PP09 (VIII)

This week we have started the analysis of interactions between point processes. The oldest model in this context is probably due to Hawkes and it is based on the following idea.

We want to model a family point processes, each with its own events

t_j^n, \qquad j,n \in \mathbb N

where j is the index for the event and n is the index for the process. The simplest idea is to assume that each of the processes is Cox one, i.e. an inhomogeneous Poisson process whose rate function changes randomly in time. We then let depend the rate function on the realization itself of the point process, by convolving the realization with a causal kernel.

This procedure is known as a Hawkes process, and it is very useful. It only has a drawback for neuronal modeling: you cannot model inhibition, since your kernels have to be positive, but if you are interested in this problem, you probably should take a look at the lecture.

Only to write that I (and many others) support pacific demonstrators in Teheran.

Auguri... (II)

Oggi mi sono ascoltato questa intervista a Maurizio Monti, del MSI. Al minuto 2:30 lui afferma, rispondendo all'intervistatrice che gli chiedeva se le divise delle ronde nere gli facessero venire in mente qualcosa

Ma sicuramente anche a me mi torna in mente qualcosa. [...] Il fascismo fa parte della storia italiana, fa parte del patrimonio del popolo italiano, è una cosa che è avvenuta tanti anni fa, è una cosa che può aver portato cose buone, e cose cattive, ma sa, i ricordi non fanno male a nessuno

Detto in altre parole: questi sono neofasciti (vedi qui), lo dicono pure, e non se ne vergognano.

PP09 (VII)

Last time we have defined the spectrum of a point process.

In general, spectral properties of stochastic processes are an interesting and difficult topic. Interesting because of the amount of information that can be extracted by analysis in frequency domain, and difficult mainly because it is quite difficult to measure the spectrum of a stochastic process.

Why it is difficult? To measure the spectrum, one idea could be to sample the stochastic process for a very long time, to plug the observed values into the definition of the Fourier transform, and take this as an estimation of the spectrum.

Does it work? If you do this, you're actually computing the periodogram of the stochastic process: as you can read in wikipedia, the periodogram has the problem to be a non-consistent estimator of the spectrum, and this fact can't be avoided but by collecting more observations of the process.

So, surprisingly, the key is not to observe the process for a long time, but to collect many independent observations of them.

Of course, observing for long times also is useful, e.g. to reduce spectral leakage, but this is another story...