Multivariate kernel density estimation: Difference between revisions

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[[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 | pages=832–837}}</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 | pages=1065–1076}}</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=0-387-94716-7}}</ref>

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==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,&nbsp;−1.5): for the one on the right, we shift the anchor point by 0.125 in both directions to (−1.625,&nbsp;−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=0-412-24620-1 | pages=7–11}}</ref>