Calculating Proportional Overlap of Convex Hulls in PCA Plot

[csho...@gmail.com]

Dear geomorph group,

I'm currently working on a project looking at species delimitation among morphologically similar pairs of endangered land snail species, utilizing historical museum collections to identify potential shifts in morphology or loss of morphological diversity that explain current trends and apparent convergence of morphology.

I have PCA plots for both contemporary populations and historical populations, with convex hulls on them to show overlap and morphological limits for the species. I'm trying to work out basically what the morphospace extent of each species (as the total area of the convex hull for each species), the overlap in convex hulls between the pairs of species, and the proportion of that overlap in relation to the entire species morphospace, with values for both historical and contemporary populations.

I've been looking through the vignette and have not been able to find anything, but does anyone know of a way to be able to calculate these values within the package? I've done morphological disparity work to look at diversity, alongside discriminant analysis and statistical delimitation, but this would be a really cool way to visually explain and explore what is happening.

Thanks in advance for your help!

Chris 

[Mike Collyer]

Chris,

You might want to check out the package, geometry, It has a vignette for the convhulln function, which can calculate hull areas/volumes.

In terms of overlap, although maybe a little inelegant, one can calculate the area of two hulls (in two dimensions), plus one hull that combines the points of the two hulls.  If there is little or no overlap, the one hull will have a much larger area than either of the two hulls, individually.  If there is considerable overlap, the area of the one hull will resemble the areas of the two hulls.  A good way to evaluate this is compare AC, the area of the third hull comprising points of both groups to AA + AB, the sum of the areas of the hulls of individual groups; i.e., AC / (AA +AB).  This value will start to exceed 1 if the hulls do not overlap, or do not overlap much, but will be considerably less than 1 if they overlap substantially.

I have to also warn against conclusions made in two-dimension projections.  Imagine two discoidal clouds of points with no overlap at all in a third PC dimension, viewed from a two-dimensional projection of PC1 and PC2.  One might conclude that the groups share the same space, when they don’t.  Luckily, convhulln can handle volumes of hulls in n dimensions.

Hope that helps,

Mike

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[Chris Hobbs]

Hi Mike,

Thank you so much for the advice, I'll give it a try!

Best wishes,

Chris

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