# Decomposing multiple poisson process

Assume a time series composed by many recurring events coming from many different poisson process each with a different rate. Lets assume for simplicity no overlap between events. Is there any math/statistical tool to decompose the given time series and get the various poisson processes. Ideally like fourier analysis can yield the harmonics composing a periodic signal, the tool should provide a list of poisson processes composing the whole sequence.

• Do you only observe their sum? – dsaxton Mar 2 '16 at 16:16
• Yes, you can only observe the sum. But events never overlaps – Davide C Mar 2 '16 at 16:49
• My first impression is that you can't. The sum itself will be Poisson$(\lambda)$, but there is an unlimited number of ways, all apparently equally valid, in which you could decompose this total count into a sum of Poisson random variables. – dsaxton Mar 2 '16 at 16:57
• Sum of Possions is Poisson. So, no, you can't decompose the sum... Unless the rates change in time in different ways, and you know something about this – Aksakal Mar 2 '16 at 16:57

Say, you observe $y_t$, where $y_t=x_t+z_t$, where $x_t\sim Poisson(\lambda^x_t)$ and $z_t\sim Poisson(\lambda^z_t)$. This means that $$y_t\sim Poisson(\lambda^y_t),$$ where $\lambda^y_t=\lambda^x_t+\lambda^z_t$
Let's say you can get $\hat\lambda^y_t$, how would you extract then $\hat\lambda^x_t,\hat\lambda^z_t$? There's no way unless you know something about the processes for $\lambda^x_t,\lambda^z_t$.