ESTIMATING GAUSSIAN MIXTURE DENSITIES

VIA A MATLAB IMPLEMETATION OF THE EXPECTATION MAXIMIZATION ALGORITHM

DR. ASOKA KORALE, C.ENG. MIET & MIESL

APPLICATIONS FOR GAUSSIAN MIXTURE DECOMPOSITION MODELING ANALYSIS

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Cluster Analysis – Data mapped to a set of Normal Densities – with

specified degree of membership – a model based clusteringCustomer Profiling – Characterizing the

Distributions encountered – Age, ARPU, Net Stay…

leading to a probabilistic description / modeling of the data Sentiment Analysis via Independent Term

Matching where each word is drawn from a specified Normal Distribution – combined by

their sum to determine overall sentiment score A model based approach to data

analysis

Goal: model arbitrary distributions as sums of Gaussian densities (with parameters estimated via expectation

maximization algorithm)– so that each data point is characterized with

respect to a distribution from which it is expected to have originated

PARAMETER ESTIMATION VIA EXPECTATION MAXIMIZATION ALGORITHM

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Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University

PARAMETER ESTIMATION VIA EXPECTATION MAXIMIZATION ALGORITHM

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Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University

RESULTS – ESTIMATING THE COMPONENT GAUSSIAN DENSITIES

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II. Standardize the Data and estimate empirical Probability

Density Function

I. Histogram of original Data – (which composite densities to

be estimated)

III. Estimate Gaussian Component Densities

(fx1/2/3) via EM Algorithm and their scaled Sum (fx)

IV. fx: Sum of the individual component densities scaled by their mixing probabilities (for comparison with II the empirical PDF of Data)

CONVERGENCE OF THE EM ALGORITHM FOR THE PARAMETERS

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RESULTS – INTERPRETATION OF CLUSTER MEMBERSHIP

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Test with one dimensional data, through EM algorithm can estimate parameters for sums of “D” dimensional data

*Applicable for multi dimensional data and need to explore correlated random variables