Projection Pursuit Density Estimation

Friedman, Jerome H.; Stuetzle, Werner; Schroeder, Anne Pauline

doi:10.2307/2288406

Cited by 81 publications

(91 citation statements)

References 5 publications

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“…They apply Bayesian networks as base learners. Projection pursuit density estimation as presented in Friedman et al (1984) constructs an estimate of product form with a stagewise algorithm minimizing the negative log-likelihood criterion. Priebe (1994) considers an iterative algorithm with a stopping rule for estimating Gaussian mixtures.…”

Section: Introductionmentioning

confidence: 99%

Density estimation with stagewise optimization of the empirical risk

Klemelä

2006

Mach Learn

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We consider multivariate density estimation with identically distributed observations. We study a density estimator which is a convex combination of functions in a dictionary and the convex combination is chosen by minimizing the L 2 empirical risk in a stagewise manner. We derive the convergence rates of the estimator when the estimated density belongs to the L 2 closure of the convex hull of a class of functions which satisfies entropy conditions. The L 2 closure of a convex hull is a large nonparametric class but under suitable entropy conditions the convergence rates of the estimator do not depend on the dimension, and density estimation is feasible also in high dimensional cases. The variance of the estimator does not increase when the number of components of the estimator increases. Instead, we control the bias-variance trade-off by the choice of the dictionary from which the components are chosen.

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Section: Introductionmentioning

confidence: 99%

Density estimation with stagewise optimization of the empirical risk

Klemelä

2006

Mach Learn

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“…Another generalization might be considered for a variant of the projection pursuit density estimation (PPDE) proposed by Friedman et al (1984). In PPDE, a sequence of augmenting functions is determined via optimization of the (marginal) Kullback-Leibler divergence.…”

Section: Discussionmentioning

confidence: 99%

“…The purpose of this study is to develop a general but practical learning method for multivariate density estimation. There have been several reports on stagewise optimization algorithms like boosting for density estimation, including Friedman et al (1984), Ridgeway (2002), Rosset and Segal (2002) and Klemelä (2007). The approaches used in these studies are based on implementing repeat optimization of a certain loss function which can be seen as an empirical form of the corresponding divergence measure: the Kullback-Leibler divergence and the L 2 norm.…”

Section: Introductionmentioning

confidence: 99%

Density estimation with minimization of U-divergence

Naito

Eguchi

2012

Mach Learn

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This paper is concerned with density estimation based on the stagewise minimization of the U -divergence. The U -divergence is a general divergence measure involving a convex function U which includes the Kullback-Leibler divergence and the L 2 norm as special cases. The algorithm to yield the density estimator is closely related to the boosting algorithm and it is shown that the usual kernel density estimator can also be seen as a special case of the proposed estimator. Non-asymptotic error bounds of the proposed estimators are developed and numerical experiments show that the proposed estimators often perform better than several existing methods for density estimation.

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“…The first contribution involves a dedicated chapter that shows how a number of state-of-the-art problems are special cases of transmetric density estimation. The examples includes distributions with elliptic level sets (Liescher, 2005), linear regression, partial linear models (Härdle et al, 2000), projection pursuit models (Friedman et al, 1984) and also an example of nonparametric functional analysis (Ferraty and Vieu, 2006). Based on these examples, it should be straightforward to see that other methods can be formalized in this way as well.…”

Section: Introductionmentioning

confidence: 99%

Properties of Transmetric Density Estimation

Hovda¹

2016

IJSP

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Transmetric density estimation is a generalization of kernel density estimation that is proposed in Hovda (2014) and Hovda (2016), This framework involves the possibility of making assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display. In this paper we show that several state-of-the-art nonparametric, semiparametric and even parametric methods are special cases of this formulation, meaning that there is a unified approach.Moreover, it is shown that parameters can be trained using unbiased cross-validation. When parameter estimation is included, the mean integrated squared error of the transmetric density estimator is lower than for the common kernel density estimator, when the number of dimensions is larger than two.

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Projection Pursuit Density Estimation

Cited by 81 publications

References 5 publications

Density estimation with stagewise optimization of the empirical risk

Density estimation with stagewise optimization of the empirical risk

Density estimation with minimization of U-divergence

Properties of Transmetric Density Estimation

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