(Prompted to some extent by the answers already given by Shane and Srikant, I've rewritten this to try to clarify what I'm getting at, if only to myself.)
Suppose we have several similar systems, each with behaviour that approximates a continuous time Markov process. That is, there are some number of discrete states the system can be in and associated probabilities of transitioning from one state to another at any instant, depending solely on the current state. For now, consider the processes to be stationary, ie the transition probabilities do not change over time, and unaffected by seasonality or other external considerations.
Unfortunately, we cannot measure the state of any system directly, but have instead to measure a proxy quantity, which varies with state but is not discrete and is subject to various sources of noise and error.
The principal question is this:
Q1: Given two data sequences produced independently from two such systems, how can we decide whether the underlying Markov processes are the same?
Now, it may be that the best way to approach this is as two separate problems:
- Convert the imperfect proxy sequence into an idealised time series of (categorical) states
- Determine whether the state sequences correspond
On the other hand, such a separation might involve discarding some information from the data in step 1 (eg, about its variability) that would be useful in step 2. Which leads to:
Q2: Does it make sense to decompose the problem in this way or is it better to compare the proxy data directly?
If such a decomposition does make sense, that opens up a whole other issue about how to do the idealisation, but that's definitely a question for another day.
Shane, below, mentions goodness-of-fit and distributional tests such as Anderson-Darling, and that seems like a promising approach. But I'd like to check I'm understanding the idea correctly.
Given sufficient samples in a sequence, we would expect the proportion of time spent in each state to tend to the stationary distribution. So one could test the distributions of occupancies in the two sequences for similarity. (I have the vague sense a two-sample Kolmogorov-Smirnov might suit for this, but please set me right about that.)
The thing is, I'm not sure how good this can be as evidence. If the distributions are very different, that seems like a reasonable strike against the underlying processes being the same, but what if they're very similar? Can we draw conclusions in that direction?
Q3: Does a good fit of occupancy distributions tell us anything useful?
It seems like there could be an infinite number of processes that will tend to the same stationary distribution. I think this is very unlikely in practice, and that different systems will tend to have distinctly different behaviour, but still it's worth considering.
Finally, we will often have a model of the underlying process that we are looking for, although it may not be perfect. So we could compare each sequence with the expected behaviour of the model, instead of with each other. We may also have more than two sequences to test.
Q4: Is it better to compare multiple sequences to a single model, even an approximate one, or are we better off comparing data directly?