Kernel regression


In statistics, Kernel regression is a non-parametric technique in statistics 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 relative to a variable may be written:
where is an unknown function.

Nadaraya–Watson kernel regression

and Watson, both in 1964, proposed to estimate as a locally weighted average, using a kernel as a weighting function. The Nadaraya–Watson estimator is:
where is a kernel with a bandwidth. The denominator is a weighting term with sum 1.

Derivation

Using the kernel density estimation for the joint distribution f and f with a kernel K,
,

we get
which is the Nadaraya–Watson estimator.

Priestley–Chao kernel estimator

where is the bandwidth.

Gasser–Müller kernel estimator

where

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. 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
library # non parametric library
data
attach
m <- npreg
plot
points

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."

Statistical implementation