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.

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...

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.

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.

PP09 (II)

Today we had another lecture about renewal processes. The focus was on variability of count statistics.

We have seen a theorem relating the asymptotics of the Fano factor, which is the normalized variance of the counts, and the coefficient of variation of the inter event times.

If you have a point process, you can think of two different types of variability:

1) variability within each realization: how inter event times do differ from each other;

2) variability across trials: how statistics change from one realization to the next.

It is quite intuitive that for renewal processes both type of variability should be somehow connected: if you have a long realization of a renewal process, you can cut that into pieces and construct many shorter realizations; the renewal property implies that the statistics of the every single realization will be independent from the others.

This is exactly the meaning of the theorem about the asymptotics of the Fano factor: in the limit, for a renewal process the Fano factor and the CV2 will coincide.

Why the exponential function?

In the first lecture of the point processes course, we have used at least twice the exponential function. It is maybe worth to shortly explain why the exponential function can be represented in different ways.
We are mainly concerned with the identities

e^{x} =  \sum_{n=0}^\infty \frac{x^n}{n!}= \lim_n (1+\frac{x}{n})^n 

Let us start by defining the exponential function to be the only function for which

f(x) = f'(x)

up to a scalar factor. So, we have to check that both definitions enjoy this property. If it is case, then they will both agree with the exponential function, assumed that everything converges.

Let us check it for the first formula. Deriving term by term yields

d/dx\sum_{n=0}^\infty \frac{x^n}{n!} = \sum_{n=1}^\infty \frac{n x^{n-1}}{n!}=\sum_{n=1}^\infty \frac{ x^{n-1}}{(n-1)!}

By calling k the term n-1 we obtain our desired identity.

For the second one, observe that

d/dx f^n(x) = nf'(x)f^{n-1}(x) 

Applying this to the product in the formula we obtain

d/dx(1+\frac{x}{n})^n  =\frac{n}{n} (1+\frac{x}{n})^{n-1}

But since

\lim_n (1+\frac{x}{n})=1

one could hope that the identity holds. But here we have some difficulties with the convergence, indeed.

PP09 (I)

Tomorrow the lecture "One-dimensional point processes" will start.

I will introduce the theory of point processes on the real line in a rather mathematically informal way. The main goal is to provide the students with some tools for modeling neural networks on the basis of point processes and to give them a feeling for what could be done with spike data, once you have collected them.

The first topic will be very basic: the time-representation of renewal processes. We will see how far we can go by describing renewal processes as sum of an i.i.d. sequence.

I will post the lectures here and will try to give some additional material here on the blog.