Help with a non-homogeneous poisson distribution please
I want to be able to model something with a poisson process with an intensity function that changes with both time and space.
Let's say for example that the time interval I'm considering is 100 hours long and I believe that the intensity function increases at a constant rate so that it's twice as big at the start of the 100th hour (t=99) as it is at the start of the 1st hour (t=0).
So I'm thinking of this as the start intensity rate (x) being continuously compounded by a rate (r) so that xe^(r*99) divided by x is 2, so r is ln(2)/99.
Using the additive nature of a poisson process and the formula for the sum of a geometric series I can find the mean between any interval. So the mean over the 100 hours is x (1-r^100)/(1-r).
Does that seem ok so far?
The problem I'm having more is with the spacial part. I'd like to model it so that when (if) the 1st event happens the intensity gets multiplied by some constant factor. So it doesn't matter what time the first event happens, the current intensity gets multiplied by some constant and then carries on changing through time as it was before. When (if) the 2nd event happens then the intensity gets multiplied by another constant, and so on and so on.
When I try thinking of these events happening in infinitesimal parts of time and how the spacial component affects my original mean before it was introduced, I keep confusing myself.
The probability mass function for poisson is (λ^k*e^-λ)/k! for k= 0,1,2,3....
So in my example is the probability of no events occurring over the 100 hours exp(-x(1-r^100)/(1-r)) or do I have to consider how introducing the spacial factors has affected the mean?
If a and b are the factors when the 1st and 2nd events occur respectively, is there a closed form way of calculating p(k=1) and p(k=2) ?
Thanks for any help.
Re: Help with a non-homogeneous poisson distribution please
If you want to make your parameter change with respect to other parameters you can do that if you wish but you will be looking at a completely different distribution.
The classic way that this is talked about in textbook, research papers, and among practitioners is by referring to it as a stochastic process. It can be in either fixed intervals (discrete time) or instantaneously changing (continuous time).
Your problem is looking at a collection of random variables in your dimensions that change over time and space.
If you want to look at say calculus or means and other moments across the means of variation (time and space) then you need to look at stochastic calculus and understand the basic concepts and techniques in that field.
The introduced method deals with Brownian Motion and Wiener Processes but you can find examples where they look at Poisson models. Your model will be a complex one and you may have to derive a lot of the stuff yourself, but the ideas in these resources should help you understand the terminology and also to get started in getting a solution to your problem.