
Not to be confused with Kernel principal component analysis.
The kernel regression is a nonparametric technique in statistics to estimate the conditional expectation of a random variable. The objective is to find a nonlinear 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.
Contents

NadarayaWatson kernel regression 1

PriestleyChao kernel estimator 2

GasserMüller kernel estimator 3

Example 4

Related 5

References 6

Statistical implementation 7

External links 8
NadarayaWatson kernel regression
Nadaraya 1964 and Watson 1964 proposed to estimate m as a locally weighted average, using a kernel as a weighting function. The NadarayaWatson estimator is:
\widehat{m}_h(x)=\frac{\sum_{i=1}^n K_h(xx_i) y_i}{\sum_{i=1}^nK_h(xx_i)}
where K is a kernel with a bandwidth h. The fraction is a weighting term with sum 1.
Derivation
\operatorname{E}(Y  X=x) = \int y f(yx) 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) = n^{1} h^{2} \sum_{i=1}^{n} K\left(\frac{xx_i}{h}\right) K\left(\frac{yy_i}{h}\right) ,
\hat{f}(x) = n^{1} h^{1} \sum_{i=1}^{n} K\left(\frac{xx_i}{h}\right)
we obtain the NadarayaWatson estimator.
PriestleyChao kernel estimator
\widehat{m}_{PC}(x) = h^{1} \sum_{i=1}^n (x_i  x_{i1}) K\left(\frac{xx_i}{h}\right) y_i
GasserMüller kernel estimator
\widehat{m}_{GM}(x) = h^{1} \sum_{i=1}^n \left[\int_{s_{i1}}^{s_i} K\left(\frac{xu}{h}\right) du\right] y_i
where s_i = \frac{x_{i1} + x_i}{2}
Example
This example is based upon Canadian crosssection 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.
We consider estimating the unknown regression function using NadarayaWatson kernel regression via the R np package that uses automatic (datadriven) bandwidth selection; see the np vignette for an introduction to the np package.
The figure below shows the estimated regression function using a second order Gaussian kernel along with asymptotic variability bounds
Estimated Regression Function.
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 Salsburg 2002, pp. 290–1, 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 densitybased regressions appear to have been developed completely independently of one another."
References

Nadaraya, E. A. (1964). "On Estimating Regression". Theory of Probability and its Applications 9 (1): 141–2.

Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press.

Simonoff, Jeffrey S. (1996). Smoothing Methods in Statistics. Springer.

Salsburg, D. (2002). The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century. W.H. Freeman.

Richard, C.; Bermudez, J.C. M.; Honeine, P. (March 2009). "Online prediction of time series data with kernels" (PDF). IEEE Transactions on Signal Processing 57 (3): 1058–67.

Parreira, W.; Bermudez, J.C. M.; Richard, C.; Tourneret, J.Y. (May 2012). "Stochastic behavior analysis of the Gaussian kernelleastmeansquare algorithm." (PDF). IEEE Transactions on Signal Processing 60 (5): 2208–2222.

Richard, C.; Bermudez, J.C. M. (November 2012). "Closedform conditions for convergence of the Gaussian kernelleastmeansquare algorithm." (PDF). Proc. of Asilomar'12: 1797–1801.
Statistical implementation
kernreg2 y x, bwidth(.5) kercode(3) npoint(500) gen(kernelprediction gridofpoints)

R: )npnpreg (package

GNU/octave mathematical program package:
External links

Scaleadaptive kernel regression (with Matlab software).

Tutorial of Kernel regression using spreadsheet (with Microsoft Excel).

An online kernel regression demonstration Requires .NET 3.0 or later.

The np package An R package that provides a variety of nonparametric and semiparametric kernel methods that seamlessly handle a mix of continuous, unordered, and ordered factor data types.
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