Sense of correlation coefficient matrix of particle filter's parameters

Question

I am using a particle filter to estimate the parameters($\Phi_{n\times1}$) of a non-linear model. Say my input (observations) is $t=1:k$, I will have a vector of length $k$ for each of the parameter estimates. Thus, at the end of the particle filter, I will end up with a parameter estimate matrix ($\Theta$) of $k\times n$ size, where the $k$ rows are the estimates at each time a new observation is made with the $k^{th}$ observation as my final estimate.

My question:

Does it make sense to find the correlation coefficient matrix ($C_{n \times n}$) of this parameter estimate matrix $\Theta$? If so, what will this represent?

--AM

edit:

1. The non-linear model is a physiological signal model with 4 states and 7 unknown parameters. These 7 parameters are treated as state variables, thus yielding a 11 state vector which needs to be estimated. This is done in a particle filter framework.

2. Yes, the final estimates are the only ones of interest but I came across a report where the correlation of the estimates of the particles in the filtering timeline (as I mentioned above) was measured. Seeing how the estimates evolve tells a great deal about the filtering process. So I am interesting in knowing what (if any) does the correlation matrix of this evolution tell.

Answer 1

The setup

Your edit suggests to me that the quantities of interest are a k x 4 matrix of states and a 7 estimated parameters, where the former are varying over time and the latter are fixed.

If that's the setup then the logical plan would seem to be to iteratively optimise the parameters using the filter to give you the likelihood term. Then take the best parameter values and use them to filter and smooth the state. Unless you're doing this online or you expect the parameters to vary I'm not sure why you would filter them too. Anyway...

The question

To the question: The only related correlation I've seen discussed in the particle filtering literature is to do with the tendency of particles to degenerate into clumps and give biased estimates of the filtered posterior. There, measures of the effective number of particles are more useful.

I do see how knowing how the estimates evolve would be useful, and not only for the degeneracy reasons mentioned above. What I don't see how that correlation matrix would tell you anything about this because it is effectively summing over the 'evolution' part.

Perhaps seeing the report where it was suggested it might put things in better context.

Full disclosure: The last thing I needed to think about on this topic was covered in Doucet et al. 2001 and I'm sure things have moved on. Perhaps some other folk are more up to date?