fitting a model to correlated data
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Trying to fit a model (bloch equations from MRI) to experimental data (12 parameters, 12 equation). Monte Carlo Multistart ('lsqcurvefit' in MATLAB) gives expected values for all parameters except for two, which are correlated:
http://imgur.com/1PRayGj (fig. 1) The fig. is a result of the Monte Carlo simulation.
Briefly about the model. It is a system of differential equations (solved numerically by Cramer's rule). The fig. 2 contains 9 diff. eq. My model is a bit more complicated and contain 12 diff. eq.
http://imgur.com/E5LiYdq (the model. fig. 2)
I am passing initial guess, lower and upper bounds for all parameters to the solver. The problem - In the optimized output, 1st parameter always takes the value of its upper limit and 2nd one the value of its lower limit (please, see the fig. 1).
Bad luck, because I am only after those two parameters. Are there any good fitting algorithm for correlated data?
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dpb
2014-1-21
Where does Fig 1 come from? If this is known a priori, recast in terms of one or the other.
As for the Q, the answer is "in general, no" because the correlation implies a nearly singular system of equations need to be solved is one way to think of the issue.
Alex
2014-1-22
dpb
2014-1-22
I seriously doubt repeating the same calculation w/ various tools is going to change anything...the problem is in the basic formulation that you need to be able to recast or add another independent expression in order to remove the correlation.
Or, potentially, as suggested before, estimate the one in terms of the other and then back-substitute that into the remainder of the system.
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