# Re: [MORPHMET] How to project shape difference onto different PC

```This really clarified my confusions on the PCA part. Thanks a lot for the
```
Sincerely.
Yinan

On Friday, May 18, 2018 at 4:10:21 PM UTC-4, f.james.rohlf wrote:
>
> Yes, the PC axes are “comparable”. I think the best way to think about
> what a PCA does is to interpret it as a projection of a multidimensional
> space down to a low dimensional space that captures as much of the overall
> variation as possible. The first axis is somewhat special because it
> represents the best 1-dimensional space. Past that one should think of 1
> and 2 giving the best 2-dimensional space and 1, 2, and 3 giving the best
> 3-dimensional space, etc. The axes themselves are not of a priori interest
> in an application – it is the space that is of interest. A consequence is
> that plots showing projections of points relative to PC1, PC2,etc. must be
> plotted to the same scale (i.e., consistent with the fact that the
> eigenvalues give the variances along each axis). If, as unfortunately often
> the case, the axes are plotted using different scales then the space has
> been distorted and is no longer the space that best accounts for the
> overall variation in the data. That also distorts the impressions one gets
> in looking at the plot as using different scales changes the relative
> distances between points.
>
>
>
> Within that reduced space one may find that particular axes can seem to be
> interpretable but one should really look at the space and decide which
> directions within the space are most interesting based on the patterns of
> the data. That is, the data need to suggest interesting direction unless
> one has some a priori groups one wishes to compares. Often the first PC is
> of special interest but that is often due to allometry and the relatively
> large impact of size variation. That is, by now, a rather boring result!
> The individual PC axes are defined based on convenient mathematical
> properties – not based on any biological models so each one should not be
> considered separately as things of special interest.
>
>
>
> The above also means that one need not just visualize variation along each
> axis separately. One can, as in tpsRelw software, visualize any specified
> point within the PC space or in any direction of interest within the PC
> space.
>
>
>
> _ _ _ _ _ _ _ _ _
>
> F. James Rohlf, Distinguished Prof. Emeritus
>
> [image: univautosig]
>
> Depts. of Anthropology and of Ecology & Evolution
>
>
>
>
>
> *From:* Yinan Hu <yinan...@gmail.com <javascript:>>
> *Sent:* Friday, May 18, 2018 2:19 PM
> *To:* MORPHMET <morp...@morphometrics.org <javascript:>>
> *Subject:* Re: [MORPHMET] How to project shape difference onto different
> PC
>
>
>
> Dear James,
>
>
>
> Thanks for the reply. Yes I have completed a PCA on a GM dataset with 11
> landmarks, and you got it exactly right that I'm trying to decompose shape
> differences onto individual PCs.
>
>
>
> The reason I was hesitating to do the vector projection is that I'm not
> sure if PC scores on different PCs are directly comparable to each other.
> For simplicity, let's say I'm only considering PC1 and PC2, which explains
> 80% of shape variation in total (60% + 20%). Group A has a mean PC1 score
> of 0.5, and PC2 score of 0.1; where as Group B has a mean PC1 score of 0.4
> and PC2 score of 0.3.  Then I'm looking at a 0.1 difference along PC1 and a
> 0.2 difference along PC2 between these two groups.
>
>
>
> Would this mean they differ twice as much along PC2 than PC1, such that in
> the 80% of shape variation explained by these two PCs, 1/3 is along PC1 and
> 2/3 is along PC2?
>
>
>
> But considering that PC1 explains three times more variation than PC2 (60%
> vs 20%), would this mean I should weigh the PC score difference (distance
> along each PC)? i.e. although the absolute difference in mean PC1 score is
> 0.1, it should be weighed three times more than the difference along PC2 so
> in the 80% of shape variation explained by these two PCs, 3/5 is along PC1
> and 2/5 is along PC2?
>
>
>
>
>
> On the other hand, I agree visualizing the shape difference along each PC
> can be helpful, and I'm pretty sure the plotRefToTarget function from the R
> package geomorph can achieve this.
>
>
>
> Thanks again.
>
> Best,
>
>
>
> Yinan
>
>
>
>
> On Friday, May 18, 2018 at 12:53:32 PM UTC-4, K. James Soda wrote:
>
> Dear Dr. Hu,
>
> Let me begin by restating how I understand the question: You have
> completed a PCA on a morphological data set in which there are two subsets
> of interest. Now you would like to decompose the difference between the two
> subsets into differences along individual PCs. Here is my two cents on the
> issue:
>
> I would say that the literal solution to this problem would probably be
> something along the lines of what you proposed. For simplicity, say that
> you summarized each subset using its mean position in the PC space. This
> would be expressed as a vector where each element is a position along a
> single PC. The difference between these two vectors would then be a
> decomposition of how far you would need to move along each PC axis to move
> from one mean to the other. You could then standardize the elements so that
> their absolute values sum to one. This would be an expression of what
> percentage of the distance is along each PC.
>
> What I perceive as the subtext of your question, though, is whether this
> sort of decomposition has a reasonable interpretation, and the answer to
> this question is somewhat trickier. Assuming this is a GM data set, the
> more relevant point might be how you convert the difference into
> visualizations. A nice feature of GM data is that each PC will correspond
> to a "type" of deformation. This feature can be used to decompose the
> difference between two shapes in a shape-PC space as well. For example,
> imagine you moved from one mean shape in the PC space to the other by only
> moving parallel to PC axes. If you are interested in two PCs, this could be
> accomplished in two ways. You could then visualize the shape at the points
> where you make a turn; that is, you would visualize how mean shape 1 would
> need to be deformed to have the same PC1 or PC2 score as mean shape 2 if
> all other PCs were held constant. The degree of deformation would then
> provide a qualitative measure of how radical each PC's contribution is to
> the shape difference. Of course, this is not a quantitative measure, as you
> requested, but I would argue it is a more helpful assessment b/c it
> directly corresponds to observable phenomena. How helpful, though, will
> depend on your research question.
>
> Hope something in there helps a little,
>
> James
>
>
>
> On Thu, May 17, 2018 at 10:15 AM, Yinan Hu <yinan...@gmail.com> wrote:
>
> Dear colleagues,
>
>
>
> I'm trying to figure out how to break down shape differences onto
> individual PC axes.  I have a morphospace where PC1 explains 60% of shape
> variation and PC2 explains 20% of variation. Two subsets of samples of
> particular interest do not differ much along PC1, but differs significantly
> along PC2. How should I project the shape difference between these subsets
> onto seperate PC axes, such that I can quantitatively show X% of shape
> difference between them are along PC1 and Y% is along PC2?
>
>
>
> A simple vector projection (i.e. using the mean difference of PC1 score
> and PC2 score) doesn't feel right to me as I don't think PC scores are
> directly comparable between different PCs. Or am I wrong?
>
> Any suggestions would be greatly appreciated.
>
>
>
> Many thanks for your time.
>
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