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Re: your response to 2A=A

Hi, Tom and other friends,

Thank you for your response. It was encouraging. After I read
your msg carefully, I decided to go little bit further: IF WE
ACCEPT THE INDEPENDENCE OF SPECIES IN VEGETATION, THEN THE PCA
MAY NOT BE CONSIDERED A WRIGHT TOOL FOR VEGETATION SCIENCE.

I posted the following discussion for your comments. It was
developed for a proposal that I submitted to Chicago University.

Thank you very much for your attention and involvement.

Jay Bai

P.S. There were "typoes" in previous mails. The species number
should be represented by m, instead of n, so the n-space and n-
vector should be m-space and m-vector, to seperate from the
sample number, n.
====================

"Species space represents the abundance values of the original
species-by-samples data matrix fully (as does samples space). The
only problem is that species space is of many dimensions,
typically dozens to handreds of dimensions, and hence impossible
to visualize or inspect. " (Gauch, 1982, pp.166). Also, it was
mentioned in the same book that one of the solution is "to
project the original high-dimensional species space onto a space
of fewer dimensions, say, two or three dimensions, so that the
distribution of sample points can be inspected..." However, "When
this is done, structure along the chosen two or three dimensions
will be visible and structure in other dimensions will be
lost...." The presumption of people can do this (PCA) is that
"..., if the structure of sample points is concentrated within
two or three dimensions (because other dimensions are correlated
with these),..." However, this presumption met a challange from
the fact that people can grow any plant without other species, as
long as supply the correct conditions. 

To avoid loosing information, MDSM project the data from the m-
space onto unit m-hypersphere. It uses m-vectors to conduct the
synthesis and analysis and to express the results: centroid
vector to represent the n points, cosine vector to express the
composition, and tangent vector to project the dynamics. As m-
vector is a 1*m matrix, it can be converted to m values in one
dimension, hence the data synthesis and analysis can be done.

Your comments are welcome.

Jay Bai
MDSM Research
P.O.Box 272628
Ft. Collins, CO 80527