# Cross Validation for Kernel Density Estimation

# Cross Validation for Kernel Density Estimation

Join the DZone community and get the full member experience.

Join For Free**Hortonworks Sandbox for HDP and HDF is your chance to get started on learning, developing, testing and trying out new features. Each download comes preconfigured with interactive tutorials, sample data and developments from the Apache community.**

In a post published in July, I mentioned the so-called Goldilocks principle, in the context of kernel density estimation, and bandwidth selection. The bandwidth should not be too small (the variance would be too large) and it should not be too large (the bias would be too large). Another standard method to select the bandwidth, as mentioned this afternoon in class is the cross-validation technique (described in Chiu (1991)). Here, we would like to minimize

The integral can be written

Since the third component is constant, we have to minimize the expected value of the sum of the first two.

The idea is to approximate it as

which can easily be computed. Consider here some sample, with 50 observations, from a Gaussian distribution,

> set.seed(1) > X=rnorm(50)

From Silverman’s rule of thumb (which should be appropriate here since the sample has a Gaussian sample) the optimal bandwidth is

> 1.06*sd(X)*length(X)^(-1/5) [1] 0.4030127

Using the cross-validation technique mentioned above, compute

> J=function(h){ + fhat=Vectorize(function(x) density(X,from=x,to=x,n=1,bw=h)$y) + fhati=Vectorize(function(i) density(X[-i],from=X[i],to=X[i],n=1,bw=h)$y) + F=fhati(1:length(X)) + return(integrate(function(x) fhat(x)^2,-Inf,Inf)$value-2*mean(F)) + } > vx=seq(.1,1,by=.01) > vy=Vectorize(J)(vx) > df=data.frame(vx,vy) > library(ggplot2) > qplot(vx,vy,geom="line",data=df)

The function has the following shape

and the optimal value is

> optimize(J,interval=c(.1,1)) $minimum [1] 0.4687553 $objective [1] -0.3355477

Note that, indeed, it is close to Siverman’s optimal bandwidth.

**Hortonworks Community Connection (HCC) is an online collaboration destination for developers, DevOps, customers and partners to get answers to questions, collaborate on technical articles and share code examples from GitHub. Join the discussion.**

Published at DZone with permission of Arthur Charpentier , DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

## {{ parent.title || parent.header.title}}

## {{ parent.tldr }}

## {{ parent.linkDescription }}

{{ parent.urlSource.name }}