Parzen Windows

Richard O. Duda, Peter E. Hart, David G. Stork

Pattern Classi�cation - 1st edition: 1973, 2nd edition: 2000

February 15, 2012

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Introduction

Parzen Windows ' Kernel density estimation

Goal: given a set of d -dimensional samples, estimate the

underlying probability distribution

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Introductory example: hypercube

We can use a d -dymensional hypercube as a window function

to describe the distribution probability

,

where hdn is the length of each edge and uj is u's j th normalized

position inside the hypercube

Now, probability distribution can be estimated as

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General window functions

For pn (x) to be a proper density function, we can use any

window function satisfying

Probability distribution can be written as

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Parzen windows: width (hn) e�ect

There's little justi�cation for a speci�c hn size or a speci�c

window function if no knowledge about the underlying

distribution is available

Usually, a Gaussian distribution is used for statistichal

independence

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Convergence of the mean

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Convergence of the variance

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Illustrations I

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Illustrations II

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Illustrations III

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Classi�cation example

A di�erent estimate of the probability distribution can be built

for each class

Given a sample, the estimated probabilities to belong to each

class can be computed

The highest probability can be chosen as the classi�er's output

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2-class classi�cation example: hn size e�ect

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Probabilistic Neural Network I

Fast implementation of parzen-window classi�cation

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PNN II

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