Ill-posed inverse problem (linear operator estimation)

[bpet]

Ok this kind of question seems to come up a lot in research and applications but has me completely stumped.

Say we have a dynamical system $$x_{t+1} = Ax_t$$ where for simplicity we'll assume A is a constant symmetric real matrix but otherwise unknown.

1. What is the "best" estimate of A, when only the vectors $$x_0$$ and $$x_1$$ are known?

2. What is the new "best" estimate of A when $$x_2$$ is observed?

Obviously it's ill-posed because there are more unknowns than data. Various generalizations of the problem could include noise, observations of only $$y=Bx$$, infinite dimensions, etc but question 1 has it in a nutshell.

Any thoughts?

 

[spaghetti3451]

No thoughts!?

I would have replied if I fully understood the problem.

 

[Charles49]

How do you define best estimate?

 

[Stephen Tashi]

1. What is the "best" estimate of A, when only the vectors $$x_0$$ and $$x_1$$ are known?

2. What is the new "best" estimate of A when $$x_2$$ is observed?

Being a Bayesian, I would look for the "prior distribution" for the matrix $A$ that had maximum entropy. Then I would do a Bayesian update of $A$ that makes it the matrix (or one of the matrices) that makes the observed $x_i$ most probable.

If that proved too involved, I'd try to find a model for the joint distribution of the first n observations $x_1,x_2,... x_n$ (assuming A is n by n ). I'd want a model that was at least good enough to eliminate things that I think are absurd. (For example, the sequence (3.9, 4.2, 0, 0, 0, 0,...,0) might be implausible based on the physics of the problem at hand.) Given the first k observations, you could pick the matrix A by various criteria. You could pick it to be one that produces a subsequent series of observations that has a high probability. Or you could pick A to be one whose predictions are the best mean square error estimator of the subsequent observations.

 

[chiro]

I'm wondering if you can use some kind of operator algebra technique to solve for A.

Essentially you are going to be given something in terms of powers of A, where you have something like:

x1 = Ax0
xn = A^nx0

If you can get some root involving the various powers of A and x0 in terms of x1,x2,...,xn, then you can get an operator relationship for A in its powers and x0. Then by using an iterative technique, you could extract a good estimate for the linear operator A.

There is already an established theory to work out functions of linear operators given that the operator has certain conditions, and with an iterative method, I think this might be useful.

You have to check though what the requirements of the operators are for the operator algebraic techniques to work and given something useful.