Looking for a modified Poisson distribution

[Hoplite]

I'm looking to model a system in which events are nearly perfectly randomly distributed but with a slight tendency for events to avoid each other. As you know, if the system were perfectly random, I could use a Poisson distribution. The probability distribution for the number of events would then be

## P(N) = \frac{\lambda^N e^{-\lambda}}{N!} . ##

And the Poisson distribution also allows us to determine the probability distributions for the locations of each event, since the Poisson distribution implies that individual events follow a uniform distribution. Hence, if every event occurs within ##(-L/2, L/2)##, and ##p(x)## is the probability distribution for the location of an individual event, then

## p(x) = \frac{1}{L}. ##

So, I'm looking for a modified Poisson distribution that allows the events to be slightly dispersed, rather than perfectly random and gives me equivalent equations for both ##P(N)## and ##p(x)##. I know of many that would give me an equation equivalent to ##P(N)##, but none that would give me an equivalent to ##p(x)##.

Naturally, any equation for ##p(x)## would have to take into account the locations of the other events, so it may only be able to produce an equation for ##p(x)## for a small number of events. For example, the first event to be placed within ##(-L/2, L/2)## has no preference for any location, so its probability density function would be

## p_1(x_1) = \frac{1}{L}. ##

The location of the second event, ##x_2##, however, would depend slightly on the location of the first, ##x_1##, and this is where it gets tricky. I don't know of any modified Poisson distribution that would allow me to determine the probability density function of the 2nd event, let alone the third or fourth.

Could anyone recommend a suitable distribution to use?

 

[mathman]

The spatial distribution of the events and the time distribution (Poisson) are independent. I suggest you work out the spatial distribution based on the model you want and leave the time distribution as is.

 

[mfb]

You can choose P(N) as you like as it has no information about the positions in your interval.

It is possible to find a function that is nearly uniform over (-L/2, L/2) but slightly smaller close to other events. There are tons of functions without a clear way to prefer one over others. Normalizing it might be ugly, and writing down a combined probability distribution ##p(x_1,x_2,x_3,...)## in that way could be even more ugly.

 

[HuskyNamedNala]

What you can try to do is develop an "intersection matrix" of events and develop your own distribution. For example suppose you are only concerned with the probability of any two events occurring at the same time. You can develop two vectors of all possible events:
(Im at work I can't do latex because of the firewall sadly)

u = first event occurring = {event 1, event 2,...}
v = second even occurring = {event 1, event 2,...}

Each "event" has a probability associated with it. Suppose you know the probability of a few intersections of u and v, then you can develop a matrix:

P(u n v) = 2D matrix corresponding to events in u and v, with n1*n2 elements where n1 and n2 are the number of events in u and v respectively.

Now this might be difficult if you have a lot of events to work with, but you know a key piece of information: the sum of all probabilities in the matrix must add up to one. You also know that the double dot product of your matrix and an event vector must result in the expected value.

I hope this helps. This strategy comes from a problem I am working on in my free time to predict a few interesting things.