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Estimating the Support of a HighDimensional Distribution
, 1999
"... Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propo ..."
Abstract

Cited by 508 (31 self)
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Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a preliminary theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled d...
Kernel estimation of density level sets
 J. Multivariate Anal
, 2006
"... Abstract. Let f be a multivariate density and fn be a kernel estimate of f drawn from the nsample X1, · · ·,Xn of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the tlevel set {f ≥ t} and its plug ..."
Abstract

Cited by 8 (1 self)
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Abstract. Let f be a multivariate density and fn be a kernel estimate of f drawn from the nsample X1, · · ·,Xn of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the tlevel set {f ≥ t} and its plugin estimator {fn ≥ t}. As a corollary, we obtain the exact rate of convergence of a plugin type estimate of the density level set corresponding to a fixed probability for the law induced by f.