February 2, 2007
This is a response to the query from Dr. (or Ms.) Cramon-Taubadel
earlier today regarding the combination of Procrustes shape coordinates
with Fourier coefficients in multivariate morphometrics
computations. I assume these are elliptic Fourier
coefficients, the Rohlf-Ferson version; if not, the comments
below have even greater force, as the interpretation in terms
of sums of squares becomes even more complicated. (For instance,
the wrong versions of Fourier analysis drop phase information, without
any good reason for doing so. I hope that's not what you've been doing.)
In my opinion what you are proposing is not a good idea. For one
thing,
the Procrustes registration used for landmarks is not the correct
registration to use if you're producing elliptic Fourier coefficients.
For another, the sum-of-squares that the Procrustes shape coordinates
are decomposing is only distantly related to the sum-of-squares that
the Fourier coefficients are decomposing. There is probably no
straightforward principal coordinates analysis of the combination
of the two analyses, whereas each one sustains its own PCOORD
separately. (But for that purpose you'd have to re-register the
elliptic Fourier coefficients using the appropriate rules for those.)
You didn't say if the curves go through the landmarks.
If they do, the similarity of your two ordinations is probably just
an artifact of the sums of squares that overlap between the two
techniques. In effect, the Procrustes analysis uses all the
Cartesian dimensions, with one weighting scheme, and the Fourier
analysis uses only some of the dimensions, with another weighting
scheme. If the schemes are close enough together, the ordinations
should be, too, but still the combination makes no algebraic sense at all.
The purpose you're seeking can probably be achieved just as
well by using landmarks along with semilandmarks (points representing the
curves in-between the landmarks) along the lines demonstrated by
my Vienna group in recent years (see papers by Gunz and by
Mitteroecker, including various chapters of the book edited by
Slice and published in 2004). The semilandmark methods seem suited
for describing either large-scale or small-scale changes in form;
the Fourier methods work only at large scales (in the applications
known to me), where (once again) the information is redundant with
what you get from the landmarks by themselves -- truly local
features are blurred by the Fourier methods, but are preserved
by the semilandmark methods to a great extent.
I hope these comments are of some help.
Fred Bookstein, [EMAIL PROTECTED]
On Fri, Feb 02, 2007 at 11:50:38AM -0500, morphmet wrote:
> Dear All
>
> I have a question regarding combining two types of morphological
> variables into the same analysis. I have a 3D landmark configuration
> and associated 3D outlines, which are registered in a common
> reference frame (via GPA). I have generated Procrustes variables from
> the landmark configuration and Fourier coefficients from the outline
> data.
>
> A) Does anyone know of a published example where procrustes variables
> have been analysed WITH Fourier coefficients in the same analysis
> (e.g. Principal Components Analysis) and B) If not, can anyone think
> of any reason why these two types of shape variables cannot be
> analysed simultaneously? I ran a dummy test using both variables at
> the same time and found that similar patterns between groups were
> produced with both variable-types, although cross-validated
> classification of groups (using Discriminant Function Analysis) was
> best when both Procrustes and Fourier variables were used together.
>
> Any further comments or suggestions would be greatly appreciated!
>
> Many thanks in advance Noreen [EMAIL PROTECTED]
>
> -- Noreen von Cramon-Taubadel Department of Biological Anthropology
> University of Cambridge The Henry Wellcome building, Fitzwilliam
> street Cambridge CB2 1QH Phone: +44 1223 764719 [EMAIL PROTECTED]
>
>
>
> -- Replies will be sent to the list. For more information visit
> http://www.morphometrics.org
>
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