-------- Original Message --------
Subject: RE: Simulating naturally occurring human craniofacial variation
Date: Sat, 5 Feb 2011 13:58:00 -0500
From: Hans Wellens <[email protected]>
To: <[email protected]>
Major thanks Dr. Polly, Brendan McCane, Hester Breman and Professor
Rohlf for their most kind and utterly helpful responses.
The idea is indeed to produce a random sample of specimens around the
population`s consensus configuration (mean) while using the same
population`s covariance structure.
After carefully reading your suggestions, I integrated them into the
following scenario:
1. Perform PCA by decomposing the population`s variance-covariance
matrix, calculated after expressing the original dataset as x1, y1, x2,
y2, x3, y3, ... xn, yn (2-dimensional data). Supposing for example a 14
x 2 configuration of landmarks per specimen (superimposed previously
using generalized partial Procrustes superimposition), this would render
a 28 x 28 variance-covariance matrix. The SVD of the variance-covariance
matrix P=t(U)WV yields the necessary eigenvectors (28 in total in left
vector U of the SVD result) and eigenvalues (middle, diagonal matrix W
in SVD result). Four eigenvectors would hold no useful information due
to translation, scaling, and rotation. The final, rotated data is
obtained by multiplying U with the transpose of the data matrix.
2. Construct a simulation data matrix, with mean 0 and identity
covariance matrix. This is performed by generating normally distributed
random noise around zero for each variable x1,y1,x2,y2 separately,
generating a x times 28 matrix (with x representing the number of
virtual observations). The range for the randomly generated numbers
would be the +/- three times the square root of the eigenvalue of each
subsequent eigenvector.
3. Premultiply the generated virtual data matrix with the
(complete)vector of eigenvectors, and add the consensus configuration
vector to the result (modified "get the old data back" procedure).
Done!
Would you agree with this interpretation of your suggestions?
Again, I`d like to thank you for the much appreciated advice!
With kind regards,
Hans Wellens
Hans Wellens, DDS
Orthodontist
Groene-Poortdreef 16
8200 Sint-Michiels
Tel: 050/39.68.36
[email protected]
