Ch3-Pattern_Classification2

Ch3-Pattern_Classification2 - Speech Recognition Pattern...

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Unformatted text preview: Speech Recognition Pattern Classification 2 February 13, 2012 Veton Kpuska 2 Pattern Classification Introduction Parametric classifiers Semi-parametric classifiers Dimensionality reduction Significance testing February 13, 2012 Veton Kpuska 3 Semi-Parametric Classifiers Mixture densities Maximum Likelihood (ML) parameter estimation Mixture implementations Expectation maximization (EM) February 13, 2012 Veton Kpuska 4 Mixture Densities PDF is composed of a mixture of m components densities { 1 ,, 2 }: Component PDF parameters and mixture weights P( 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 ) ( ) | ( ) ( x x ) , ( ~ ) | ( k k k N p x February 13, 2012 Veton Kpuska 5 Gaussian Mixture Example: One Dimension p(x)=0.6p 1 (x)+0.4p 2 (x) p1(x)~N(- , 2 ) p 2 (x) ~N(1.5 , 2 ) February 13, 2012 Veton Kpuska 6 Gaussian Example First 9 MFCCs from [s]: Gaussian PDF February 13, 2012 Veton Kpuska 7 Independent Mixtures [s]: 2 Gaussian Mixture Components/Dimension February 13, 2012 Veton Kpuska 8 Mixture Components [s]: 2 Gaussian Mixture Components/Dimension February 13, 2012 Veton Kpuska 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 February 13, 2012 Veton Kpuska 10 Gaussian Mixtures: ML Parameter Estimation The maximum likelihood solutions are of the form: February 13, 2012 Veton Kpuska 11 Gaussian Mixtures: ML Parameter Estimation The ML solutions are typically solved iteratively: Select a set of initial estimates for P ( k ) , k , k Use a set of n samples to re-estimate the mixture parameters until some kind of convergence is found Clustering procedures are often used to provide the initial parameter estimates Similar to K-means clustering procedure February 13, 2012 Veton Kpuska 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| 1 )~N(1,1), p(x| 2 )~N(-1,1), P( i )=0.5 3. Estimate: 4. Recompute mixture parameters (only shown for 1 ): x 1 x 2 x 3 x 4 P( 1 |x) 0.98 0.88 0.12 0.02 P(...
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Ch3-Pattern_Classification2 - Speech Recognition Pattern...

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