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where V is the volume of some neighborhood(say A) around x and k denotes the number of observations that are contained within the neighborhood. | where V is the volume of some neighborhood(say A) around x and k denotes the number of observations that are contained within the neighborhood. | ||
| − | The basic idea of k-NN is to extend the neighborhood, until the k nearest values are included. If we consider the neighborhood around x as a sphere, for the given N Observations, we pick an integer, <br/> < | + | The basic idea of k-NN is to extend the neighborhood, until the k nearest values are included. If we consider the neighborhood around x as a sphere, for the given N Observations, we pick an integer, <br/> |
| + | |||
| + | {an equation goes here} <br/> | ||
| + | |||
| + | If x<sub>l</sub> is the k<sup>th</sup> closest sample point to x, then h<sub>k</sub> = ||x<sub>l</sub> - x||<br/> | ||
| + | |||
| + | {equation of estimated density at x here} | ||
| + | |||
| + | We approximate the density p(x) by, | ||
| + | {equation here } <br/> | ||
| + | |||
Revision as of 19:18, 24 April 2014
K-Nearest Neighbors Density Estimation
A slecture by CIT student Raj Praveen Selvaraj
Partly based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
Introduction
This slecture discusses about the K-Nearest Neighbors(k-NN) approach to estimate the density of a given distribution. The approach of K-Nearest Neighbors is very popular in signal and image processing for clustering and classification of patterns. It is an non-parametric density estimation technique which lets the region volume be a function of the training data. We will discuss the basic principle behind the k-NN approach to estimate density at a point X and then move on to building a classifier using the k-NN Density estimate.
Basic Principle
The general formulation for density estimation states that, for N Observations x1,x2,x3,...,xn the density at a point x can be approximated by the following function,
where V is the volume of some neighborhood(say A) around x and k denotes the number of observations that are contained within the neighborhood.
The basic idea of k-NN is to extend the neighborhood, until the k nearest values are included. If we consider the neighborhood around x as a sphere, for the given N Observations, we pick an integer,
{an equation goes here}
If xl is the kth closest sample point to x, then hk = ||xl - x||
{equation of estimated density at x here}
We approximate the density p(x) by,
{equation here }
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