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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
Slide | 2
Cluster Analysis – Data mapped to a set of
Normal Densities – with specified degree of
membership – a model based clustering
Customer Profiling – Characterizing the
Distributions encountered – Age, ARPU, Net
Stay…
leading to a probabilistic description /
modeling of the dataSentiment 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
Slide | 3
Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University
PARAMETER ESTIMATION VIA EXPECTATION MAXIMIZATION ALGORITHM
Slide | 4
Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University
RESULTS – ESTIMATING THE COMPONENT GAUSSIAN DENSITIES
Slide |
5
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
Slide |
6
RESULTS – INTERPRETATION OF CLUSTER MEMBERSHIP
Slide |
7
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

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Estimating Gaussian Mixture Densities via an implemetation of the Expectaation Maximization Algorithm

  • 1. ESTIMATING GAUSSIAN MIXTURE DENSITIES VIA A MATLAB IMPLEMETATION OF THE EXPECTATION MAXIMIZATION ALGORITHM DR. ASOKA KORALE, C.ENG. MIET & MIESL
  • 2. APPLICATIONS FOR GAUSSIAN MIXTURE DECOMPOSITION MODELING ANALYSIS Slide | 2 Cluster Analysis – Data mapped to a set of Normal Densities – with specified degree of membership – a model based clustering Customer Profiling – Characterizing the Distributions encountered – Age, ARPU, Net Stay… leading to a probabilistic description / modeling of the dataSentiment 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
  • 3. PARAMETER ESTIMATION VIA EXPECTATION MAXIMIZATION ALGORITHM Slide | 3 Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University
  • 4. PARAMETER ESTIMATION VIA EXPECTATION MAXIMIZATION ALGORITHM Slide | 4 Ref: Estimating Gaussian Mixture Densities with EM, Carlo Tomasi, Duke University
  • 5. RESULTS – ESTIMATING THE COMPONENT GAUSSIAN DENSITIES Slide | 5 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)
  • 6. CONVERGENCE OF THE EM ALGORITHM FOR THE PARAMETERS Slide | 6
  • 7. RESULTS – INTERPRETATION OF CLUSTER MEMBERSHIP Slide | 7 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