Help with a non-homogeneous poisson distribution please

[ChickenOrient]

Hello,

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 spatial 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 spatial 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 spatial 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.

 

[mmwave]

Thank you for your question. It seems like you have a good understanding of the concepts involved in modeling a Poisson process with a changing intensity function. Your calculations for the mean over the 100 hours and the probability mass function for no events occurring are correct.

To incorporate the spatial component into your model, you can use the concept of conditional probability. This means that the probability of an event happening at a certain time and space is dependent on the previous events that have occurred. So for example, if the first event occurs at a certain time and space, the intensity at that time and space will be multiplied by a constant factor. The probability of this event occurring can be calculated by multiplying the probability of no events occurring before it (which you have already calculated) with the probability of the first event occurring at that specific time and space, which can be calculated using the Poisson probability mass function.

Similarly, for the second event occurring at a different time and space, you would multiply the probability of no events occurring before it with the probability of the second event occurring at that specific time and space. This would give you the probability of k=1 and k=2 events occurring in your model.

In summary, to incorporate the spatial component into your model, you would need to calculate the conditional probabilities of events occurring at specific times and spaces, and then use them to calculate the overall probabilities of k=1 and k=2 events occurring.

I hope this helps. Good luck with your modeling!