Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels
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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. © Institute of Mathematical Statistics, 2011.
Published Version (Please cite this version)10.1214/11-AOS889
Publication InfoWolpert, RL; Clyde, MA; & Tu, C (2011). Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels. Annals of Statistics, 39(4). pp. 1916-1962. 10.1214/11-AOS889. Retrieved from https://hdl.handle.net/10161/8885.
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Professor of Statistical Science
Model uncertainty and choice in prediction and variable selection problems for linear, generalized linear models and multivariate models. Bayesian Model Averaging. Prior distributions for model selection and model averaging. Wavelets and adaptive kernel non-parametric function estimation. Spatial statistics. Experimental design for nonlinear models. Applications in proteomics, bioinformatics, astro-statistics, air pollution and health effects, and environmental sciences.
Professor Emeritus of Statistical Science
I'm a stochastic modeler-- I build computer-resident mathematical modelsfor complex systems, and invent and program numerical algorithms for makinginference from the models. Usually this involves predicting things thathaven't been measured (yet). Always it involves managing uncertainty andmaking good decisions when some of the information we'd need to be fullycomfortable in our decision-making is unknown. Originally trained as a mathematician specializing
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