The two negative eigenvalues indicate that the genetic distances are not Euclidean, that is, no configuration of points can reproduce D exactly. Fortunately, the negative eigenvalues are small relative to the largest positive ones, and the reduction to the first two columns of Y should be fairly accurate.<\/p><\/blockquote>\n
This isn’t the case for my dataset. The first (i.e. largest-magnitude) 16 eigenvalues are all of the same order of magnitude (between 0.6 and 0.1), and the 40 largest-magnitude negative eigenvalues are just 1 order of magnitude less than those largest positive eigenvalues. This means that there’s no euclidean representation of the relative positions of the genera in morphospace. This might not come as a great surprise, since the matrix is still quite sparse and very high-dimensional. I’m not trying to represent geographic locations of points on a globe on a map\u2014the sort of exercise in the Matlab example\u2014but points in a much, much higher-dimensional space…<\/p>\n
Here’s a plot of the eigenvalues:<\/p>\n
This is actually not too bad. The first two eigenvalues don’t necessarily explain a huge hog of the total <\/em>variance, but they’re definitely substantially bigger than any of the others, so that’s good. It’s at least clear that the third doesn’t explain much more than the fourth or fifth, so it doesn’t necessarily make sense to start going down the list and including more dimensions. That’s a good thing, I think.<\/p>\n So what does the plot for the morphospace look like for this new culled dataset?<\/p>\n Interestingly, the groups seem to be better separated now\u2014less overlap. Is the same true if we look at the subgroups, which were completely overlapped when using the full set of characters and taxa? Well, there’s still a lot of overlap, but at least there are some areas that seem to be differentiated.<\/p>\n![\"\"](\"http:\/\/blogs.law.harvard.edu\/kotrc\/files\/2011\/09\/EigenvaluesCulled50.png\")
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