Editing Kernel (statistics)

{{short description|Window function}}
{{other uses|Kernel (disambiguation)}}

{{refimprove|date=May 2012}}
The term '''kernel''' is used in [[statistics|statistical analysis]] to refer to a [[window function]]. The term "kernel" has several distinct meanings in different branches of statistics.

==Bayesian statistics==
In statistics, especially in [[Bayesian statistics]], the kernel of a [[probability density function]] (pdf) or [[probability mass function]] (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted.<ref>{{cite journal |last1=Schuster |first1=Eugene |title=Estimation of a probability density function and its derivatives |journal=The Annals of Mathematical Statistics |date=August 1969 |volume=40 |issue=4 |page=1187-1195 |doi=10.1214/aoms/1177697495|doi-access=free }}</ref>  Note that such factors may well be functions of the [[parameter]]s of the pdf or pmf.  These factors form part of the [[normalization factor]] of the [[probability distribution]], and are unnecessary in many situations.  For example, in [[pseudo-random number sampling]], most sampling algorithms ignore the normalization factor.  In addition, in [[Bayesian analysis]] of [[conjugate prior]] distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered.  At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated.  Otherwise, it may be unnecessary (for example, if the distribution only needs to be sampled from).

For many distributions, the kernel can be written in closed form, but not the normalization constant.

An example is the [[normal distribution]].  Its [[probability density function]] is

:<math>p(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}</math>

and the associated kernel is

:<math>p(x|\mu,\sigma^2) \propto e^{-\frac{(x-\mu)^2}{2\sigma^2}}</math>

Note that the factor in front of the exponential has been omitted, even though it contains the parameter <math>\sigma^2</math> , because it is not a function of the domain variable <math>x</math> .

==Pattern analysis==
The kernel of a [[reproducing kernel Hilbert space]] is used in the suite of techniques known as [[kernel methods]] to perform tasks such as [[statistical classification]], [[regression analysis]], and [[cluster analysis]] on data in an implicit space. This usage is particularly common in [[machine learning]].

==Nonparametric statistics==
{{further|Kernel smoothing}}

In [[nonparametric statistics]], a kernel is a weighting function used in [[non-parametric]] estimation techniques. Kernels are used in [[kernel density estimation]] to estimate [[random variable]]s' [[density function]]s, or in [[kernel regression]] to estimate the [[conditional expectation]] of a random variable. Kernels are also used in [[time-series]], in the use of the [[periodogram]] to estimate the [[spectral density]] where they are known as [[window functions]]. An additional use is in the estimation of a time-varying intensity for a [[point process]] where window functions (kernels) are convolved with time-series data.

Commonly, kernel widths must also be specified when running a non-parametric estimation.

===Definition===
{{further|Integral kernel}}
A kernel is a [[non-negative]] [[real-valued function|real-valued]] [[integrable]] function ''K.'' For most applications, it is desirable to define the function to satisfy two additional requirements:

*[[Normalizing constant|Normalization]]:
:<math>\int_{-\infty}^{+\infty}K(u)\,du = 1\,;</math>

*[[Even_and_odd_functions|Even-function]] Symmetry:
:<math>K(-u) = K(u) \mbox{ for all values of } u\,.</math>
The first requirement ensures that the method of kernel density estimation results in a [[probability density function]]. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.

If ''K'' is a kernel, then so is the function ''K''* defined by ''K''*(''u'') = λ''K''(λ''u''), where λ > 0. This can be used to select a scale that is appropriate for the data.

===Kernel functions in common use===
[[File:Kernels.svg|thumb|500px|All of the kernels below in a common coordinate system.]]

Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov,<ref>Named for {{cite journal |last=Epanechnikov |first=V. A. |year=1969 |title=Non-Parametric Estimation of a Multivariate Probability Density |journal=Theory Probab. Appl. |volume=14 |issue=1 |pages=153–158 |doi=10.1137/1114019 }}</ref> quartic (biweight), tricube,<ref>{{cite journal|author=Altman, N. S.|author-link=Naomi Altman|year=1992|title=An introduction to kernel and nearest neighbor nonparametric regression|journal=The American Statistician|volume=46|issue=3|pages=175–185|doi=10.1080/00031305.1992.10475879|hdl=1813/31637|hdl-access=free}}</ref> triweight, Gaussian, quadratic<ref>{{cite journal|author1=Cleveland, W. S.|author1-link= William S. Cleveland  |author2=Devlin, S. J.|author2-link= Susan J. Devlin |year=1988|title=Locally weighted regression: An approach to regression analysis by local fitting|journal=Journal of the American Statistical Association|volume=83|issue=403|pages=596–610|doi=10.1080/01621459.1988.10478639}}</ref> and cosine. 

In the table below, if <math>K</math> is given with a bounded [[Support (mathematics)|support]], then <math> K(u) = 0 </math> for values of ''u'' lying outside the support.
<!--

The gallery style has the problem that the maths is cut off

<gallery Caption="Commonly used Kernel Functions" style="background:white">
Image:Kernel_uniform.svg      |Uniform                             <br/> <math>K(u) = \frac{1}{2}\ 1_{(|u|\leq1)}</math>
Image:Kernel_triangle.svg     |Triangle                            <br/> <math>K(u) = (1-|u|)\ 1_{(|u|\leq1)}</math>
Image:Kernel_epanechnikov.svg |[[V. A. Epanechnikov|Epanechnikov]] <br/> <math>K(u) = \frac{3}{4}(1-u^2)\ 1_{(|u|\leq1)}</math>
Image:Kernel_quartic.svg      |Quartic                             <br/> <math>K(u) = \frac{15}{16}(1-u^2)^2\ 1_{(|u|\leq1)}</math>
Image:Kernel_triweight.svg    |Triweight                           <br/> <math>K(u) = \frac{35}{36}(1-u^2)^3\ 1_{(|u|\leq1)}</math>
Image:Kernel_exponential.svg  |[[Normal_distribution|Gaussian]]    <br/> <math>K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}</math>
Image:Kernel_cosine.svg       |Cosine                              <br/> <math>K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right)1_{(|u|\leq1)}</math>
Image:Kernel logistic.svg     |[[Logistic distribution|Logistic]]  <br/> <math>K(u) = \frac{1}{e^{u}+2+e^{-u}}</math>
Image:Kernel Silverman.svg    |Silverman kernel                    <br/> <math>K(u) = \frac{1}{2} e^{-\frac{|u|}{\sqrt{2}}}\cdot \sin\left( \frac{|u|}{\sqrt{2}}+\frac{\pi}{4}\right)</math>
</gallery>

-->

{| class="wikitable sortable" style="background-color:white;text-align:left"
!colspan=3| Kernel Functions, ''K''(''u'')
! <math>\textstyle \int u^2K(u)du</math>
! <math>\textstyle \int K(u)^2 du</math>
! Efficiency<ref>Efficiency is defined as <math>\sqrt{\int u^2 K(u) \, d u} \int K(u)^2 \, d u</math>.</ref> relative to the Epanechnikov kernel
|-
! Uniform ("rectangular window")
| <math>K(u) = \frac12 </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel uniform.svg|150px]]
"[[Boxcar function]]"
|data-sort-value="0.3333333333333333"| &nbsp; <math>\frac13</math>
|data-sort-value="0.5"| &nbsp; <math>\frac12</math>
| 92.9%
|-
! Triangular
| <math>K(u) = (1-|u|) </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel triangle.svg|150px]]
|data-sort-value="0.16666666666666666"| &nbsp; <math>\frac{1}{6}</math>
|data-sort-value="0.6666666666666666"| &nbsp; <math>\frac{2}{3}</math>
| 98.6%
|-
! [[Epanechnikov_distribution|Epanechnikov]]
(parabolic)
| <math>K(u) = \frac{3}{4}(1-u^2) </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel epanechnikov.svg|150px]]
|data-sort-value="0.2"| &nbsp; <math>\frac{1}{5}</math>
|data-sort-value="0.6"| &nbsp; <math>\frac{3}{5}</math>
| 100%
|-
! Quartic <br />(biweight)
| <math>K(u) = \frac{15}{16}(1-u^2)^2 </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel quartic.svg|150px]]
|data-sort-value="0.14285714285714285"| &nbsp; <math>\frac{1}{7}</math>
|data-sort-value="0.7142857142857143"| &nbsp; <math>\frac{5}{7}</math>
| 99.4%
|-
! Triweight
| <math>K(u) = \frac{35}{32}(1-u^2)^3 </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel triweight.svg|150px]]
|data-sort-value="0.1111111111111111"| &nbsp; <math>\frac{1}{9}</math>
|data-sort-value="0.8158508158508159"| &nbsp; <math>\frac{350}{429}</math>
| 98.7%
|-
! Tricube
| <math>K(u) = \frac{70}{81}(1- {\left| u \right|}^3)^3</math>
Support: <math> |u|\leq1</math>
| [[File:Kernel tricube.svg|150px]]
|data-sort-value="0.1440329218106996"| &nbsp; <math>\frac{35}{243}</math>
|data-sort-value="0.708502024291498"| &nbsp; <math>\frac{175}{247}</math>
| 99.8%
|-
! [[Normal distribution|Gaussian]]
| <math>K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}</math>
| [[File:Kernel exponential.svg|150px]]
|data-sort-value="1"| &nbsp; <math>1\,</math>
|data-sort-value="0.28209479177387814"| &nbsp; <math>\frac{1}{2\sqrt\pi}</math>
| 95.1%
|-
! Cosine
| <math>K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right) </math>
Support: <math> |u|\leq1</math>
| [[File:Kernel cosine.svg|150px]]
|data-sort-value="0.1894305308612978"| &nbsp; <math>1-\frac{8}{\pi^2}</math>
|data-sort-value="0.6168502750680849"| &nbsp; <math>\frac{\pi^2}{16}</math>
| 99.9%
|-
! [[Logistic distribution|Logistic]]
| <math>K(u) = \frac{1}{e^{u}+2+e^{-u}}</math>
| [[File:Kernel logistic.svg|150px]]
|data-sort-value="3.289868133696453"| &nbsp; <math>\frac{\pi^2}{3}</math>
|data-sort-value="0.16666666666666666"| &nbsp; <math>\frac{1}{6}</math>
| 88.7%
|-
! [[Sigmoid function]]
| <math>K(u) = \frac{2}{\pi}\frac{1}{e^{u}+e^{-u}}</math>
| [[File:Kernel logistic.svg|150px]]
|data-sort-value="2.4674011002723395"| &nbsp; <math>\frac{\pi^2}{4}</math>
|data-sort-value="0.20264236728467555"| &nbsp; <math>\frac{2}{\pi^2}</math>
| 84.3%

|-
! Silverman kernel<ref>{{cite book |last=Silverman |first=B. W. |year=1986 |title=Density Estimation for Statistics and Data Analysis | publisher = Chapman and Hall, London|bibcode=1986desd.book.....S }}</ref>
| <math>K(u) = \frac{1}{2} e^{-\frac{|u|}{\sqrt{2}}} \cdot \sin\left( \frac{|u|}{\sqrt{2}}+\frac{\pi}{4}\right)</math>
| [[File:Kernel Silverman.svg|150px]]
|data-sort-value="0"| &nbsp; <math>0</math>
|data-sort-value="0.26516504294495535"| &nbsp; <math>\frac{3\sqrt{2}}{16}</math>
| not applicable
|}

==See also==
*[[Kernel density estimation]]
*[[Kernel smoother]]
*[[Stochastic kernel]]
*[[Positive-definite kernel]]
*[[Density estimation]]
*[[Multivariate kernel density estimation]]
*[[Kernel method]]

{{More footnotes|date=May 2012}}

==References==
{{Reflist}}

*{{cite book
  | last = Li
  | first = Qi
  |author2=Racine, Jeffrey S.
   | title = Nonparametric Econometrics: Theory and Practice
  | publisher = Princeton University Press
  | year = 2007
  | isbn =  978-0-691-12161-1}}

*{{cite web
|last=Zucchini
|first=Walter
|title=APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation
|url=http://staff.ustc.edu.cn/~zwp/teach/Math-Stat/kernel.pdf|access-date=6 September 2018}}

*{{cite journal|year=2002
|first1=D|last1=Comaniciu|first2= P|last2= Meer
|title=Mean shift: A robust approach toward feature space analysis
|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence| volume= 24| issue= 5|pages= 603–619
|id = {{citeseerx|10.1.1.76.8968}} |doi=10.1109/34.1000236}}

[[Category:Nonparametric statistics]]
[[Category:Time series]]
[[Category:Point processes]]
[[Category:Bayesian statistics]]