Gaussian Mixtures and Their Applications to Signal Processing
Konstantinos N. Plataniotis, Dimitris Hatzinakos · 2017
K o s ta n tin o s N . Nomenclature 3*1 Plataniotis Abstract 3-2 University o f Toronto 3.1 Introduction 3-2. . . . 3.2 Mathematical Aspects of Gaussian M ixtures 3-4 Dimitris Hatzmakos The Approximation Theorem • The Identifiability Problem University o f Toronto 3.3 Methodologies for Mixture Parameter Estimation 3-73.4 Computer Generation o f Mixture Variables 3-13 3.5 Mixture Applications 3-153.6 Concluding Remarks 3-32 References 3-32Nomenclature Family of distributions F(x10 ) Conditional distribution 0 Unknown parameter 0 Estimated value N(x; jLlj, Xj) d -Dimensional Gaussian p Mean value X Covariance Wi Mixing coefficient ML Maximum likelihood EM Expectation maximization logA Log likelihood Z Missing data (EM algorithm) LRT Likelihood ratio test Ng Number o f mixture components RBF Radial-basis functions PNN Probabilistic neural network x(k) State vectorZk = z( 1), z(2), z{k)y ... Observation recordx (k\k) = E(x(k)\Zk) Mean-squared-error filtered estimate EKF Extended Kalman filter Jh(x (k)) Jacobian matrix AGSF Adaptive Gaussian sum filter CMKF Converted measurement Kalman filter e-mixture e-Contaminated Gaussian mixture model i(k) Inter-symbol inference n(k) Thermal noise NMSE Normalized mean square error DS/SS Direct-sequence spread-spectrum HR Infinite-duration impulse responseAbstract There are a number o f engineering applications in which a function should be estimated from data. Mixtures of distributions, especially Gaussian mixtures, have been used extensively as models in such problems where data can be viewed as arising from two or more populations mixed in varying propor tions.1-3 The objective of this chapter is to highlight the use of mixture models as a way to provide efficient and accurate solutions to problems o f important engineering significance. Using the Gaussian mixture formulation, problems are treated from a global viewpoint that readily yields and unifies previous, seemingly unrelated results. This chapter reviews the existing methodologies, examines current trends, provides connections with other methodologies and practices, and discusses application areas.