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Say we have $N$ birds, $r$ is the probability that one bird sings. What is the probability $p$ that any of $N$ birds sings?

If we assume independence, there is a simple model describing the situation:

$$p = 1 - (1 - r)^N$$

But in reality, the independence is invalid - if one bird sings, the others which are nearby are more likely to sing as well. How can this be modelled? We would for sure need at least one another parameter in the model representing the autocorrelation of the events. But I have no clue how to do that... thanks for your help.

P.S.: There are certainly many ways how to build such a model. I would be grateful for very simple one (very simple formula like above, adding perhaps one extra parameter) and then perhaps gradually adding complexity. Thanks.

EDIT: I am actually interested in birds. The data I have is basically $N_i$, which is number of singing birds within the radius of 100 meters from point $i$. I do not have particular coordinates of the singing individuals, therefore I want the model as simple as possible.

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  • $\begingroup$ This question is extremely broad, because it (implicitly) covers all of time-series analysis and spatial analysis, yet it doesn't even describe what kinds of data you might have. Are you actually interested in birds or are you using them as a metaphor for some other problem? In either case, please edit your question to add information useful to narrow your question suitably and guide the construction of an appropriate model. $\endgroup$ – whuber Mar 1 '15 at 21:10
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This looks like a simple Bayes' theorem. You just need the probability of how likely a bird will sing if a nearby bird starts singing first.

You can also make this probability dynamic regarding the distance between the birds - the closer they are, the higher the probability.

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  • $\begingroup$ thanks alesc, but these are quite obvious ideas. I am interested in how to parametrize the model, how to write it. That's where I am clueless. Ideas are quite clear, I need formulas now :-) $\endgroup$ – Curious Mar 1 '15 at 14:30

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