1. ## Covariance matrix

Hello everybody,

I have a problem and I am really in need of your comments. My problem can be described as follow.

INPUT: Many sets of observed temporal data with $T$ elements ( $T$ is constant). Each set corresponds to a pair of number of entities and size, denoted by $n$ and $s$. For example, if $n=50$, s=1000, and T=1000, I have one set.

OUTPUT: A statistical model illustrating the observed data. In addition, that model should be able to predict value of output data with different pairs of (n, s). It means that if we test $n = 60$ and s=1500, we should have output with same shape as observed data and appropriate values.

I use a nonlinear regression model with normal distributed error term and the model has some, e.g. 2 $(a, b)$, coefficients. These coefficients can be expressed as functions of (n, s) with new coefficients, however. Therefore, these "mother" coefficients have "sibling" coefficients. For example, consider coefficient a = f_1(n, s) = a_1*n + a_2*s; that means $a_1$ and $a_2$ are sibling coefficients of $a$.

I want to use Markov chain Monte Carlo method to generate values for these coefficients. In MCMC method, suppose that $(a, b)$ are multivariate normal distributed, we need a covariance matrix. There are some studies on this stuff. Nevertheless, the posed situation in my problem is a little different as it has siblings. Because of this reason, I do not know how to compute covariance matrix, which needs to take care of all sibling coefficients, not just 2 mother coefficients.

Could you please kindly recommend me something to do with this problem?

I would highly appreciate every comment that you can give me. Thank you very much.

Regards,
Iaoh.

P.S: I am very sorry, I tried to put all mathematical symbols into MATH tag but some did not work.