Stochastic processes where we know that ancient history is still relevant for the future predictions, even if we know the recent history; how do we analyse such things?
I haven’t said anything about the types of generating process here. Maybe there is an explicit long-time dependency in your process which is not mediated through a hidden state; can we statistically distinguish these cases? Sometimes, e.g. Küns86, but generally you probably want a model, or how will you do it?
There are some obvious process model which have a long-memory in this sense, such as stack automata, or Hawkes processes with non-exponential kernels.
Note that “long memory” can work with not only time series but any random field, e.g. spatial, or random fields indexed by any number of dimensions, over whatever topology, causal or non-causal. I guess you’d need a notion of distance to make this meaningful, so let’s presume we are on a metric space at least.
Questions for me:
- Can we use the “memory length” of a system to infer the number of hidden states for some class of interesting systems, or vice versa?
- If so, when?
- Is the length even computable? Under what assumptions?
- Can we infer the memory length alone as a parameter of interest in some model classes? (need making precise).
For now, see learning theory for dependent data, which has some concrete results for predictive time series.
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- Beran, J. (1994) Statistics for Long-Memory Processes. . CRC Press
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