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<title>Kernel (statistics)</title>

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Kernel (statistics)

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Kernel_(disambiguation)" class="mw-redirect mw-disambig" title="Kernel (disambiguation)">Kernel (disambiguation)</a>.</div>

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The term kernel is used in <a href="Statistics" title="Statistics">statistical analysis</a> to refer to a <a href="Window_function" title="Window function">window function</a>. The term "kernel" has several distinct meanings in different branches of statistics.

<div class="mw-heading mw-heading2"><h2 id="Bayesian_statistics">Bayesian statistics</h2></div>

In statistics, especially in <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, the kernel of a <a href="Probability_density_function" title="Probability density function">probability density function</a> (pdf) or <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> (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.<a href="#cite_note-1">[1]</a> Note that such factors may well be functions of the <a href="Parameter" title="Parameter">parameters</a> of the pdf or pmf. These factors form part of the <a href="Normalization_factor" class="mw-redirect" title="Normalization factor">normalization factor</a> of the <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, and are unnecessary in many situations. For example, in <a href="Pseudo-random_number_sampling" class="mw-redirect" title="Pseudo-random number sampling">pseudo-random number sampling</a>, most sampling algorithms ignore the normalization factor. In addition, in <a href="Bayesian_analysis" class="mw-redirect" title="Bayesian analysis">Bayesian analysis</a> of <a href="Conjugate_prior" title="Conjugate prior">conjugate prior</a> 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 <a href="Normal_distribution" title="Normal distribution">normal distribution</a>. Its <a href="Probability_density_function" title="Probability density function">probability density function</a> is

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<annotation encoding="application/x-tex">{\displaystyle p(x|\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}</annotation>

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</math><img src="./6b717991d23902abe1754bb3cfc0cd2e4b6b5ca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:28.49ex; height:7.176ex;" alt="{\displaystyle p(x|\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}" loading="lazy"></dd></dl>

and the associated kernel is

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<annotation encoding="application/x-tex">{\displaystyle p(x|\mu ,\sigma ^{2})\propto e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}</annotation>

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</math><img src="./3e7fcb89965d2ac9face5709e24c70c8a6ce5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.839ex; height:5.176ex;" alt="{\displaystyle p(x|\mu ,\sigma ^{2})\propto e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}" loading="lazy"></dd></dl>

Note that the factor in front of the exponential has been omitted, even though it contains the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>

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</math><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy"> , because it is not a function of the domain variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> .

<div class="mw-heading mw-heading2"><h2 id="Pattern_analysis">Pattern analysis</h2></div>

The kernel of a <a href="Reproducing_kernel_Hilbert_space" title="Reproducing kernel Hilbert space">reproducing kernel Hilbert space</a> is used in the suite of techniques known as <a href="Kernel_methods" class="mw-redirect" title="Kernel methods">kernel methods</a> to perform tasks such as <a href="Statistical_classification" title="Statistical classification">statistical classification</a>, <a href="Regression_analysis" title="Regression analysis">regression analysis</a>, and <a href="Cluster_analysis" title="Cluster analysis">cluster analysis</a> on data in an implicit space. This usage is particularly common in <a href="Machine_learning" title="Machine learning">machine learning</a>.

<div class="mw-heading mw-heading2"><h2 id="Nonparametric_statistics">Nonparametric statistics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Kernel_smoothing" class="mw-redirect" title="Kernel smoothing">Kernel smoothing</a></div>

In <a href="Nonparametric_statistics" title="Nonparametric statistics">nonparametric statistics</a>, a kernel is a weighting function used in <a href="Non-parametric" class="mw-redirect" title="Non-parametric">non-parametric</a> estimation techniques. Kernels are used in <a href="Kernel_density_estimation" title="Kernel density estimation">kernel density estimation</a> to estimate <a href="Random_variable" title="Random variable">random variables</a>' <a href="Density_function" class="mw-redirect" title="Density function">density functions</a>, or in <a href="Kernel_regression" title="Kernel regression">kernel regression</a> to estimate the <a href="Conditional_expectation" title="Conditional expectation">conditional expectation</a> of a random variable. Kernels are also used in <a href="Time_series" title="Time series">time series</a>, in the use of the <a href="Periodogram" title="Periodogram">periodogram</a> to estimate the <a href="Spectral_density" title="Spectral density">spectral density</a> where they are known as <a href="Window_functions" class="mw-redirect" title="Window functions">window functions</a>. An additional use is in the estimation of a time-varying intensity for a <a href="Point_process" title="Point process">point process</a> where window functions (kernels) are convolved with time-series data.

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

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Integral_kernel" class="mw-redirect" title="Integral kernel">Integral kernel</a></div>

A kernel is a <a href="Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> <a href="Real-valued_function" title="Real-valued function">real-valued</a> <a href="Integrable" class="mw-redirect" title="Integrable">integrable</a> function K. For most applications, it is desirable to define the function to satisfy two additional requirements:

<ul><li><a href="Normalizing_constant" title="Normalizing constant">Normalization</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }K(u)\,du=1\,;}</annotation>

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</math><img src="./58133348a2172dc741a7208339292866dac2d9a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.438ex; height:6.176ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }K(u)\,du=1\,;}" loading="lazy"></dd></dl>

<ul><li><a href="Even_and_odd_functions" title="Even and odd functions">Even-function</a> Symmetry:</li></ul>

<mtext> for all values of </mtext>

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<annotation encoding="application/x-tex">{\displaystyle K(-u)=K(u){\mbox{ for all values of }}u\,.}</annotation>

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</math><img src="./e6c8121d4bacfdd92a526adcbf6d123727656677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.978ex; height:2.843ex;" alt="{\displaystyle K(-u)=K(u){\mbox{ for all values of }}u\,.}" loading="lazy"></dd></dl>

The first requirement ensures that the method of kernel density estimation results in a <a href="Probability_density_function" title="Probability density function">probability density function</a>. 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) = λK(λu), where λ > 0. This can be used to select a scale that is appropriate for the data.

<div class="mw-heading mw-heading3"><h3 id="Kernel_functions_in_common_use">Kernel functions in common use</h3></div>

Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov,<a href="#cite_note-2">[2]</a> quartic (biweight), tricube,<a href="#cite_note-3">[3]</a> triweight, Gaussian, quadratic<a href="#cite_note-4">[4]</a> and cosine.

In the table below, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is given with a bounded <a href="Support_(mathematics)" title="Support (mathematics)">support</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(u)=0}">

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<annotation encoding="application/x-tex">{\displaystyle K(u)=0}</annotation>

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</math><img src="./35f02ccd32f987de18ce3dd5df2f3b57825b52b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.466ex; height:2.843ex;" alt="{\displaystyle K(u)=0}" loading="lazy"> for values of u lying outside the support.

<th colspan="3">Kernel Functions, K(u)

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \int u^{2}K(u)du}</annotation>

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</math><img src="./2ab683409a300e73999db2a8f5caa9a0f1110abc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.939ex; height:3.176ex;" alt="{\displaystyle \textstyle \int u^{2}K(u)du}" loading="lazy">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \int K(u)^{2}du}</annotation>

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</math><img src="./0ca32ef82d40e859081a0823d0b71b5fdc45d1d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.61ex; height:3.176ex;" alt="{\displaystyle \textstyle \int K(u)^{2}du}" loading="lazy">

</th>

<th>Efficiency<a href="#cite_note-5">[a]</a> relative to the Epanechnikov kernel

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<th>Uniform ("rectangular window")

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<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {1}{2}}}</annotation>

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</math><img src="./033151f8f275ba16a1c0eea7024ee0b7074635c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.302ex; height:5.176ex;" alt="{\displaystyle K(u)={\frac {1}{2}}}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

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</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

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<td>

"<a href="Boxcar_function" title="Boxcar function">Boxcar function</a>"

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}}</annotation>

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</math><img src="./7093420fb1b77a06432a4e0d9eba91705cef6d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}}" loading="lazy">

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

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</math><img src="./a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy">

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<td>92.9%

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<th>Triangular

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<annotation encoding="application/x-tex">{\displaystyle K(u)=(1-|u|)}</annotation>

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</math><img src="./c2261b6870aad5902b6a9a80b604ebf58215b019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.739ex; height:2.843ex;" alt="{\displaystyle K(u)=(1-|u|)}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

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</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}}</annotation>

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</math><img src="./ba1caf4c96d913f6aafa9da0634f069fa42e0290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}}" loading="lazy">

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}}</annotation>

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</math><img src="./19eee5d63f2cf9106dc531cdfdea8cfb8f34b2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}}" loading="lazy">

</td>

<td>98.6%

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<tr>

<th><a href="Epanechnikov_distribution" class="mw-redirect" title="Epanechnikov distribution">Epanechnikov</a>

(parabolic)

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<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {3}{4}}(1-u^{2})}</annotation>

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</math><img src="./491890437994a1dbbe96c452c91af92796de960e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.498ex; height:5.176ex;" alt="{\displaystyle K(u)={\frac {3}{4}}(1-u^{2})}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

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</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

<td>

</td>

<mfrac>

</mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}}</annotation>

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</math><img src="./1b977656fee2fb2e66210e174d78e2a484f89e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}}" loading="lazy">

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<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {3}{5}}}</annotation>

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</math><img src="./f254c32b4a96c635a14bd99af108d576de7091a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{5}}}" loading="lazy">

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<td>100%

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<th>Quartic (biweight)

</th>

<mfrac>

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<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {15}{16}}(1-u^{2})^{2}}</annotation>

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</math><img src="./b0c8b60cf84bc6bcdaa124b727e382b0716c033d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.715ex; height:5.176ex;" alt="{\displaystyle K(u)={\frac {15}{16}}(1-u^{2})^{2}}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

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</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

<td>

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{7}}}</annotation>

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</math><img src="./7488185acac9cec8180b1f6a1f3c13b72d59b395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.999ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{7}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {5}{7}}}</annotation>

</semantics>

</math><img src="./a2aa194f8d741a32708fa8caa8760ecbc7f0acb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:1.999ex; height:5.343ex;" alt="{\displaystyle {\frac {5}{7}}}" loading="lazy">

</td>

<td>99.4%

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<tr>

<th>Triweight

</th>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {35}{32}}(1-u^{2})^{3}}</annotation>

</semantics>

</math><img src="./3b809e8bf45d204e4f61753db083c56abed5bed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.715ex; height:5.176ex;" alt="{\displaystyle K(u)={\frac {35}{32}}(1-u^{2})^{3}}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

</semantics>

</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

<td>

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{9}}}</annotation>

</semantics>

</math><img src="./f5e41e551372333b5c0d7d525ecf3797d67f980a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{9}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {350}{429}}}</annotation>

</semantics>

</math><img src="./6036cedefc51911f26167d2b4ece82641461bc93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.323ex; height:5.343ex;" alt="{\displaystyle {\frac {350}{429}}}" loading="lazy">

</td>

<td>98.7%

</td></tr>

<tr>

<th>Tricube

</th>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {70}{81}}(1-{\left|u\right|}^{3})^{3}}</annotation>

</semantics>

</math><img src="./b4702b920836bbc3332b764aaebd6d069b47ed58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.008ex; height:5.176ex;" alt="{\displaystyle K(u)={\frac {70}{81}}(1-{\left|u\right|}^{3})^{3}}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

</semantics>

</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

<td>

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {35}{243}}}</annotation>

</semantics>

</math><img src="./7b739b0291d86ef50bd739b5e2f7f6c06478b9d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.323ex; height:5.343ex;" alt="{\displaystyle {\frac {35}{243}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {175}{247}}}</annotation>

</semantics>

</math><img src="./b063300097e11623be75f8d3e12ddfe556624546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.323ex; height:5.343ex;" alt="{\displaystyle {\frac {175}{247}}}" loading="lazy">

</td>

<td>99.8%

</td></tr>

<tr>

<th><a href="Normal_distribution" title="Normal distribution">Gaussian</a>

</th>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}}</annotation>

</semantics>

</math><img src="./2823201b13dc500e1a48d7907c3c64c2ad82395d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.439ex; height:6.176ex;" alt="{\displaystyle K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}}" loading="lazy">

</td>

<td>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation>

</semantics>

</math><img src="./bfd1e7984fe6e1b79a26404a8138a6c6ee41a476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}" loading="lazy">

</td>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2{\sqrt {\pi }}}}}</annotation>

</semantics>

</math><img src="./283e1b6bfa6c589d65fc584778ba60fa2b9d6d06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:5.266ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2{\sqrt {\pi }}}}}" loading="lazy">

</td>

<td>95.1%

</td></tr>

<tr>

<th>Cosine

</th>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)}</annotation>

</semantics>

</math><img src="./8580d3f2956fde0e2748538936a395a1af4fe417.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.243ex; height:4.843ex;" alt="{\displaystyle K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)}" loading="lazy">

Support: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|\leq 1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |u|\leq 1}</annotation>

</semantics>

</math><img src="./63aeef556c12f0f18861dbeee4d1342237989334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |u|\leq 1}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-{\frac {8}{\pi ^{2}}}}</annotation>

</semantics>

</math><img src="./f75d0f667134602a81d2bd72b4268b16ccdc5719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.227ex; height:5.509ex;" alt="{\displaystyle 1-{\frac {8}{\pi ^{2}}}}" loading="lazy">

</td>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{16}}}</annotation>

</semantics>

</math><img src="./aedd09d89d9445c2f2adf79371059f3f0987bda3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.225ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi ^{2}}{16}}}" loading="lazy">

</td>

<td>99.9%

</td></tr>

<tr>

<th><a href="Logistic_distribution" title="Logistic distribution">Logistic</a>

</th>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {1}{e^{u}+2+e^{-u}}}}</annotation>

</semantics>

</math><img src="./b444948aecc4b16f507248a337a7b0063015556c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.773ex; height:5.509ex;" alt="{\displaystyle K(u)={\frac {1}{e^{u}+2+e^{-u}}}}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{3}}}</annotation>

</semantics>

</math><img src="./9d3b7c9d6cc8a585ec603653327b50f53530e4b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.225ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi ^{2}}{3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}}</annotation>

</semantics>

</math><img src="./ba1caf4c96d913f6aafa9da0634f069fa42e0290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}}" loading="lazy">

</td>

<td>88.7%

</td></tr>

<tr>

<th><a href="Sigmoid_function" title="Sigmoid function">Sigmoid function</a>

</th>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {2}{\pi }}{\frac {1}{e^{u}+e^{-u}}}}</annotation>

</semantics>

</math><img src="./86964ddf33b30dc29d31efba233a3532f95d941f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.939ex; height:5.509ex;" alt="{\displaystyle K(u)={\frac {2}{\pi }}{\frac {1}{e^{u}+e^{-u}}}}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\pi ^{2}}{4}}}</annotation>

</semantics>

</math><img src="./d5b105b0cb1d043fcbe48795f98ce5e5165d01f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.225ex; height:5.676ex;" alt="{\displaystyle {\frac {\pi ^{2}}{4}}}" loading="lazy">

</td>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\pi ^{2}}}}</annotation>

</semantics>

</math><img src="./2bc571f715ad5f5300b5254c2a524e4eb9e6ccfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.225ex; height:5.509ex;" alt="{\displaystyle {\frac {2}{\pi ^{2}}}}" loading="lazy">

</td>

<td>84.3%

</td></tr>

<tr>

<th>Silverman kernel<a href="#cite_note-6">[5]</a>

</th>

<mfrac>

</mfrac>

</mrow>

<msup>

<mfrac>

<mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(u)={\frac {1}{2}}e^{-{\frac {|u|}{\sqrt {2}}}}\cdot \sin \left({\frac {|u|}{\sqrt {2}}}+{\frac {\pi }{4}}\right)}</annotation>

</semantics>

</math><img src="./be34c87bf29026d1add0165eb6f1fbc7121d5476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.41ex; height:7.009ex;" alt="{\displaystyle K(u)={\frac {1}{2}}e^{-{\frac {|u|}{\sqrt {2}}}}\cdot \sin \left({\frac {|u|}{\sqrt {2}}}+{\frac {\pi }{4}}\right)}" loading="lazy">

</td>

<td>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

</td>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {3{\sqrt {2}}}{16}}}</annotation>

</semantics>

</math><img src="./2872e024f50c97f7d0f9a26c3ad193a774724e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.097ex; height:5.843ex;" alt="{\displaystyle {\frac {3{\sqrt {2}}}{16}}}" loading="lazy">

</td>

<td>not applicable

</td></tr></tbody></table>

<ul><li><a href="Kernel_density_estimation" title="Kernel density estimation">Kernel density estimation</a></li>

<li><a href="Kernel_smoother" title="Kernel smoother">Kernel smoother</a></li>

<li><a href="Stochastic_kernel" class="mw-redirect" title="Stochastic kernel">Stochastic kernel</a></li>

<li><a href="Positive-definite_kernel" title="Positive-definite kernel">Positive-definite kernel</a></li>

<li><a href="Density_estimation" title="Density estimation">Density estimation</a></li>

<li><a href="Multivariate_kernel_density_estimation" title="Multivariate kernel density estimation">Multivariate kernel density estimation</a></li>

<li><a href="Kernel_method" title="Kernel method">Kernel method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Efficiency is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du}</annotation>

</semantics>

</math><img src="./8ea52ccbc81da0fcbc6afabc141b1d84cc4d6ded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.586ex; height:7.509ex;" alt="{\displaystyle {\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du}" loading="lazy">.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFSchuster1969" class="citation journal cs1">Schuster, Eugene (August 1969). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177697495">"Estimation of a probability density function and its derivatives"</a>. The Annals of Mathematical Statistics. 40 (4): 1187-1195. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177697495">10.1214/aoms/1177697495</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Named for <cite id="CITEREFEpanechnikov1969" class="citation journal cs1">Epanechnikov, V. A. (1969). "Non-Parametric Estimation of a Multivariate Probability Density". Theory Probab. Appl. 14 (1): 153–158. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1114019">10.1137/1114019</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFAltman,_N._S.1992" class="citation journal cs1"><a href="Naomi_Altman" title="Naomi Altman">Altman, N. S.</a> (1992). "An introduction to kernel and nearest neighbor nonparametric regression". The American Statistician. 46 (3): 175–185. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1992.10475879">10.1080/00031305.1992.10475879</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1813%2F31637">1813/31637</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFCleveland,_W._S.Devlin,_S._J.1988" class="citation journal cs1"><a href="William_S._Cleveland" title="William S. Cleveland">Cleveland, W. S.</a>; <a href="Susan_J._Devlin" title="Susan J. Devlin">Devlin, S. J.</a> (1988). "Locally weighted regression: An approach to regression analysis by local fitting". Journal of the American Statistical Association. 83 (403): 596–610. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1988.10478639">10.1080/01621459.1988.10478639</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFSilverman1986" class="citation book cs1">Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986desd.book.....S">1986desd.book.....S</a>.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFLiRacine,_Jeffrey_S.2007" class="citation book cs1">Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-12161-1</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFZucchini" class="citation web cs1">Zucchini, Walter. <a rel="nofollow" class="external text" href="http://staff.ustc.edu.cn/~zwp/teach/Math-Stat/kernel.pdf">"APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation"</a> (PDF). Retrieved 6 September 2018.</cite></li></ul>

<ul><li><cite id="CITEREFComaniciuMeer2002" class="citation journal cs1">Comaniciu, D; Meer, P (2002). "Mean shift: A robust approach toward feature space analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence. 24 (5): 603–619. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.8968">10.1.1.76.8968</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F34.1000236">10.1109/34.1000236</a>.</cite></li></ul></div><div><div class="zim-footer">

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