Chile1 - 1 Bayes Optimality of Wavelet-Based Discrimination...

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Unformatted text preview: 1 Bayes Optimality of Wavelet-Based Discrimination Woojin Chang, Seong-Hee Kim, and Brani Vidakovic Seoul University and Georgia Institute of Technology ISBA 2004 Vi ˜ na del Mar, Chile May 25, 2004 2 Overview Talk about classifying Y into one of two classes labeled by or 1 , by taking into account predictor X . Definitions and Notation. Bayes Discriminators Wavelet-Based Approximation Bayes Optimality (or IL 2-Consistency) of the Wavelet-based Classifier. Simulations and Paper Production Example 3 Definitions ( X,Y ) ∈ IR d ×{ , 1 } . μ ( A ) = P ( X ∈ A ) , A ∈ B ; η ( x ) = P ( Y = 1 | X = x ) = E ( Y | X = x ) . Pair ( μ,η ) uniquely determines joint distribution of ( X,Y ) . Any function g : IR d → { , 1 } is a classifier. Bayes Classifier: g * ( x ) = 1 ( η ( x ) > 1 / 2). L ( g ) = P ( g ( X ) 6 = Y ) . [Error, Risk, Misclassification Probability] Result: ( ∀ g ) L ( g * ) ≤ L ( g ) . L * = L ( g * ) Bayes Error [Risk, Probability]. 4 Definitions, contd Assume density of X exists, X ∼ f. Let f and f 1 be class-conditional densities, i.e., densities for X | Y = 0 and X | Y = 1 . Let π = P ( Y = 1) and 1- π = P ( Y = 0) be class-probabilities. Function α ( x ) = πf 1 ( x )- (1- π ) f ( x ) has representation (2 η ( x )- 1) f ( x ) . Bayes Classifier: g * ( x ) = 1 ( α ( x ) > 0). • L * = 1 / 2- 1 / 2 E ( | 2 η ( X )- 1 | ) • L * = R ((1- π ) f ∧ πf 1 ) dx • π = 1 / 2 , L * = 1 / 2- 1 / 4 R | f ( x )- f 1 ( x ) | dx 5 Definitions, contd D n = { ( X 1 ,Y 1 ) ,..., ( X n ,Y n ) } training set . Let X be a new observation. g n ( X ) = g n ( X,D n ), a sequence of classification rules. L n = P ( Y 6 = g n ( X,D n ) | D n ) . IE L n = P ( Y 6 = g n ( X )) determined by distribution ( X,Y ) and classifier g n . Classifier g n is consistent (weakly): lim n →∞ IE L n = L * ....
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Tech.

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Chile1 - 1 Bayes Optimality of Wavelet-Based Discrimination...

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