Multivariate kernel density estimation: Difference between revisions

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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=978-0-412-24620-3 | pages=[https://archive.org/details/densityestimatio00silv_0/page/7 7–11] | url-access=registration | url=https://archive.org/details/densityestimatio00silv_0/page/7 }}</ref>

[[File:Synthetic data 2D histograms.png|thumb|center|500px|alt=Left. Histogram with anchor point at (−1.5,&nbsp;-1.5). Right. Histogram with anchor point at (−1.625,&nbsp;−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,&nbsp;-1.5). Right. Histogram with anchor point at (−1.625,&nbsp;−1.625). Both histograms have a bin widthbindwidth of 0.5, so differences in appearances of the two histograms are due to the placement of the anchor point.]]

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.