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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 ..."
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Cited by 766 (29 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...
Learning minimum volume sets
 J. Machine Learning Res
, 2006
"... Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence region ..."
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Cited by 41 (9 self)
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Given a probability measure P and a reference measure µ, one is often interested in the minimum µmeasure set with Pmeasure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure µ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency and an oracle inequality. Estimators based on histograms and dyadic partitions illustrate the proposed rules. 1
Testing stationarity with surrogates: A timefrequency approach
 IEEE TRANS. SIGNAL PROCESSING
, 2009
"... An operational framework is developed for testing stationarity relatively to an observation scale, in both stochastic and deterministic contexts. The proposed method is based on a comparison between global and local timefrequency features. The originality is to make use of a family of stationary s ..."
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Cited by 11 (4 self)
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An operational framework is developed for testing stationarity relatively to an observation scale, in both stochastic and deterministic contexts. The proposed method is based on a comparison between global and local timefrequency features. The originality is to make use of a family of stationary surrogates for defining the null hypothesis of stationarity and to base on them two different statistical tests. The first one makes use of suitably chosen distances between local and global spectra, whereas the second one is implemented as a oneclass classifier, the timefrequency features extracted from the surrogates being interpreted as a learning set for stationarity. The principle of the method and of its two variations is presented, and some results are shown on typical models of signals that can be thought of as stationary or nonstationary, depending on the observation scale used.
A review of RKHS methods in machine learning
, 2006
"... Over the last ten years, estimation and learning methods utilizing positive definite kernels have become rather popular, particularly in machine learning. Since these methods have a stronger mathematical slant than earlier machine learning methods (e.g., neural networks), there is also ..."
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Cited by 2 (1 self)
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Over the last ten years, estimation and learning methods utilizing positive definite kernels have become rather popular, particularly in machine learning. Since these methods have a stronger mathematical slant than earlier machine learning methods (e.g., neural networks), there is also
Visualizing Densities
, 1994
"... This paper focuses on visualizing densities. We first give a small generalization of kernel density estimators which is appropriate for smoothing general point masses including statistical data, but also more general data forms. We give a heuristic discussion to show that our smoother has some desir ..."
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This paper focuses on visualizing densities. We first give a small generalization of kernel density estimators which is appropriate for smoothing general point masses including statistical data, but also more general data forms. We give a heuristic discussion to show that our smoother has some desirable approximation properties. We also show that for this class of kernel smoother the 1 2 or 3dimensional marginal densities of a highdimensional kernel density approximator have the same formula as the 1 2 or 3dimensional kernel density approximator. We conclude that, for visualization purposes, it is unnecessary to compute kernel density approximators in higher than three dimensions. We also develop the formula for partial derivatives of the kernel density approximator. We develop the relationship between the isopleths, the gradient and the surface normals for a density with twodimensional support. We show that these form a trihedron. We also develop the algorithm for computing the surface normal for the isopleths of a density with threedimensional support. With this information in hand, we discuss rendering and lighting models, contouring algorithms, and stereoscopic display algorithms. We conclude with some examples and a discussion of our experiences in using rendering and lighting, transparency, stereoscopy, dynamic rotation and dynamic thresholding techniques to visualize densities.
Testing Stationarity with Surrogates: A TimeFrequency Approach
, 2010
"... An operational framework is developed for testing stationarity relatively to an observation scale, in both stochastic and deterministic contexts. The proposed method is based on a comparison between global and local timefrequency features. The originality is to make use of a family of stationary ..."
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An operational framework is developed for testing stationarity relatively to an observation scale, in both stochastic and deterministic contexts. The proposed method is based on a comparison between global and local timefrequency features. The originality is to make use of a family of stationary surrogates for defining the null hypothesis of stationarity and to base on them two different statistical tests. The first one makes use of suitably chosen distances between local and global spectra, whereas the second one is implemented as a oneclass classifier, the timefrequency features extracted from the surrogates being interpreted as a learning set for stationarity. The principle of the method and of its two variations is presented, and some results are shown on typical models of signals that can be thought of as stationary or nonstationary, depending on the observation scale used.
Contents I This is a Part 1 1 TimeFrequency Learning Machines For Nonstationarity De tection Using Surrogates 3
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Minimum Volume Peeling: a Multivariate Mode Estimator
"... Abstract A quite extensive literature exists on the estimation of the mode of univariate distributions. On the contrary, few works deal with mode estimation in the multivariate case. With this work, we introduce a procedure to estimate the mode of an unimodal multivariate distribution, that we call ..."
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Abstract A quite extensive literature exists on the estimation of the mode of univariate distributions. On the contrary, few works deal with mode estimation in the multivariate case. With this work, we introduce a procedure to estimate the mode of an unimodal multivariate distribution, that we call minimum volume peeling. Properties, robustness, and implementing issues are briefly discussed. Key words: Convex hull volume, FastMCD, Robust location. 1