In
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, kernel regression is a
non-parametric technique to estimate the
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. The objective is to find a non-linear relation between a pair of random variables ''X'' and ''Y''.
In any
nonparametric regression
Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
, the
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
of a variable
relative to a variable
may be written:
:
where
is an unknown function.
Nadaraya–Watson kernel regression
Nadaraya and
Watson
Watson may refer to:
Companies
* Actavis, a pharmaceutical company formerly known as Watson Pharmaceuticals
* A.S. Watson Group, retail division of Hutchison Whampoa
* Thomas J. Watson Research Center, IBM research center
* Watson Systems, make ...
, both in 1964, proposed to estimate
as a locally weighted average, using a
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
as a weighting function. The Nadaraya–Watson estimator is:
:
where
is a kernel with a bandwidth
.
Derivation
:
Using the
kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on '' kernels'' as ...
for the joint distribution ''f''(''x'',''y'') and ''f''(''x'') with a kernel ''K'',
:
:
we get
:
which is the Nadaraya–Watson estimator.
Priestley–Chao kernel estimator
:
where
is the bandwidth (or smoothing parameter).
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 (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."
Statistical implementation
*
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lan ...
mathematical program package
*
JuliaKernelEstimator.jl*
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
: A free MATLAB toolbox with implementation of kernel regression, kernel density estimation, kernel estimation of hazard function and many others is available o
these pages(this toolbox is a part of the book
).
*
Python: the
KernelReg
/code> class for mixed data types in the
/code> sub-package (includes other kernel density related classes), the packag
kernel_regression
as an extension of scikit-learn (inefficient memory-wise, useful only for small datasets)
* R: the function npreg
of the ''np'' package can perform kernel regression.
* Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
npregress
See also
* Kernel smoother
* Local regression
References
Further reading
*
*
*
* {{cite book , last=Simonoff , first=Jeffrey S. , title=Smoothing Methods in Statistics , publisher=Springer , year=1996 , isbn=0-387-94716-7 , url=https://books.google.com/books?id=dgHaBwAAQBAJ
External links
Scale-adaptive kernel regression
(with Matlab software).
(with Microsoft Excel
Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
).
An online kernel regression demonstration
Requires .NET 3.0 or later.
Kernel regression with automatic bandwidth selection
(with Python)
Nonparametric regression
Articles with example R code