Ch3-Pattern_Classification

Ch3-Pattern_Classification - Speech Recognition Pattern...

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Speech Recognition Pattern Classification
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February 13, 2012 Veton Këpuska 2 Pattern Classification  Introduction  Parametric classifiers  Semi-parametric classifiers  Dimensionality reduction  Significance testing 
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February 13, 2012 Veton Këpuska 3 Pattern Classification Goal:  To classify objects (or patterns) into categories (or  classes)  Types of Problems:  1. Supervised Classes are known beforehand, and data samples of  each class are available  2. Unsupervised Classes (and/or number of classes) are not known  beforehand, and must be inferred from data  Feature  Extraction Classifier Class ϖ i Feature Vectors x Observation s
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February 13, 2012 Veton Këpuska 4 Probability Basics  Discrete probability mass function (PMF):  P ( ω i ) Continuous probability density function (PDF):  p(x) Expected value:  E(x)   = i i P 1 ) ( ϖ = 1 ) ( dx x p = dx x xp x E ) ( ) (
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February 13, 2012 Veton Këpuska 5 Kullback-Liebler Distance  Can be used to compute a distance between two probability  mass distributions,  P ( z i ), and  Q ( z i ) Makes use of inequality log  ≤  - 1  Known as relative entropy in information theory The divergence of  P ( z i ) and  Q ( z i is the symmetric sum ( 29 ( 29 ( 29 ( 29 0 log || = i i i i z Q z P z P Q P D ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 = - = - i i i i i i i i i i i z P z Q z Q z P z P z Q z P z P 0 1 log ( 29 ( 29 P Q D Q P D || || +
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February 13, 2012 Veton Këpuska 6 Bayes Theorem Define: { ϖ i } a set of M mutually exclusive classes P( ϖ i ) a priori  probability for class  ϖ i p( x | ϖ i ) PDF for feature vector  x  in class  ϖ i P( ϖ i | x ) A posteriori probability of  ϖ i  given  x
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February 13, 2012 Veton Këpuska 7 Bayes Theorem From Bayes Rule: Where:   ) ( ) ( ) | ( ) | ( x p P x p x P i i i ϖ = = = M i i i P x p x p 1 ) ( ) | ( ) (
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February 13, 2012 Veton Këpuska 8 Bayes Decision Theory  The probability of making an error given  is:   P(error| x )=1-P( ϖ i | x ) if decide class  ϖ i To minimize  P ( error | x ) (and  P ( error )):   Choose  ϖ if  P( ϖ i | x )>P( ϖ j | x )   j ≠i
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February 13, 2012 Veton Këpuska 9 Bayes Decision Theory For a two class problem this decision rule means:  Choose  ϖ 1   if else  ϖ This rule can be expressed as a likelihood ratio:  ) ( ) ( ) | ( ) ( ) ( ) | ( 2 2 1 1 x p P x p x p P x p ϖ ) ( ) ( ) | ( ) | ( 1 2 2 1 P P x p x p
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February 13, 2012 Veton Këpuska 10 Bayes Risk  Define cost function  λ ij   and conditional risk  R ( ω i | x ):  λ ij   is cost of classifying  as  ω i   when it is really 
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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.

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Ch3-Pattern_Classification - Speech Recognition Pattern...

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