I have a sequence of event timings, $t_1, t_2, t_3, ..., t_n$ where there are a total number of $n$ events happening. For example, they are the timings when 911 was called in a city.
I know these events is a non-homogeneous (or inhomogeneous) Poisson process, where $\lambda=\lambda(t)$ where $\lambda$ is a function of time $t$. For example, when a riot is on, the $\lambda$ will surge up for a while, and then gradually decay to its normal frequency, as the effect of the riot fade away.
I have a model for this process, where given parameters $x$, I can generate a curve for $\lambda(t)$. Now the task is: fit the curve to actual data in order to obtain $x$. So here is what I did, which did not work:
1) I generated instantaneous event rate, by getting inter-event intervals and taking the inverse;
2) Use the least-square fit, where I minimize the squared sum of error between the generated $\lambda(t)$ curve and the actual instantaneous event rate data.
Well - there is a reason that this won't work: the least-square method is assuming normal distribution of error. When $\lambda$ is small, the Poisson distribution is obviously asymmetric, and hence cannot be fitted nicely with least squares method.
Given that - what should I do?
