| Line 23: | Line 23: | ||
It can be rewritten as | It can be rewritten as | ||
<div style="text-align: center;"> <math>\int_R p(\textbf{x}') d\textbf{x}' \simeq p(\textbf{x})V \simeq \frac{k}{n}</math>, </div> | <div style="text-align: center;"> <math>\int_R p(\textbf{x}') d\textbf{x}' \simeq p(\textbf{x})V \simeq \frac{k}{n}</math>, </div> | ||
| − | if we assume a small local region | + | if we assume a small local region <math>R</math>, a large number of samples <math>n</math>, and <math>k</math> of <math>n</math> falling in <math>R</math>.\\ |
| − | Suppose that the region | + | Suppose that the region <math>R</math> is a <math>d</math>-dimensional hypercube around <math>\textbf{x}_i \in \mathbb{R}^n</math> in the rest of this slecture, and let the volume <math>V_n</math>: |
<div style="text-align: center;"> <math>V_n = h_n^d</math> </div> | <div style="text-align: center;"> <math>V_n = h_n^d</math> </div> | ||
| − | where | + | where <math>h_n</math> is the length of an edge. Then the window function for this hypercube can be defined by |
<div style="text-align: center;"> <math>\varphi(\textbf{u}) = \left\{ \begin{array}{ccc} 1, & |u_j| \leq \frac{1}{2} & j = 1, ..., d \\ 0, & else. & \end{array} \right.</math> </div> | <div style="text-align: center;"> <math>\varphi(\textbf{u}) = \left\{ \begin{array}{ccc} 1, & |u_j| \leq \frac{1}{2} & j = 1, ..., d \\ 0, & else. & \end{array} \right.</math> </div> | ||
Revision as of 07:03, 30 April 2014
Parzen window method and classification
A slecture by ECE student Chiho Choi
Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
in progess....
Unlike parametric density estimation methods, non-parametric approaches locally estimate density function by a small number of neighboring samples [4] and therefore show less accurate estimation results. In spite of their accuracy, however, the performance of classifiers designed using these estimates is very satisfactory.
The basic idea for estimating unknown density function is based on the fact that the probability $ P $ that a vector x belongs to a region $ R $ [1]:
It can be rewritten as
if we assume a small local region $ R $, a large number of samples $ n $, and $ k $ of $ n $ falling in $ R $.\\ Suppose that the region $ R $ is a $ d $-dimensional hypercube around $ \textbf{x}_i \in \mathbb{R}^n $ in the rest of this slecture, and let the volume $ V_n $:
where $ h_n $ is the length of an edge. Then the window function for this hypercube can be defined by
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$ \rightarrow $
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