To all,
Thanks so much for all your ideas and insights thus far. To those who have
suggested a Baysean approach, I am interested, but I am weeks away from
understanding it well enough to figure out if I can use it. Also, I think I am
close to developing a usable technique along my current line. The only
constrain on my parameters is that they remain positive. Occassionally one will
approach zero, not often. I am reposting because I have another focused
question stemming from the same problem.
MY SITUATION:
I am studying a time-dependent stochastic Markov process. The conventional
method involved fitting data to exponential decay equations and using the
F-test to determine the number of components required. The problem (as I am
sure you all see) is that the F-test assumes the data is iid, and conflicting
results are often observed. As a first step, I have been attempting to fit
similar (simulated) data directly to Markov models using the Q-matrix and
maximum likelihood methods. The likelihood function is:
L= (1/Sqrt( | CV-Matrix |))*exp((-1/2)*(O-E).(CV-Matrix^-1).(O-E))
Where | CV-Matrix | is the determinant of the Covariance matrix, (O) is the
vector of observed values in time order and (E) is the vector of the values
predicted by the Markov model for the corresponding times. The Covariance
matrix is generated by the Markov model.
My two objectives are to determine the number of free parameters, and to
estimate the values of the parameters. Because the data is simulated I know
what the number of parameters and their values are.
MY PROBLEM:
I have been using the Log(Likelihood) method to compare the results of fitting
to the correct model and to a simpler sub-hypothesis (H0). I am getting very
small Log(Likelihood ratio)?��s when I know the more complex model is correct
(i.e. H0 should be rejected). When I first observed this I tried increasing the
N values, and found a decrease rather than an
increase in the Log(Likelihood ratio). When I look at the likelihood function,
the weighted Sum of Squares factor : ( (O-E).CV^-1.(O-E) ) is very different
between the two hypotheses (i.e. favoring rejection of H0), but difference in
the determinant portion ( (1/Sqrt( | CV-Matrix |)) ) is in the opposite
direction. As a result, the Log(Likelihood ratio) is below that needed to
reject H0.
I asked about just fitting (O-E).CV^-1.(O-E) and was reminded that without the
determinant factor, the likelihood would be maximized by simply increasing the
variance. This appears to be true in practice.
In learning about the quadratic form, I read in several places that, for the
distribution to approach a chi square distribution, the Covariance Matrix must
be idempotent (CV^2 = CV). I am almost certain this is not the case.
I am hoping to get feedback on this idea:
THE QUESTION: Following maximization of the full likelihood function (
(1/Sqrt( | CV-Matrix |))*exp((-1/2)*(O-E).(CV-Matrix^-1).(O-E)) ) for both
models, can I use the F-test to compare the weighted Sum of Squares (i.e.
(O-E).CV^-1.(O-E) ) of the two models, rather than the likelihood ratio
test. In other words, does correcting each (O-E) for its variance and
covariance legitimize the F-test?
Any insight is greatly appreciated. Thanks for your patience and consideration.
James Celentano
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