I have implemented algorithm 1 and 2 in this paper http://www.lse.ac.uk/statistics/documents/researchreport61.pdf for the analysis/simulation hidden states for some time series. The reason why I am attempting to do this is so that I can automate the pre-match betting on sporting events and this is part of a much larger code base.
Now, I have implemented the algorithms outlined in the paper and the output of these is of course stochastic. For algorithm two I have the following pretty picture for a 1D observation vector time-series and a 1D state vector
- Nile = actual data
- alpha_hat = smoothed state
- alpha_tilde = simulated smoothed state
Ok, the problem I have is that qualitatively I can see that the simulated smoothed state "alpha_tilde" is behaving "reasonably" and roughly following "alpha_hat" and indeed the observations, which I want it to do. However, when this is re-run I get a different, but also (via my eye) reasonable output.
I have some experience in analysis of periodicity of time-series (FFT, Wavelets, structure functions et al.), but what I want here is to determine quantitatively if my simulated state is correct. This is difficult because of it's random nature and it is not as simple as merely applying univariate statistics to the two series. I thought of using a variogram but I don't see how I can use it to compare the simulated series with the expected series and get a meaningful correlation?
Any advice on how to compare two time-series for correlation would be appreciated.