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
X
\operatorname{E}(Y\midX)=m(X)
where
m
Nadaraya and Watson, both in 1964, proposed to estimate
m
| \widehat{m} | ||||||||||||||||||||||
|
where
Kh(t)=
| 1 | K\left( | |
| h |
| t | |
| h |
\right)
h
K( ⋅ )
| infty | |
| \int | |
| -infty |
uK(u)du=0
Starting with the definition of conditional expectation,
\operatorname{E}(Y\midX=x)=\intyf(y\midx)dy=\inty
| f(x,y) | |
| f(x) |
dy
we estimate the joint distributions f(x,y) and f(x) using kernel density estimation with a kernel K:
\hat{f}(x,y)=
| 1 | |
| n |
| n | |
| \sum | |
| i=1 |
Kh(x-xi)Kh(y-yi),
\hat{f}(x)=
| 1 | |
| n |
| n | |
| \sum | |
| i=1 |
Kh(x-xi),
We get:
\begin{align} \operatorname{\hatE}(Y\midX=x)&=\inty
| \hat{f | |
| (x,y)}{\hat{f}(x)} |
dy,\\[6pt] &=\inty
| ||||||||||
|
dy,\\[6pt] &=
| ||||||||||
|
,\\[6pt] &=
| ||||||||||
|
, \end{align}
which is the Nadaraya–Watson estimator.
\widehat{m}PC(x)=h-1
| n | |
| \sum | |
| i=2 |
(xi-xi-1)K\left(
| x-xi | |
| h |
\right)yi
where
h
\widehat{m}GM(x)=h-1
| n | |
| \sum | |
| i=1 |
| si | ||
| \left[\int | K\left( | |
| si-1 |
| x-u | |
| h |
\right)du\right]yi
where
si=
| xi-1+xi | |
| 2 |
.
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.
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.
m <- npreg(logwage~age)
plot(m, plot.errors.method="asymptotic", plot.errors.style="band", ylim=c(11, 15.2))
points(age, logwage, cex=.25)detach(cps71)
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."[5]
[http://www.statsmodels.org/stable/generated/statsmodels.nonparametric.kernel_regression.KernelReg.html KernelReg] class for mixed data types in the [http://www.statsmodels.org/stable/nonparametric.html statsmodels.nonparametric] sub-package (includes other kernel density related classes), the package kernel_regression as an extension of scikit-learn (inefficient memory-wise, useful only for small datasets)npreg of the np package can perform kernel regression.[7] [8]