# Talk:Multivariate kernel density estimation

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Field: Probability and statistics
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## Images

The article is looking pretty good now. I have a number of concerns about the images.

1. In the images in the Motivation section, some of the points are very close together, such that they could be mistaken for a single point. This makes it harder to see how the points are converted into the histogram.
2. I'm not sure about some of the resulting densities shown. Are they actually population densities, rather than density estimates?

Yaris678 (talk) 17:59, 18 September 2010 (UTC)

1. Smaller dot size used and very close points separated.
2. All data sets are random samples rather than populations so they necessarily result in density estimates rather than population densities.

Drleft (talk) 10:30, 20 September 2010 (UTC)

1. That is much better.
2. I can see how the density estimate has changed in response to the moved point, so I can now see that it is an estimate.
Good work! Yaris678 (talk) 09:18, 21 September 2010 (UTC)

I like the Motivation section's intent to give an intuitive idea of kernel density estimation. Some comments:

1. Caption on first figure: "Left. Histogram with anchor point at (−1.5, 1.5)." Should that be (-1.5, -1.5)? (ie: second number negative?)

2. Description of second figure: "dashed grey lines" -- there are no dashed grey lines, just ovals with solid grey lines.

3. In the second figure, whose purpose it is to introduce kernels, I think it's a red herring to start with a kernel that has a diagonal major axis. Later the article discusses this as a special case having merit in some situations, but I suggest it's not the mainstream case that beginners would encounter first. Also, the "contour lines" on the kernels are somewhat oddly spaced (not evocative of a normal curve, unlike the subsequent figure showing three different normal kernels).

4. The Motivation section elides two separate issues: (a) comparison of discrete binning versus a more continuous measure of density. (b) visualization of an actual set of data, versus an estimate about a larger population from which a set of data was drawn. ("underlying density function" etc.)

There's no reason that a kernel density figure can't be just about a set of data (as it would be if it described an entire population), and not an estimate.

• Fixed 1. and 2. As for 3., part of the aim of this wikipage is to frame the kernel estimation problem in a more general context as befitting an encylopedia entry, so the most general kernels with the diagonal orientation are presented first. Also, the contours of the individual kernels are the 25, 50, 75% contours which correspond to those presented on the right for the density estimate. As for the sample vs population question in 4., the theory for kernel density estimation presented here is for data samples drawn from an infinite population. So in terms of mathematical correctness, it's better to state all results in terms of sample estimates until it has been verified that it is sensible for finite populations too. Drleft (talk) 15:31, 13 November 2010 (UTC)

Finally, in the section on Density estimatin in Matlab: "the Matlab routine for 2-dimensional data". With all respect to the author of that routine, it is one of a few contributed and available routines that address kde (Ie: it's "a" routine rather than "the" routine). Also, it takes a somewhat novel approach that's not exactly in line with this article: it proceeds by smoothing a binned histogram rather than using kernels, so far as I can see. (I haven't got to the bottom of whether that amounts to the same thing or not!) Gwideman (talk) 11:03, 10 November 2010 (UTC)

• I'm not the author of these Matlab code nor did I add the relevant sections to the wikipage. They were already deleted from the univariate kernel density estimation page since it concerns only bivariate estimation, and they have reappeared here. I haven't read this article thoroughly either, but my feeling is that the smoothing histograms would be asymptotically equivalent to standard kernel estimators. Drleft (talk) 15:31, 13 November 2010 (UTC)

## Silverman's rule of thumb - reference

The articles gives Silverman's rule of thumb, but no reference for it. Is it the same reference as Silverman cited earlier? That seems to be a highly regarded work on kernel density estimation, judging from Google scholar. But I couldn't find it in there. --91.52.32.135 (talk) 10:25, 28 October 2012 (UTC)