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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...
Generalizing the Convex Hull of a Sample: The R Package alphahull
"... This vignette presents the R package alphahull which implements the αconvex hull and the αshape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the αconvex hull and the αshape ..."
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Cited by 1 (0 self)
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This vignette presents the R package alphahull which implements the αconvex hull and the αshape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the αconvex hull and the αshape are able to reconstruct nonconvex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations.
ARTICLE Communicated by Vladimir Vapnik Estimating the Support of a HighDimensional Distribution
"... Suppose you are given some data set 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 equals some a priori speci�ed value between 0 and 1. We propose a method to ..."
Abstract
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Suppose you are given some data set 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 equals some a priori speci�ed value between 0 and 1. We propose a method to approach this problem by trying to estimate a function f that 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 coef�cients 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 theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabeled data. 1