Kernel regression

In statistics, Kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables X and Y.

In any nonparametric regression, the conditional expectation of a variable $Y$ relative to a variable $X$ may be written:

$\operatorname {E} (Y|X)=m(X)$ where $m$ is an unknown function.

Nadaraya and Watson, both in 1964, proposed to estimate $m$ as a locally weighted average, using a kernel as a weighting function. The Nadaraya–Watson estimator is:

${\widehat {m}}_{h}(x)={\frac {\sum _{i=1}^{n}K_{h}(x-x_{i})y_{i}}{\sum _{i=1}^{n}K_{h}(x-x_{i})}}$ where $K_{h}$ is a kernel with a bandwidth $h$ .

Derivation

$\operatorname {E} (Y|X=x)=\int yf(y|x)dy=\int y{\frac {f(x,y)}{f(x)}}dy$ Using the kernel density estimation for the joint distribution f(x,y) and f(x) with a kernel K,

${\hat {f}}(x,y)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}\left(x-x_{i}\right)K_{h}\left(y-y_{i}\right)$ ,
${\hat {f}}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}\left(x-x_{i}\right)$ ,

we get

{\begin{aligned}\operatorname {\hat {E}} (Y|X=x)&=\int {\frac {y\sum _{i=1}^{n}K_{h}\left(x-x_{i}\right)K_{h}\left(y-y_{i}\right)}{\sum _{j=1}^{n}K_{h}\left(x-x_{j}\right)}}dy,\\&={\frac {\sum _{i=1}^{n}K_{h}\left(x-x_{i}\right)\int y\,K_{h}\left(y-y_{i}\right)dy}{\sum _{j=1}^{n}K_{h}\left(x-x_{j}\right)}},\\&={\frac {\sum _{i=1}^{n}K_{h}\left(x-x_{i}\right)y_{i}}{\sum _{j=1}^{n}K_{h}\left(x-x_{j}\right)}},\end{aligned}} Priestley–Chao kernel estimator

${\widehat {m}}_{PC}(x)=h^{-1}\sum _{i=2}^{n}(x_{i}-x_{i-1})K\left({\frac {x-x_{i}}{h}}\right)y_{i}$ where $h$ is the bandwidth (or smoothing parameter).

Gasser–Müller kernel estimator

${\widehat {m}}_{GM}(x)=h^{-1}\sum _{i=1}^{n}\left[\int _{s_{i-1}}^{s_{i}}K\left({\frac {x-u}{h}}\right)du\right]y_{i}$ where $s_{i}={\frac {x_{i-1}+x_{i}}{2}}$ Example

This example is based upon Canadian cross-section wage data consisting of a random sample taken from the 1971 Canadian Census Public Use Tapes for male individuals having common education (grade 13). There are 205 observations in total.

The figure to the right shows the estimated regression function using a second order Gaussian kernel along with asymptotic variability bounds

Script for example

The following commands of the R programming language use the npreg() function to deliver optimal smoothing and to create the figure given above. These commands can be entered at the command prompt via cut and paste.

install.packages("np")
library(np) # non parametric library
data(cps71)
attach(cps71)

m <- npreg(logwage~age)

plot(m, plot.errors.method="asymptotic",
plot.errors.style="band",
ylim=c(11, 15.2))

points(age, logwage, cex=.25)

Related

According to David Salsburg, the algorithms used in kernel regression were independently developed and used in fuzzy systems: "Coming up with almost exactly the same computer algorithm, fuzzy systems and kernel density-based regressions appear to have been developed completely independently of one another."