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Title:  High order Parzen windows and randomized sampling 
Other Titles:  Gao jie Parzen windows ji sui ji cai yang 高階 Parzen windows 及隨機采樣 
Authors:  Zhou, Xiangjun (周祥軍) 
Department:  Department of Mathematics 
Degree:  Doctor of Philosophy 
Issue Date:  2009 
Publisher:  City University of Hong Kong 
Subjects:  Sampling (Statistics) Computational learning theory. Multivariate analysis. Approximation theory. 
Notes:  CityU Call Number: QA276.6 .Z45 2009 v, 62 leaves : ill. 30 cm. Thesis (Ph.D.)City University of Hong Kong, 2009. Includes bibliographical references (leaves [57]62) 
Type:  thesis 
Abstract:  In the thesis, high order Parzen windows are studied for understanding some algorithms
in learning theory and randomized sampling in multivariate approximation. Our
ideas are from Parzen window method for density estimation and sampling theory.
First, we define basic window functions to construct our high order Parzen windows.
We derived learning rates for the leastsquare regression and density estimation
on bounded domains under some decay conditions near the boundary on the marginal
distribution of the probability measure for sampling. These rates can be almost optimal
when the marginal distribution decays fast and the order of the Parzen windows is large
enough. Compared with standard Parzen windows for density estimation, the high order
Parzen window estimator is not a density function when the order J is greater than
2.
Then for randomized sampling in shiftinvariant spaces, we investigate the approximation
of functions on the whole space Rn. We consider the situation when the
sampling points are neither i.i.d. nor regular, but are noised from regular grids hZn
for some constant h > 0 by probability density functions. We assume some decay and
regularity conditions for the noise probability function and the approximated function
on Rn. Under suitable choices of the scaling parameter, the approximation orders are
estimated by means of regularity of the approximated function, the density function
and the order of the Parzen windows.
Next we study the approximation of multivariate functions in Sobolev spaces by
high order Parzen windows in a nonuniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by nonuniform 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. Our main result provides bounds for the approximation of the target
function on Rn in Sobolev spaces.
Finally, we provide an experiment example from a real application. We use second
order basic window functions to construct a second order Parzen windows. The
algorithm works well both in artificial data and in the real application. 
Online Catalog Link:  http://lib.cityu.edu.hk/record=b3008230 
Appears in Collections:  MA  Doctor of Philosophy

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