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Unformatted text preview: Speech Recognition Pattern Classification 2 February 13, 2012 Veton Këpuska 2 Pattern Classification Introduction Parametric classifiers Semiparametric classifiers Dimensionality reduction Significance testing February 13, 2012 Veton Këpuska 3 SemiParametric Classifiers Mixture densities Maximum Likelihood (ML) parameter estimation Mixture implementations Expectation maximization (EM) February 13, 2012 Veton Këpuska 4 Mixture Densities PDF is composed of a mixture of m components densities { ϖ 1 ,…, ϖ 2 }: Component PDF parameters and mixture weights P( ϖ j ) are typically unknown, making parameter estimation a form of unsupervised learning . Gaussian mixtures assume Normal components: ∑ = = m j j j P p p 1 ) ( )  ( ) ( ϖ x x ) , ( ~ )  ( k k k N p Σ μ x February 13, 2012 Veton Këpuska 5 Gaussian Mixture Example: One Dimension p(x)=0.6p 1 (x)+0.4p 2 (x) p1(x)~N(σ , σ 2 ) p 2 (x) ~N(1.5 σ , σ 2 ) February 13, 2012 Veton Këpuska 6 Gaussian Example First 9 MFCC’s from [s]: Gaussian PDF February 13, 2012 Veton Këpuska 7 Independent Mixtures [s]: 2 Gaussian Mixture Components/Dimension February 13, 2012 Veton Këpuska 8 Mixture Components [s]: 2 Gaussian Mixture Components/Dimension February 13, 2012 Veton Këpuska 9 ML Parameter Estimation: 1D Gaussian Mixture Means ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = == = = = = = ⇒ = == ∂ ∂= ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ = = n i i k n i i i k k i k i k k i n i k i k k i i k k k k i k k i x k k k k i n i k k i k i n i i k k k n i m j j j i n i i k x P x x P x P x p P x p x p x x p P L x p x e x p P x p x p x p L P x p x p L k k i 1 1 1 2 2 2 1 1 1 1 1     since  log  2 1   1 log log  log log log 2 2 ϖ μ σ π February 13, 2012 Veton Këpuska 10 Gaussian Mixtures: ML Parameter Estimation The maximum likelihood solutions are of the form: February 13, 2012 Veton Këpuska 11 Gaussian Mixtures: ML Parameter Estimation The ML solutions are typically solved iteratively: Select a set of initial estimates for P ( ϖ k ) , µ k , Σ k Use a set of n samples to reestimate the mixture parameters until some kind of convergence is found Clustering procedures are often used to provide the initial parameter estimates Similar to Kmeans clustering procedure ˆ ˆ ˆ February 13, 2012 Veton Këpuska 12 Example: 4 Samples, 2 Densities 1. Data: X = { x 1 ,x 2 ,x 3 ,x 4 } = {2 , 1 ,1 ,2} 2. Init: p(x ϖ 1 )~N(1,1), p(x ϖ 2 )~N(1,1), P( ϖ i )=0.5 3. Estimate: 4. Recompute mixture parameters (only shown for ϖ 1 ): x 1 x 2 x 3 x 4 P( ϖ 1 x) 0.98 0.88 0.12 0.02 P(...
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This note was uploaded on 02/11/2012 for the course ECE 5526 taught by Professor Staff during the Summer '09 term at FIT.
 Summer '09
 Staff

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