Get A Probabilistic Theory of Pattern Recognition PDF

By Luc Devroye

ISBN-10: 1461207118

ISBN-13: 9781461207115

ISBN-10: 146126877X

ISBN-13: 9781461268772

Pattern popularity offers the most major demanding situations for scientists and engineers, and lots of diversified methods were proposed. the purpose of this publication is to supply a self-contained account of probabilistic research of those methods. The publication contains a dialogue of distance measures, nonparametric equipment in response to kernels or nearest buddies, Vapnik-Chervonenkis thought, epsilon entropy, parametric class, mistakes estimation, unfastened classifiers, and neural networks. anyplace attainable, distribution-free homes and inequalities are derived. a considerable element of the implications or the research is new. Over 430 difficulties and routines supplement the material.

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Take logarithms and note that g*(x) = 1 if and only if (x- mJ)TL~ 1 (x- mJ)- 21ogp + log(det(LJ)) < (x- mo)rL 01(x- mo)- 2log(l- p) + log(det(Lo)). In practice, one might wish to estimate m 1, m 0 , L 1 , Lo and p from the data and use these estimates in the formula for g*. Interestingly, as (x - m; )r L;-l (x - m;) is the squared Mahalanobis distance from tom; in class i (called the Bayes rule is simply x g*(x) = { 1 0 r? rJ - 2log((l - if < otherwise. ), + log(det(Lo)/ det(L 1)) In particular, when p = 1/2, Lo = L 1 = L, we have *x = { I g ( ) 0 'f 2 rl < ro2 otherwise; I just classify according to the class whose mean is at the nearest Mahalanobis distance from x.

For any a ::;: 1, find a sequence of distributions of (Xn, Yn) having expected conditional entropies En and Bayes errors L~ such that L~ -+ 0 as n -+ oo, and En decreases to zero at the same rate as ( L~ )"'. 5. CONCAVITY OF ERROR MEASURES. Let Y denote the mixture random variable taking the value Y1 with probability p and the value Y2 with probability 1- p. Let X be a fixed Rd-valued random variable, and define ry 1(x) = P{Y1 = IIX = x }, ry 2 (x) = P{Y2 = IIX = x}, where Y1 , Y2 are Bernoulli random variables.

O 30 3. 8 The Mahalanobis Distance Two conditional distributions with about the same covariance matrices and means that are far away from each other are probably so well separated that L * is small. An interesting measure of the visual distance between two random variables Xo and X 1 is the so-called Mahalanobis distance (Mahalanobis, (1936)) given by wherem, =EX~omo=EXo,arethemeans,:E 1 =E{(X 1 -m 1)(X 1 -m 1l}and :Eo = E vxo- mo)(Xo- mol} are the covariance matrices, :E = p:E 1 + (l p):Eo, 0 is the transpose of a vector, and p = I -pis a mixture parameter.

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A Probabilistic Theory of Pattern Recognition by Luc Devroye

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