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# Extracting system behaviour from lots of individual transitions

During my offline time I've still been thinking about modelling populations and similar effects. I've decided on a slightly different simulation approach than I used previously, and it's thrown up a question which might already have been studied. If anyone is already aware of work in this area I'd be very interested.

Suppose that we've got a stochastic dynamical system which is essentially only dependent on a finite "memory" of parameters (hence only influenced by "absolute time" in that certain system input variables might be dependent on absolute time, eg, that a catastrophic forest fire reducing food supplies happens in year 20 after the simulation starts). Then this for any set of data comprising a state (which may include historical datums or generalised velocities, etc) we can approximate the probability distribution of next states by running the simulation. If we do this for lots of states that we think are "likely states" then we've built up a lot of piecemeal information about the system, but is there any existing algorithms for calculating long-term probabilites of die-off, natural periods, etc? (For a discrete Markov chain you can look at things like power series in the transition matrix, but that assumes that you've got a finite set of states and have computed transition probs for all of them: is there a way more adapted to "sparsely sampled" transitions (since it's going to be computationally prohibitive to just approximate each transition by this method.)

Many thanks for any thoughts,

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1.

If you had Monte Carlo samples from the posterior distribution of possible transition matrices, then you could calculate powers of each sample matrix and get a distribution of long-term behaviors. There are algorithms to compute such posteriors; perhaps this could be made more efficient using a sequential Monte Carlo method to update the posterior after each measured state. But I imagine that still would be an extremely inefficient way to do this, not really exploiting the sparsity of the measurements.

Comment Source:If you had Monte Carlo samples from the posterior distribution of possible transition matrices, then you could calculate powers of each sample matrix and get a distribution of long-term behaviors. There are algorithms to compute such posteriors; perhaps this could be made more efficient using a sequential Monte Carlo method to update the posterior after each measured state. But I imagine that still would be an extremely inefficient way to do this, not really exploiting the sparsity of the measurements.
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2.

A Dirichlet process might be useful. I don't understand them, but I think they are used in this kind of problem. I have seen them used in modelling molecular evolution, especially amino acid sequences.

Comment Source:A [Dirichlet process](http://en.wikipedia.org/wiki/Dirichlet_process) might be useful. I don't understand them, but I think they are used in this kind of problem. I have seen them used in [modelling molecular evolution](http://www2.lirmm.fr/mab/IMG/pdf/phylobayes2.3.pdf), especially amino acid sequences.