World Library  
Flag as Inappropriate
Email this Article

Kernel regression

Article Id: WHEBN0008771825
Reproduction Date:

Title: Kernel regression  
Author: World Heritage Encyclopedia
Language: English
Subject: Semiparametric regression, Virtual sensing, Hat matrix, Asymptotic theory (statistics), Kernel (statistics)
Collection: Nonparametric Statistics
Publisher: World Heritage Encyclopedia

Kernel regression

Not to be confused with Kernel principal component analysis.

The 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 Y relative to a variable X may be written:

\operatorname{E}(Y | X) = m(X)

where m is an unknown function.


  • Nadaraya-Watson kernel regression 1
    • Derivation 1.1
  • Priestley-Chao kernel estimator 2
  • Gasser-Müller kernel estimator 3
  • Example 4
    • Script for example 4.1
  • Related 5
  • References 6
  • Statistical implementation 7
  • External links 8

Nadaraya-Watson kernel regression

Nadaraya 1964 and Watson 1964 proposed to estimate m as a locally weighted average, using a kernel as a weighting function. The Nadaraya-Watson estimator is:

\widehat{m}_h(x)=\frac{\sum_{i=1}^n K_h(x-x_i) y_i}{\sum_{i=1}^nK_h(x-x_i)}

where K is a kernel with a bandwidth h. The fraction is a weighting term with sum 1.


\operatorname{E}(Y | X=x) = \int y f(y|x) 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{x-x_i}{h}\right) K\left(\frac{y-y_i}{h}\right) ,
\hat{f}(x) = n^{-1} h^{-1} \sum_{i=1}^{n} K\left(\frac{x-x_i}{h}\right)

we obtain the Nadaraya-Watson estimator.

Priestley-Chao kernel estimator

\widehat{m}_{PC}(x) = h^{-1} \sum_{i=1}^n (x_i - x_{i-1}) K\left(\frac{x-x_i}{h}\right) y_i

Gasser-Müller kernel estimator

\widehat{m}_{GM}(x) = h^{-1} \sum_{i=1}^n \left[\int_{s_{i-1}}^{s_i} K\left(\frac{x-u}{h}\right) du\right] y_i

where s_i = \frac{x_{i-1} + x_i}{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.

We consider estimating the unknown regression function using Nadaraya-Watson kernel regression via the R np package that uses automatic (data-driven) 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.

library(np) # non parametric library

m <- npreg(logwage~age)




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 density-based regressions appear to have been developed completely independently of one another."


  • 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 kernel-least-mean-square algorithm." (PDF). IEEE Transactions on Signal Processing 60 (5): 2208–2222.  
  • Richard, C.; Bermudez, J.-C. M. (November 2012). "Closed-form conditions for convergence of the Gaussian kernel-least-mean-square 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

  • Scale-adaptive 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.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.