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

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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
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2 Overview Talk about classifying Y into one of two classes labeled by 0 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
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3 Definitions ( X, Y ) IR d × { 0 , 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 → { 0 , 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].
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4 Definitions, contd Assume density of X exists, X f. Let f 0 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 0 ( 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 0 πf 1 ) dx π = 1 / 2 , L * = 1 / 2 - 1 / 4 R | f 0 ( x ) - f 1 ( x ) | dx
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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 * . Classifier g n is consistent (strongly): lim n →∞ L n = L * , a.s. Devroye, Gy¨orfi, Lugosi (1996)
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6 Fourier Series Classifiers Assume f IL 2 . f IL 2 α = πf 1 - (1 - π ) f 0 IL 2 .
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