# Linear Gaussian parameter estimation

1. Oct 29, 2008

### aydos

Hi,

I have a multivariate linear-gaussian model and I am trying to estaimte a particular scalar set of parameters of the model.
I know how to derive the MLE in order to find the matrices A and Q (linear transfer function and covariance respectively).
I take the log of the joint distribution, find the above parameter derivatives, equal the expression to zero and solve for each of the 2 above parameters. The expression for A only depends on the data. The expression of Q depends on A and on the data. So I solve A first and then I can solve Q.

However, I have a specific application where I do not want to estimate the entire matrix A. Some scalar elements of the matrix I know a priori and some other scalar elements are the parameters to be estimated.

This is where my problems start:
1- The matrix calculus gets very hairy and I do not know how to solve this symbolically.
2- I tried to skip the step above and expanded the linear equation into a scalar expression with Matlab Symbolic toolbox since the parameters to be estimated are now scalars. I then went through Matlab differentiation and solving tools as well. It seems to work in principle.
3- On the original problem (the one I know how to solve), the expressions for A do not depend on Q. But now the Matlab solution shows that my estimated scalar Aij parameters do depend on Q and I believe it is correct. So now I have a set of parameters that all depend on each other and I am not sure what to do to solve them.

Any light on what I might need here would be appreciated.

Regards,
Carlos

BTW. this is my first post here, how do I insert latex expressions in these posts?

2. Oct 29, 2008

### Pere Callahan

Welcome to PF, Carlos.

You can insert latex into your post by typing

[ tex] your code here [ /tex]
or
[ itex] your code here [ /itex] (for an inline formular)

When you click on the $\Sigma$ symbol you are provided a short latex reference.

As for your question. Have you tried to solve your equations in the special case of a bivariate Gaussian, just to get a feeling of how the calculations go and if they can be done at all

Last edited: Oct 29, 2008
3. Oct 29, 2008