[[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 generalizationgeneralisation 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="simonoff1996">{{Cite book| author=Simonoff, J.S. | title=Smoothing Methods in Statistics | publisher=Springer | year=1996 | isbn=978-0-387-94716-7}}</ref>
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">{{Cite book| author1=Wand, M.P | author2=Jones, M.C. | title=Kernel Smoothing | publisher=Chapman & Hall/CRC | ___location=London | year=1995 | isbn = 978-0-412-55270-0}}</ref>{{rp|36–39}} That the bandwidth matrix also induces an orientation is a basic difference between multivariate kernel density estimation from its univariate analogue since orientation is not defined for 1D kernels. This leads to the choice of the parameterizationparametrisation of this bandwidth matrix. The three main parameterizationparametrisation classes (in increasing order of complexity) are ''S'', the class of positive scalars times the identity matrix; ''D'', diagonal matrices with positive entries on the main diagonal; and ''F'', symmetric positive definite matrices. The ''S'' class kernels have the same amount of smoothing applied in all coordinate directions, ''D'' kernels allow different amounts of smoothing in each of the coordinates, and ''F'' kernels allow arbitrary amounts and orientation of the smoothing. Historically ''S'' and ''D'' kernels are the most widespread due to computational reasons, but research indicates that important gains in accuracy can be obtained using the more general ''F'' class kernels.<ref>{{cite journal | author1=Wand, M.P. | author2=Jones, M.C. | title=Comparison of smoothing parameterizations in bivariate kernel density estimation | journal=Journal of the American Statistical Association | year=1993 | volume=88 | issue=422 | pages=520–528 | doi=10.1080/01621459.1993.10476303 | jstor=2290332}}</ref><ref name="DH2003">{{Cite journal| doi=10.1080/10485250306039 | author1=Duong, T. | author2=Hazelton, M.L. | title=Plug-in bandwidth matrices for bivariate kernel density estimation | journal=Journal of Nonparametric Statistics | year=2003 | volume=15 | pages=17–30}}</ref>
[[File:Kernel parameterizationparametrisation class.png|thumb|center|500px|alt=Comparison of the three main bandwidth matrix parameterizationparametrisation 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 parameterizationparametrisation 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.]]
