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(first-time post alert)

Hello All,

I have an empirical binary sequence (i.e. 0/1) of observations taken for equal discrete time intervals. (A typical length of such a sequence would be about 120). The probability for a "1" may decrease over time.

Now I'd like to code a procedure which generates "similar" sequences, so that I can obtain longer series with "essentially" the same behaviour; and in order to do that I'd need parameters describing the sequences. So the goal here is "merely" description, not "real statistics" in the sense of trying to estimate any "true" parameters from those sample sequences.

In which general direction should I be looking? Moving averages? Autocorrelation? Markov chains? (Probably not the latter: it may well be that event(i) "looks back further" than event(i-1) ) Something completely different?

I'm aware it's a rather broad (and probably ill-defined :-) question, but I am not expecting detailed analyses or guidelines; I'm asking "just" for some appropriate pointers or keywords to get me started, so that I do not take off into a completely wrong direction.

Essentially, I am trying to find the right terms for the search engines, and then will take it from there... :-)

Thanks for your help!

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    @Dilip Sarwate: "behaviour": things like e.g. probabilities (over the whole sequence or parts thereof), change of those probabilities over time, number / length of "runs", etc. Finding out what "essentially the same" actually means is just the question. The practical goal is to generate a sequence which in its characteristics is "indistinguishable" from the observed ones. I do realize these questions may well be un-answerable. (It's not about _checking data against_ a hypothesis, it's about _finding_ one.2011-12-05

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For some arbitrarily chosen (but not too big) $m$, consider the sequence as an order-$m$ Markov chain. That is, you look at the conditional probabilities for $a_t$ given $a_{t-1}, a_{t-2}, \ldots, a_{t-m}$.

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    Thanks, this sounds useful. I didn't realize (or rather, had forgotten) that there are M-chains which actually "look back" _more_ than just 1 step.2011-12-05
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Example program to simulate flip of a biased coin in python. You can design a function that decreases probability of getting one with each toss.

import random  def flip(p):         return 1 if random.random() < p else 0  p = 0.5 # 0 and 1 are equally likely   for i in range(120):         p = p - 0.5/120  # example function that decreases probability of getting 1          print flip(p), 
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    It will. Just change 120 to some other length.2011-12-05