Estimating Poisson process intensity using GLM

Question

Suppose I want to build an explanatory model for events generated by an inhomogeneous Poisson process with unknown intensity $\lambda$. Each entry in my dataset represents the registration of an event: time $t_i$, location $(x_i,y_i)$ and some explanatory variables $z_{i1},...,z_{ip}, i=1,...,n$. The entries are in chronological order. I want to estimate the effect of the explanantory variables on the intensity of the Poisson process using GLM.

The entries cannot be grouped, because this inevitably leads to loss of explanatory information. Hence a count model with poisson regression does not work in this case.

However, if we assume that the sojourn times $s_i$ (which we can compute in this dataset) are exponentially i.i.d., we can apply Gamma regression. Please ignore for the moment that we lose one observation due to differencing. Using a log link this becomes

$E[s_i|\textbf{z}_i] = exp(\beta_0 + \beta_1 z_{i1} + ... + \beta_p z_{ip}) = \lambda(z_{i1},...,z_{ip})$

(Edit) Since the explanatory data analysis reveals that the process is inhomogeneous both in time and two-dimensional space, I also include the location coordinates in the linear component:

$E[s_i|\textbf{z}_i,x_i,y_i] = exp(\beta_0 + \beta_1 z_{i1} + ... + \beta_p z_{ip} + \gamma_1 x_{i} + \gamma_2 y_{i}) = \lambda(z_{i1},...,z_{ip},x_i,y_i)$

Questions:

1. Is this approach valid?
2. Can I assume the sojourn times to be exponentially i.i.d. given that the Poisson process is inhomogeneous?
3. Does $E[s_i|\textbf{z}_i,x_i,y_i]$ indeed model the intensity of the process?
4. Am I modeling a one-dimensional (time), three-dimensional (time and location) or a $p+2$ dimensional Poisson process?

Answer 1

Can I assume the sojourn times to be exponentially i.i.d. given that the Poisson process is inhomogeneous?

In general the inter-event intervals will not be exponentially distributed. This paper by Yakovlev et al. (2008) derives an expression for the inter-event distribution for one-dimensional non-homogeneous Poisson distributions (Equation 6) and gives counterexamples (e.g., Equation 8).

Am I modeling a one-dimensional (time), three-dimensional (time and location) or a $p + 2$ dimensional Poisson process?

I'm guessing you want to model a three-dimensional process. It's not one-dimensional, because you will have different rates $\lambda(t, x, y)$ for different positions $(x, y)$. On the other hand, it's not $(p + 2)$-dimensional, since you probably observe only one $\mathbf{z}$ for every time and position, so that knowing $(t, x, y)$ implies knowing the value of $\mathbf{z}$. Of course you might then assume something like $\lambda(t, x, y) = \lambda(\mathbf{z}_{txy})$ or $\lambda(t, x, y) = \lambda(\mathbf{z}_{txy}, x, y)$, but I would still call that a three-dimensional process.