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{{Short description|Concept in statistics mathematics}}
[[Kernel density estimation]] is a [[nonparametric]] technique for [[density estimation]] i.e., estimation of [[probability density function]]s, which is one of the fundamental questions in [[statistics]]. It can be viewed as a generalisation of [[histogram]] density estimation with improved statistical properties. Apart from histograms, other types of density estimators include [[parametric statistics|parametric]], [[spline interpolation|spline]], [[wavelet]] and [[Fourier series]]. Kernel density estimators were first introduced in the scientific literature for [[univariate]] data in the 1950s and 1960s<ref>{{Cite journal| doi=10.1214/aoms/1177728190 | last=Rosenblatt | first=M.| title=Remarks on some nonparametric estimates of a density function | journal=Annals of Mathematical Statistics | year=1956 | volume=27 | issue=3 | pages=832–837| doi-access=free }}</ref><ref>{{Cite journal| doi=10.1214/aoms/1177704472| last=Parzen | first=E.| title=On estimation of a probability density function and mode | journal=Annals of Mathematical Statistics| year=1962 | volume=33 | issue=3 | pages=1065–1076| doi-access=free }}</ref> and subsequently have been widely adopted. It was soon recognised that analogous estimators for multivariate data would be an important addition to [[multivariate statistics]]. Based on research carried out in the 1990s and 2000s, '''multivariate kernel density estimation''' has reached a level of maturity comparable to its univariate counterparts.<ref name="WJ1995">{{Cite book| author1=Wand, M.P | author2=Jones, M.C. | title=Kernel Smoothing | publisher=Chapman & Hall/CRC | ___location=London | year=1995 | isbn = 9780412552700}}</ref><ref name="simonoff1996">{{Cite book| author=Simonoff, J.S. | title=Smoothing Methods in Statistics | publisher=Springer | year=1996 | isbn=
==Motivation==
We take an illustrative [[Synthetic data|synthetic]] [[bivariate data|bivariate]] data set of 50 points to illustrate the construction of histograms. This requires the choice of an anchor point (the lower left corner of the histogram grid). For the histogram on the left, we choose (−1.5, −1.5): for the one on the right, we shift the anchor point by 0.125 in both directions to (−1.625, −1.625). Both histograms have a binwidth of 0.5, so any differences are due to the change in the anchor point only. The colour-coding indicates the number of data points which fall into a bin: 0=white, 1=pale yellow, 2=bright yellow, 3=orange, 4=red. The left histogram appears to indicate that the upper half has a higher density than the lower half, whereas the reverse is the case for the right-hand histogram, confirming that histograms are highly sensitive to the placement of the anchor point.<ref>{{Cite book | author=Silverman, B.W. | title=Density Estimation for Statistics and Data Analysis | publisher=Chapman & Hall/CRC | year=1986 | isbn=
[[File:Synthetic data 2D histograms.png|thumb|center|500px|alt=Left. Histogram with anchor point at (−1.5, -1.5). Right. Histogram with anchor point at (−1.625, −1.625). Both histograms have a bin width of 0.5, so differences in appearances of the two histograms are due to the placement of the anchor point.|Comparison of 2D histograms. Left. Histogram with anchor point at (−1.5, -1.5). Right. Histogram with anchor point at (−1.625, −1.625). Both histograms have a
One possible solution to this anchor point placement problem is to remove the histogram binning grid completely. In the left figure below, a kernel (represented by the grey lines) is centred at each of the 50 data points above. The result of summing these kernels is given on the right figure, which is a kernel density estimate. The most striking difference between kernel density estimates and histograms is that the former are easier to interpret since they do not contain artifices induced by a binning grid.
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* <math>K_\mathbf{H}(\mathbf{x})=|\mathbf{H}|^{-1/2}K(\mathbf{H}^{-1/2}\mathbf{x} )</math>.
The choice of the kernel function ''K'' is not crucial to the accuracy of kernel density estimators, so we use the standard [[multivariate normal distribution|multivariate normal]] kernel throughout: <math display="inline">K_\mathbf{H}(\mathbf{x})={(2 \pi)^{-d/2}} \mathbf{|H|}^{-1/2} e^{ -\frac{1}{2}\mathbf{x^T}\mathbf{H^{-1}}\mathbf{x} }</math>, where H plays the role of the [[covariance matrix]]. On the other hand, the choice of the bandwidth matrix <strong>H</strong> is the single most important factor affecting its accuracy since it controls the amount and orientation of smoothing induced.<ref name="WJ1995"
[[File:Kernel parametrisation class.png|thumb|center|500px|alt=Comparison of the three main bandwidth matrix parametrisation classes. Left. S positive scalar times the identity matrix. Centre. D diagonal matrix with positive entries on the main diagonal. Right. F symmetric positive definite matrix.|Comparison of the three main bandwidth matrix parametrisation classes. Left. ''S'' positive scalar times the identity matrix. Centre. ''D'' diagonal matrix with positive entries on the main diagonal. Right. ''F'' symmetric positive definite matrix.]]
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H <- Hpi(x=faithful)
fhat <- kde(x=faithful, H=H)
plot(fhat, display="filled.
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where, ''N'' is the number of data points, ''d'' is the number of dimensions (variables), and <math>I_{\vec{A}}(\vec{t})</math> is a filter that is equal to 1 for 'accepted frequencies' and 0 otherwise. There are various ways to define this filter function, and a simple one that works for univariate or multivariate samples is called the 'lowest contiguous hypervolume filter'; <math>I_{\vec{A}}(\vec{t})</math> is chosen such that the only accepted frequencies are a contiguous subset of frequencies surrounding the origin for which <math>|\hat{\varphi}(\vec{t})|^2 \ge 4(N-1)N^{-2}</math> (see <ref name=":22"/> for a discussion of this and other filter functions).
Note that direct calculation of the ''empirical characteristic function'' (ECF) is slow, since it essentially involves a direct Fourier transform of the data samples. However, it has been found that the ECF can be approximated accurately using a [[Non-uniform discrete Fourier transform|non-uniform fast Fourier transform]] (nuFFT) method,<ref name=":1" /><ref name=":22"/> which increases the calculation speed by several orders of magnitude (depending on the dimensionality of the problem). The combination of this objective KDE method and the nuFFT-based ECF approximation has been referred to as ''[https://
[[File:FastKDE_example.jpg|alt=A demonstration of fastKDE relative to a sample PDF. (a) True PDF, (b) a good representation with fastKDE, and (c) a slightly blurry representation.|none|thumb|664x664px|A non-trivial mixture of normal distributions: (a) the underlying PDF, (b) a fastKDE estimate on 1,000,000 samples, and (c) a fastKDE estimate on 10,000 samples.]]
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==External links==
* [http://www.mvstat.net/
* [http://www.mathworks.com/matlabcentral/fileexchange/17204-kernel-density-estimation kde2d.m] A [[Matlab]] function for bivariate kernel density estimation.
* [http://libagf.sf.net libagf] A [[C++]] library for multivariate, [[variable bandwidth kernel density estimation]].
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