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.
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 $\lambda^x_t,\lambda^z_t$.