote Sensing Images". IEEE Transactions on Geoscience and Remote Sensing 46 (5): 1428–1441. doi:10.1109/tgrs.2008.916476. ISSN 0196-2892. Bibcode: 2008ITGRS..46.1428M. http://dx.doi.org/10.1109/tgrs.2008.916476. 
  20. ↑ Sundaresan, Ashok; Varshney, Pramod K.; Rao, Nageswara S. V. (2011). "Copula-Based Fusion of Correlated Decisions". IEEE Transactions on Aerospace and Electronic Systems 47 (1): 454–471. doi:10.1109/taes.2011.5705686. ISSN 0018-9251. Bibcode: 2011ITAES..47..454S. http://dx.doi.org/10.1109/taes.2011.5705686. 



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