From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: singular value decomposition and 3D matrices
Date: 1 Feb 2001 11:54:28 GMT
Newsgroups: sci.math.num-analysis
Summary: What can the SVD decomposition accomplish?

original aticle not included because of space limitations.
I think you have understood the functionality of the svd correctly.
It goes as follows:
you have an input Y (say). In the usual notation this is 
represented as a vector, but it is on your side to renumber any input-data,
(say it are 3d-matrices) as a vector. You also know the representation of
an ideal input Y_true as a sum of basic elements B_i, for which you have also 
the representation as vectors (of the same length). that is 
 
Y_true = \sum_{i=1,...,n} a_i B_i 

where a_i are the unknown coefficients (making the mix) and n is in principle
any number. For the estimation procedure for the a_i to make sense,
n must be much smaller than the number of inputs (the length of Y).
Now, since you have not Y_true , but rather Y, where

   Y = Y_true +noise
you want to estimate the a_i from Y with minimal influence of the noise. 
Now we must make a lot of (maybe unrealistic) assumptions on the nature of
the noise and then come up after some manipulations with an equation for the 
a_i_estimated:

   a_estimated = (pseudoinverse of the matrix (B_1,...,B_n)) * Y
and the SVD gives you the best numerical way to compute this 
pseudoinverse. the mathematical details are of no interest for you. 

But this is a completely static relation. 

In your case, Y=Y(t) is observed over time, the "ingredients" B_i 
are time-independent (??) (these are the possible states) and the a_i are 
hence time-dependent. you could of course try to compute the a_i(t)
pointwise over time following the above formula. But is this what you 
required? Reading your posting i first thought about some kind
of generalized fourier-analysis with a generalized fourierseries, 
the basic elements being the appropriate descriptions of the different 
states , now these as functions of time , yielding constant a_i's,
but i suspect that this is incorrect. If the first idea meets
your intentions, then a trouble comes up: it is by no means clear
that this pointwise consideration at single instances of time give what you 
desire. Should there be a better measure of error, like the time-integral of
the error squared? and the functions a_i(t): they seem to have jump
discontinuities now (due to discontinuous change of atet of molecules?)
this would change the estimation procedure considerably.

should not the work of an enzyme be described by equations for
chemical reactions? and these are modeled by differential equations?
and you come up with an identification problem for the parameters in 
this differential equation. and conclude finally from these parameters
how the enzyme works?


hope that helps for finding the correct mathematical model of your problem.
(having done this, the rest is on the side of numerical analysis and computers)

hope this helps
peter