Parameter optimization for a model that relates two datasets

Hello! This is a seemingly simple problem I have been stuck on for some time now. I would greatly appreciate any ideas or help!

I have a model of the form:

Y = A*X^-1 + B*X^4*F(X)

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A and B are unknown parameters.

Y is a measured frequency distribution dataset of the form (Y(i), f(Y(i))).

X is another measured frequency distribution dataset (X(i), g(X(i))).

F is a known (albeit complicated) function of X.

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I would like to find values for A and B such that the right hand side of the function transforms the frequency distribution (X,g(X)) into something that looks very nearly like the measured (Y,f(Y)).

I have tried a nested for-loop "brute-force" method which tests all combinations of A and B within reasonable bounds, but it has proven too sensitive to the discretization scale chosen for A and B. Is there any matlab solution to this sort of problem? Thank you in advance!

Andrew

Answer 1

[Jeremy Kemmerer]

Do you know for certain that the parameters A and B exist? It sounds like you may have an underdetermined system.

One quick thing to try is to form a matrix Z with three columns: Y, X^-1, and X^4*F(X) and check the rank of the matrix using the function “ rank ”.

If Z is full rank (rank is equal to 3 in this case), this tells you that the columns are not linearly dependent, and hence the parameters you are looking for won’t exist.

For more information on the command “ rank ”, please see the documentation pages: http://www.mathworks.com/help/matlab/ref/rank.html

[Andrew Johnson]

Thank you, Jeremy. I had suspected this might be the case. I appreciate your answer!

Best, Andrew