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

[Yinan Hu]

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 > 
> *Sent:* Friday, May 18, 2018 2:19 PM
> *To:* MORPHMET <morp...@morphometrics.org >
> *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 

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

[F. James Rohlf]

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



Depts. of Anthropology and of Ecology & Evolution

 

 

From: Yinan Hu <yinanhu...@gmail.com> 
Sent: Friday, May 18, 2018 2:19 PM
To: MORPHMET <morphmet@morphometrics.org>
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 

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

[Yinan Hu]

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  > 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 

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

[K. James Soda]

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  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+unsubscr...@morphometrics.org.
>

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[MORPHMET] How to project shape difference onto different PC

[Yinan Hu]

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+unsubscr...@morphometrics.org.