This really clarified my confusions on the PCA part. Thanks a lot for the reply, I really appreciate it.
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. > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+u...@morphometrics.org. > > > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+u...@morphometrics.org <javascript:>. > -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. 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