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== Motivation ==
We take a illustrative synthetic [[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 it is the reverse is the case for the right-hand histogram, confirming that histograms are highly sensitive the placement of the anchor point.<ref>{{cite
[[Image:Synthetic data 2D histograms.png|center|600px]]
One possible solution to this anchor point placement problem to remove the histogram binning grid completely. In the left figure below, a kernel (represented by the dashed 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 coloured contours correspond to the smallest regions which contains that corresponding probability mass: red = 25%, orange + red = 50%, yellow + orange + red = 75%, thus indicating that a single central region contains the highest density.
[[Image:Synthetic data 2D KDE.png|center|600px]]
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The choice of the kernel function <em>K</em> is not crucial to the accuracy of kernel density estimators, so we use the standard [[multivariate normal distribution|multivariate normal]] or Gaussian density function as our kernel <em>K</em> throughout: <math>K (\bold{x}) = (2\pi)^{-d/2} \exp(-\tfrac{1}{2} \, \bold{x}^T \bold{x})</math>. Whereas the choice of the bandwidth matrix <strong>H</strong> is the single most important factor affecting its accuracy since it controls the amount of 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 | date=1995 | isbn = 0412552701}}</ref>(pp. 36-39).
== Optimal bandwidth matrix selection ==
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== Computer implementation==
The [http://cran.r-project.org/web/packages/ks/index.html ks package]<ref>{{cite journal | author1=Duong, T. | title=ks: Kernel density estimation and kernel discriminant analysis in R | journal=Journal of Statistical Software | year=2007 | volume=21(7) | url=http://www.jstatsoft.org/v21/i07}}</ref> in [[R programming language|R]] implements the plug-in and smoothed cross validation selectors (amongst others). This dataset (included in the base distribution of R) contains
272 records with two measurements each: the duration time of an eruprion (minutes) and the
waiting time until the next eruption (minutes) of the [[Old Faithful Geyser]] in Yellowstone National Park, USA
The code fragment computes the kernel density estimate with the plug-in bandwidth matrix <math>\hat{\bold{H}}_\operatorname{PI} = \begin{bmatrix}0.052 & 0.510 \\ 0.510 & 8.882\end{bmatrix}.</math>
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