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Topic: Least-squares criteria for multiple experiments fit simultaneously?
Replies: 0

 Woody Posts: 32 Registered: 7/29/09
Least-squares criteria for multiple experiments fit simultaneously?
Posted: Jan 9, 2012 4:30 AM

[previously posted to sci.math, no responses]

The project I am working on involves fitting experimental data to a
model function. The model has multiple parameters, and is non-linear.
My questions involve the proper criterion for optimization of fit when
there are multiple experiments.

The model function is of the form m(a; p; x,t). with a a vector of
fixed parameters, p the parameters to be fitted, and x and t the
independent variables (assumed to have no error).

We're using least squares as our measure of best fit for a single
experiment; i e, minimize SUM((m(x_i,t_i) - y_i)^2)/n. This is the
form when all n experimental points (t_i,x_i,y_i) are weighted
equally. When they are not, each point is assigned a weight estimated
as 1/variance, and the denominator replaces n with SUM(w_i).

When there is more than one experiment to be fit simultaneously with
the same model function, what should the best-fit criterion be? There
are two typical situations:

1) A small number of either fixed or variable parameters are different
for each experiment, but the remainder are the same. This may occur,
for example, if the initial conditions differ for each experiment, but
the physical phenomena are the same.

2) The scaling of experiments differs. This will be taken into account
by the weighting. This occurs when the same physical phenomena are
measured by two different methods.