Difference between revisions of "Parzen Windows" - Rhea

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Introduction

As a Local Density Estimation method, Parzen Windows, focuses on estimating the density in a small region and making a decision based on that specific region. The density estimation relies on the specific test point being classified and the region around it. The reasoning behind it is to not focus on the accuracy of density estimation, but on the accuracy of decision making.

The idea is to create a smaller region, with the test point at its center, within the full region of the training data, and estimate the density in that small region and classify that test point based on the class with the highest density.

The method relies on the type or shape of the window used to form the smaller region.

Parzen Windows Method with a Cube Window

As mentioned before the reasoning behind it is to accurately make decisions and classify data. Thus the ideal method to explain how it works is to start with the classification process.

Given a set of training data in a region $ \Re^n $, and a point $ x_0 $ in the region $ \Re^n $.

a unit cube region $ R $ is created around $ x_0 $

using the following window function to count the samples in $ R $

$ \phi(x) = \rho(x_1,x_2,...,x_n) = \begin{cases} 1 \quad if \left\vert x_1 \right\vert < \frac{1}{2}, \left\vert x_2 \right\vert < \frac{1}{2},..., \left\vert x_n \right\vert < \frac{1}{2} \\ 0 \quad else \\ \end{cases} $

[[Image:File:Aalshai fig1.jpg]]

Questions and comments

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