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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 propose a metho ..."
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Cited by 501 (32 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...
Extreme values and Haar series estimates of point processes boundaries
 Scandinavian Journal of Statistics
, 2002
"... We present a new method for estimating the edge of a twodimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkab ..."
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Cited by 7 (3 self)
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We present a new method for estimating the edge of a twodimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation. Keywords: Haar basis, Extreme values, Poisson process, Shape estimation. AMS Subject Classification: Primary 60G70; Secondary 62M30, 62G05, 62G20. 1 1 Introduction Many proposals are given in the literature for estimating a bounded set S of R 2 , given a finite random set N of points drawn from the interior. Such a diversity follows from crossing properties of the observed random set N (sample, point process, random field on a grid, ...), properties of the unknown bounded set S (convex sets, starshaped domains, pieces of support under a given curve, images, ...), ...
Extreme Values and Projection Estimates of Point Processes Boundaries
, 2000
"... We present a method for estimating the edge of a twodimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on projection on C 1 bases and extreme values of the point process. We give conditions on the Dirichlet's kernel associated to the ..."
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We present a method for estimating the edge of a twodimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on projection on C 1 bases and extreme values of the point process. We give conditions on the Dirichlet's kernel associated to the C 1 basis for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias, illustrated by a simulation. Keywords: Projection on C 1 bases, Extreme values, Poisson process, Shape estimation. AMS Subject Classification: Primary 60G70; Secondary 62M30, 62G05, 62G20. 1 1 Introduction We address the problem of estimating a bounded set S of R 2 given a finite random set N of points drawn from the interior. This kind of problem arises in various frameworks such as classification [10], image processing [14] or econometrics problems [3]. A lot of different solutions were proposed since [7] and [16] depending on the properties of the observed ran...