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Ch3-Pattern_Classification1

# Ch3-Pattern_Classification1 - 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
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 ) ( ) (
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 Q Q P D ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - = - i i i i i i i i i i i z Q z P z Q z P z Q z Q z P z Q 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
February 13, 2012 Veton Këpuska 7 Bayes Theorem From Bayes Rule: Where:   ) ( ) ( ) | ( ) | ( x p P x p x P i i i ϖ ϖ ϖ = = = M i i i P x p x p 1 ) ( ) | ( ) ( ϖ ϖ

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Bayesian Decision Theory Reference: Pattern Classification – R. Duda, P. Hard & D.  Stork, Wiley & Sons, 2001
February 13, 2012 Veton Këpuska 9 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 10 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 ϖ ϖ ϖ ) ( ) ( ) | ( ) | ( 1 2 2 1 ϖ ϖ ϖ ϖ P P x p x p
February 13, 2012 Veton Këpuska 11 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  ω j   R ( ω i | x ) is the risk for classifying  as class  ω i Bayes risk  is the minimum risk which can be achieved: Choose  ω i   if   R(

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Ch3-Pattern_Classification1 - Speech Recognition Pattern...

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