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| The term '''kernel''' has two separate meanings in [[statistics]].
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| ==In Bayesian statistics==
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| 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.{{Citation needed|date=May 2012}} 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).
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| For many distributions, the kernel can be written in closed form, but not the normalization constant.
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| An example is the [[normal distribution]]. Its [[probability density function]] is
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| :<math>p(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}</math> | |
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| and the associated kernel is
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| :<math>p(x|\mu,\sigma^2) \propto e^{-\frac{(x-\mu)^2}{2\sigma^2}}</math>
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| 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> .
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| ==In non-parametric statistics==
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| In [[non-parametric]] 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]]. An additional use is in the estimation of a time-varying intensity for a [[point process]].
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| Commonly, kernel widths must also be specified when running a non-parametric estimation.
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| ===Definition===
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| A kernel is a [[non-negative]] [[real-valued function|real-valued]] [[integrable]] function ''K'' satisfying the following two requirements:
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| *<math>\int_{-\infty}^{+\infty}K(u)\,du = 1\,;</math>
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| *<math>K(-u) = K(u) \mbox{ for all values of } u\,.</math>
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| 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.
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| 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.
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| ===Kernel functions in common use===
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| 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.|year=1992|title=An introduction to kernel and nearest
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| neighbor nonparametric regression|journal=The American Statistician|volume=46|issue=3|pages=175–185}}</ref> triweight, Gaussian,quadratic<ref>{{cite journal|author=Cleveland, W. S. & Devlin, S. J.|year=1988|title=Locally weighted regression: An approach to regression analysis by local fitting|journal=Journal of the American Statistical Association|volume=83|pages=596–610}}</ref> and cosine.
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| In the table below, '''1'''<sub>{…}</sub> is the [[indicator function]].
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| <!--
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| The gallery style has the problem that the maths is cut off
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| <gallery Caption="Commonly used Kernel Functions" style="background:white">
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| Image:Kernel_uniform.svg |Uniform <br/> <math>K(u) = \frac{1}{2}\ 1_{(|u|\leq1)}</math>
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| Image:Kernel_triangle.svg |Triangle <br/> <math>K(u) = (1-|u|)\ 1_{(|u|\leq1)}</math>
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| Image:Kernel_epanechnikov.svg |[[V. A. Epanechnikov|Epanechnikov]] <br/> <math>K(u) = \frac{3}{4}(1-u^2)\ 1_{(|u|\leq1)}</math>
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| Image:Kernel_quartic.svg |Quartic <br/> <math>K(u) = \frac{15}{16}(1-u^2)^2\ 1_{(|u|\leq1)}</math>
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| Image:Kernel_triweight.svg |Triweight <br/> <math>K(u) = \frac{35}{32}(1-u^2)^3\ 1_{(|u|\leq1)}</math>
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| Image:Kernel_exponential.svg |[[Normal_distribution|Gaussian]] <br/> <math>K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}</math>
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| Image:Kernel_cosine.svg |Cosine <br/> <math>K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right)1_{(|u|\leq1)}</math>
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| </gallery>
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| -->
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| {| class="wikitable" style="background-color:white;text-align:left"
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| !colspan=3| Kernel Functions, ''K''(''u'')
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| ! <math>\textstyle \int u^2K(u)du</math>
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| ! <math>\textstyle \int K^2(u)du</math>
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| |-
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| ! Uniform
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| | <math>K(u) = \frac12 \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel uniform.svg|150px]]
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| | <math>\frac13</math>
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| | <math>\frac12</math>
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| |-
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| ! Triangular
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| | <math>K(u) = (1-|u|) \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel triangle.svg|150px]]
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| | <math>\frac{1}{6}</math>
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| | <math>\frac{2}{3}</math>
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| |-
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| ! Epanechnikov
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| | <math>K(u) = \frac{3}{4}(1-u^2) \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel epanechnikov.svg|150px]]
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| | <math>\frac{1}{5}</math>
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| | <math>\frac{3}{5}</math>
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| |-
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| ! Quartic <br />(biweight)
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| | <math>K(u) = \frac{15}{16}(1-u^2)^2 \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel quartic.svg|150px]]
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| | <math>\frac{1}{7}</math>
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| | <math>\frac{5}{7}</math>
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| |-
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| ! Triweight
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| | <math>K(u) = \frac{35}{32}(1-u^2)^3 \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel triweight.svg|150px]]
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| | <math>\frac{1}{9}</math>
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| | <math>\frac{350}{429}</math>
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| |-
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| ! Tricube
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| | <math>K(u) = \frac{70}{81}(1- {\left| u \right|}^3)^3 \,\mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel tricube.svg|150px]]
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| | <math>\frac{35}{243}</math>
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| | <math>\frac{175}{247}</math>
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| |-
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| ! [[Normal distribution|Gaussian]]
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| | <math>K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}</math>
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| | [[File:Kernel exponential.svg|150px]]
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| | <math>1\,</math>
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| | <math>\frac{1}{2\sqrt\pi}</math>
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| |-
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| ! Cosine
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| | <math>K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right) \mathbf{1}_{\{|u|\leq1\}}</math>
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| | [[File:Kernel cosine.svg|150px]]
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| | <math>1-\frac{8}{\pi^2}</math>
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| | <math>\frac{\pi^2}{16}</math>
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| |-
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| ! [[Logistic distribution|Logistic]]
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| | <math>K(u) = \frac{1}{e^{u}+2+e^{-u}}</math>
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| | [[File:Kernel exponential.svg|150px]]
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| | <math>\frac{\pi^2}{3}</math>
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| | <math>\frac{1}{6}</math>
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| |}
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| ====All of the above Kernels in a Common Coordinate System====
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| [[File:Kernels.svg|590px|All of the above kernels in a common coordinate system]]
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| ==See also==
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| *[[Kernel density estimation]]
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| *[[Kernel smoother]]
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| *[[Stochastic kernel]]
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| *[[Density estimation]]
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| {{More footnotes|date=May 2012}}
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| ==References==
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| {{Reflist}}
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| *{{cite book
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| | last = Li
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| | first = Qi
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| | authorlink =
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| | coauthors = Racine, Jeffrey S.
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| | title = Nonparametric Econometrics: Theory and Practice
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| | publisher = Princeton University Press
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| | year = 2007
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| | location =
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| | pages =
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| | url =
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| | doi =
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| | id =
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| | isbn = 0-691-12161-3}}
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| *{{cite web
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| |last=Zucchini
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| |first=Walter
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| |title=APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation
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| |url=http://isc.temple.edu/economics/Econ616/Kernel/ast_part1.pdf|accessdate=28 March 2012}}
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| *{{cite journal|year=2002
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| |first1=D|last1=Comaniciu|first2= P|last2= Meer
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| |title=Mean shift: A robust approach toward feature space analysis
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| |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence| volume= 24| issue= 5|pages= 603–619
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| |id = {{citeseerx|10.1.1.76.8968}} |doi=10.1109/34.1000236}}
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| [[Category:Non-parametric statistics]]
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| [[Category:Time series analysis]]
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| [[Category:Point processes]]
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