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 formula_1 relative to a variable formula_2 may be written:
formula_3
where formula_4 is an unknown function.
Nadaraya–Watson kernel regression.
Nadaraya and Watson, both in 1964, proposed to estimate formula_4 as a locally weighted average, using a kernel as a weighting function. The Nadaraya–Watson estimator is:
formula_6
where formula_7 is a kernel with a bandwidth formula_8. The denominator is a weighting term with sum 1.
Derivation.
formula_9
Using the kernel density estimation for the joint distribution "f(x,y)" and "f(x)" with a kernel K,
formula_10,
formula_11,
we get
formula_12
which is the Nadaraya–Watson estimator.
Priestley–Chao kernel estimator.
formula_13
where formula_14 is the bandwidth (or smoothing parameter).
Gasser–Müller kernel estimator.
formula_15
where formula_16
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 codice_1 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."