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Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
  Published:2011-03-05
 Printed: Sep 2011
Canadian Mathematical Bulletin
  • Xiang-Jun Zhou,
    Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China
  • Lei Shi,
    Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China
  • Ding-Xuan Zhou,
    Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China
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Abstract

We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.
Keywords: multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence rates multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence rates
MSC Classifications: 68T05, 62J02 show english descriptions Learning and adaptive systems [See also 68Q32, 91E40]
General nonlinear regression 68T05 - Learning and adaptive systems [See also 68Q32, 91E40]
62J02 - General nonlinear regression
 
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