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== Basic Principle == | == Basic Principle == | ||
The general formulation for density estimation states that, for N Observations x<sub>1</sub>,x<sub>2</sub>,x<sub>3</sub>,...,x<sub>n</sub> the density at a point x can be approximated by the following function, | The general formulation for density estimation states that, for N Observations x<sub>1</sub>,x<sub>2</sub>,x<sub>3</sub>,...,x<sub>n</sub> the density at a point x can be approximated by the following function, | ||
| − | [[Image:knn1.jpg|border]] | + | <center>[[Image:knn1.jpg|border]]</center> |
| + | |||
| + | 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 k <math>${\bf k \ge 2}$</math> | ||
Revision as of 17:46, 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 k $ ${\bf k \ge 2}$ $
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