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PREVENTING THE DIRAC DISASTER: Wavelet Based Density Estimation
 J. Ital. Statist. Soc
, 1998
"... This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known that projection estimators tend to overfit the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases t ..."
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Cited by 8 (3 self)
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This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known that projection estimators tend to overfit the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases the estimator is close to the Dirac function concentrated at the observations. We propose a roughness measure of wavelet estimators and establish a data driven method for determining the number of levels to be included in the estimate. Our method exploits the idea of Good and Gaskins (1971) who used the Fisher information functional as a roughness penalty measure. The method is demonstrated on simulated data. Key words and phrases: Fisher Information, Wavelets, Density Estimation. AMS Subject Classification: 62G07, 42A06. Abbreviated title: Linear Wavelet Density Estimators. 1 Introduction It is a well known fact that taking too many basis functions in the orthogonal density estimate will...
Mean squared error minimization for inverse moment problems
 Applied Mathematics and Optimization
, 2014
"... Abstract We consider the problem of approximating the unknown density u ∈ L 2 (Ω, λ) of a measure µ on Ω ⊂ R n , absolutely continuous with respect to some given reference measure λ, from the only knowledge of finitely many moments of µ. Given d ∈ N and moments of order d, we provide a polynomial p ..."
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Cited by 4 (2 self)
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Abstract We consider the problem of approximating the unknown density u ∈ L 2 (Ω, λ) of a measure µ on Ω ⊂ R n , absolutely continuous with respect to some given reference measure λ, from the only knowledge of finitely many moments of µ. Given d ∈ N and moments of order d, we provide a polynomial p d which minimizes the mean square error (u − p) 2 dλ over all polynomials p of degree at most d. If there is no additional requirement, p d is obtained as solution of a linear system. In addition, if p d is expressed in the basis of polynomials that are orthonormal with respect to λ, its vector of coefficients is just the vector of given moments and no computation is needed. In general nonnegativity of p d is not guaranteed even though u is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing p d ≥ 0 that minimizes (u − p) 2 dλ now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating u.
B.: Wavelet Density Estimators over Data Streams
 In: Proc. of SAC. (2005
"... Many scientific and commercial applications rely on an immediate processing of transient data streams. In addition to processing queries over streams, their continuative analysis has received more attention recently. Due to specific characteristics of data streams, common analysis techniques known f ..."
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Cited by 4 (3 self)
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Many scientific and commercial applications rely on an immediate processing of transient data streams. In addition to processing queries over streams, their continuative analysis has received more attention recently. Due to specific characteristics of data streams, common analysis techniques known from the area of data mining are not directly applicable to streams. One of the core operations in data analysis is density estimation, which is used for capturing an unknown distribution in various analysis tasks. Modern density estimates base either on kernel functions or wavelets, whereas the waveletbased ones profit from their ability to identify discontinuities as well as local oscillations. In this paper, we present a new approach to computing wavelet density estimators over data streams. Our estimators allow continuous updates by arrival of new data and provide accurate analytical results, while consuming only a constant amount of memory. Moreover, our estimators are adaptive according to memory as well as CPU usage, i. e., we may change the memory size as well as its computing overhead at runtime. An experimental evaluation proves the feasibility of our approach and shows the superiority of wavelet density estimators compared to their kernelbased counterparts. 1
unknown title
, 2005
"... Abstract Assume that (Xn)n2Z is a real valued stationary time series admitting a common density f. To estimate f in an independent and identically distributed setting, Donoho, Johnstone, Kerkyacharian & Picard (1996) proposed a quasiminimax method based on thresholding wavelets. The aim of th ..."
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Abstract Assume that (Xn)n2Z is a real valued stationary time series admitting a common density f. To estimate f in an independent and identically distributed setting, Donoho, Johnstone, Kerkyacharian & Picard (1996) proposed a quasiminimax method based on thresholding wavelets. The aim of the present work is to extendthis methodology to the dependent case. For this purpose, we introduce the new \Phi weak dependence based on a probability inequality, which includes a large spectrumof classical weak dependence cases. Actually, we establish a link between this condition and the ~OEdependence of Dedecker & Prieur (2004) and the jweak dependence\Phi condition introduced by Doukhan & Louhichi (1999). The estimator we propose adapts the threshold to the dependence of the observations. We obtain near minimaxconvergence rates for Lp losses, p> = 1. We thus apply this method on simulations ofnon stationary but geometrically ergodic cases like dynamical systems and Markovian
Adaptative density estimation with dependent observations
, 2005
"... Assume that (Xn)n∈Z is a real valued stationary time series admitting a common density f. To estimate f in an independent and identically distributed setting, Donoho, Johnstone, Kerkyacharian & Picard (1996) proposed a quasiminimax method based on thresholding wavelets. The aim of the present w ..."
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Assume that (Xn)n∈Z is a real valued stationary time series admitting a common density f. To estimate f in an independent and identically distributed setting, Donoho, Johnstone, Kerkyacharian & Picard (1996) proposed a quasiminimax method based on thresholding wavelets. The aim of the present work is to extend this methodology to the dependent case. For this purpose, we introduce the new Φweak dependence based on a probability inequality, which includes a large spectrum of classical weak dependence cases. Actually, we establish a link between this condition and the ˜ φdependence of Dedecker & Prieur (2004) and the ηweak dependence Φ condition introduced by Doukhan & Louhichi (1999). The estimator we propose adapts the threshold to the dependence of the observations. We obtain near minimax convergence rates for L p losses, p ≥ 1. We thus apply this method on simulations of non stationary but geometrically ergodic cases like dynamical systems and Markovian fields on the line.
PREPRINT: PLEASE DO NOT DISTRIBUTE OR CITE Maximum Likelihood Wavelet Density Estimation with Applications to Image and Shape Matching
, 2007
"... Density estimation for observational data plays an integral role in a broad spectrum of applications, e.g. statistical data analysis and informationtheoretic image registration. Of late, wavelet based density estimators have gained in popularity due to their ability to approximate a large class of ..."
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Density estimation for observational data plays an integral role in a broad spectrum of applications, e.g. statistical data analysis and informationtheoretic image registration. Of late, wavelet based density estimators have gained in popularity due to their ability to approximate a large class of functions; adapting well to difficult situations such as when densities exhibit abrupt changes. The decision to work with wavelet density estimators (WDE) brings along with it theoretical considerations (e.g. nonnegativity, integrability) and empirical issues (e.g. computation of basis coefficients) that must be addressed in order to obtain a bona fide density. In this paper, we present a new method to accurately estimate a nonnegative density which directly addresses many of the problems in practical wavelet density estimation. We cast the estimation procedure in a maximum likelihood framework which estimates the square root of the density √ p; allowing us to obtain the natural nonnegative density representation � √ p � 2. Analysis of this method will bring to light a remarkable theoretical connection with the Fisher information of the density and consequently lead to an efficient constrained optimization procedure to estimate the wavelet coefficients. We illustrate the effectiveness of the algorithm by evaluating its performance on mutual information based image registration, shape point set alignment and empirical comparisons to known densities. The present method is also compared to fixed and variable bandwidth kernel density estimators (KDE).
ADAPTIVE DENSITY ESTIMATION UNDER WEAK DEPENDENCE
, 2008
"... Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue’s measure. Donoho et al. (1996) propose nearminimax estimators bfn based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. T ..."
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Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue’s measure. Donoho et al. (1996) propose nearminimax estimators bfn based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators b fn depend on weak dependence properties of the sequence (Xt)t∈Z through the constant. If these properties are unknown, we propose crossvalidation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.
Adaptive density estimation under dependence
, 2006
"... Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue measure. Donoho et al. (1996) propose a nearminimax method based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of ..."
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Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue measure. Donoho et al. (1996) propose a nearminimax method based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend this methodology to different weakly dependent cases. Bernstein’s type inequalities are proved to be sufficient to extend nearminimax results. Assumptions are detailed for dynamical systems and under the ηweak dependence condition from Doukhan & Louhichi (1999). The threshold levels in our estimator integrates the dependence structure of the sequence (Xt)t∈Z through one parameter γ. The near minimaxity is obtained for ̷L pconvergence rates (p≥1). An estimator of γ is obtained by a crossvalidation procedure. The procedure is illustrated via a simulation study of some dynamical systems and non Markovian ηweakly dependent sequences.
ADAPTIVE DENSITY ESTIMATION UNDER WEAK DEPENDENCE
"... Abstract. Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue’s measure. Donoho et al. (1996) propose nearminimax estimators f̂n based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed ..."
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Abstract. Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue’s measure. Donoho et al. (1996) propose nearminimax estimators f̂n based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators f̂n depend on weak dependence properties of the sequence (Xt)t∈Z through the constant. If these properties are unknown, we propose crossvalidation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds. 1.