PP09 (VI)

Finally, the update of the lecture.

I will skip lecture VI: I was somehow busy last week, and I could not update the blog.

So, tuesday we started to learn something about networks. In particular, we discussed what can be said about the statistics of the superposition of renewal processes.

Superposition is not a renewal process.

Assume that you have two different processes and fix a spike at time -1 from the first process, the previous one being in a late past. Then, depending on when the last spike of the second process was, the hazard of the superposition will be different, and so it can't be a renewal process. The exception is of course the superposition of Poisson processes for which the hazard is constant.

Superposition of large numbers of renewal processes in equilibrium has exponential distributed intervals

How to see it? Formal method: compute the ISI distribution of the superposition, see the slides or the book by Cox. More intuitevly, you can reason in the following manner: first discretize with width dt the time and assume that a time 0 you had a spike from neuron 1. Now, the probability of having a spike at time 1 will be dt(r_1+r_2+...+r_n) approximately for all subsequent bins: this is because we are superimposing a large number of processes which all are in equilibrium. So, it is a Poisson process in discrete time, and going to the limit dt-->0 we obtain what we are looking for.

Auguri...

...a tutti quelli che vivono ancora in Italia e che mi chiedono perchè non voglio tornare. Quello che parla nel video linkato è il signor Borghezio.

PP09 (IV)

So, today we started a new topic: that of non stationary point processes. In particular, we looked at inhomogeneous Poisson processes and Cox processes. The former are Poisson processes where the rate change in time, in a deterministic manner. The latter are the same, only the rate is itself a stochastic process.

As usual, the lecture is here.

Inhomogeneous Poisson processes can be seen also from another point of view: instead of thinking of a process with changing rate. one can think of a standard Poisson process with intensity 1, where the time is distorted. The distortion is, of course, proportional to the rate.

The advantage of thinking that way is to define non stationary renewal process: a method is, in fact, to define a stationary process and then to operate a time distortion, which leads to a non stationary process.

And this is exactly what we will do next time.

PP09 (III)

There was an error in the derivation of the partial differential equation for the age distribution in the slides. I have changed it, and now the slides are correct.

The derivation goes as follows; assume that we have a renewal process with age distribution u and hazard rate f. After a short time dt you will have that units with age s+dt at time t+dt are exactly the units that at time t had age s and that did not produce an event. Locally on time, the probability of not emitting an event is

P(\mbox{no event in (t,t+dt}) \approx e^{-\phi(s(t)) dt}

so, plugging this into the age distribution we obtain

u(t+dt,s+dt) \approx e^{-\phi(s) dt} u(t,s)

Fine. We now subtract u(t,s+dt) from both sides of the equation and expand the exponential function up to first order. This leads to

u(t+dt,s+dt)-u(t,s+dt) \approx (1-\phi(s) dt) u(t,s) - u(t,s+dt)

Rearrangin and dividing by dt leads to

\frac{u(t+dt,s+dt)-u(t,s+dt)}{dt} \approx \frac{u(t,s)- u(t,s+dt)}{dt}- \phi(s) u(t,s)  

Of course this means

\partial_t u(t,s) \approx -\partial_s u(t,s) - \phi(s) u(t,s)  

complemented with some initial and boundary condition.

QED

Ieri ho notato che l'elettrodinamica quantistica e il quod erat demonstrandum hanno la stessa abbreviazione

Cosa significa?