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August 2011 Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels
Robert L. Wolpert, Merlise A. Clyde, Chong Tu
Ann. Statist. 39(4): 1916-1962 (August 2011). DOI: 10.1214/11-AOS889


This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. Lévy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the Lévy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the Lévy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.


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Robert L. Wolpert. Merlise A. Clyde. Chong Tu. "Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels." Ann. Statist. 39 (4) 1916 - 1962, August 2011.


Published: August 2011
First available in Project Euclid: 24 August 2011

zbMATH: 1227.62030
MathSciNet: MR2893857
Digital Object Identifier: 10.1214/11-AOS889

Primary: 62G08
Secondary: 60E07

Keywords: Bayes , Besov , kernel regression , LARK , Lévy random field , Nonparametric regression , relevance vector machine , reversible jump Markov chain Monte Carlo , splines , Support Vector Machine , Wavelets

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 4 • August 2011
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