27 Nov 2007 20:11

Random Time Changes for Stochastic Processes

There are a class of results about transforming one stochastic process into another by stretching and shrinking the time-scale, sometimes in a deterministic manner, more often by means of a random change of time-scale which depends on the realized trajectory of the process you started with.

The easiest example might be from point processes (which deserve a notebook of their own, some day). The simplest point process is the homogeneous Poisson process with "unit intensity": there is a constant probability per unit time of an event happening, which we can normalize to 1 (if necessary by changing our time unit). Events are thus laid down completely independently of one another. The consequence is that the distance between successive events is exponentially distributed, with mean 1. So you can imagine someone creating a realization of a homogeneous Poisson process with unit intensity by winding up a kitchen timer for a random, exponentially-distributed amount of time, and then putting down a point whenever it buzzes, at which point it is immediately reset to a new, totally independent count.

Now suppose you have a homogeneous Poisson process which does not have unit intensity, but one whose probability per unit time of an event is $\lambda$ . To make a realization of this process look like a realization of the standard Poisson process, simply take the time-axis and rescale it by $\lambda$ --- stretch out the separation between points if the process has high intensity (so that points fall close together), or compress them if the process has low intensity (so that points are widely spaced). Symbolically, one defines a new time, $\tau = t \lambda$ , and now $X(\tau)$ looks like a realization of the standard Poisson process, even though $ X(t) $ does not.

If the Poisson process is inhomogeneous, so that the probability per unit time follows some fixed intensity function $\lambda(t)$ , then the idea is similar, though the implementation is more complicated. We want the new time to run slow and stretch out when the intensity is high, and we want it to run fast and compress events when the intensity is low. It turns out that the right definition is $\tau(t) = \int_{0}^{t}{\lambda(t) dt}$ More specifically, if $t_1, t_2, \ldots$ are the times of the events, then the transformed times $\tau(t_1), \tau(t_2), \ldots$ come from a standard Poisson process, and $\tau(t_{i+1}) - \tau(t_i)$ are IID and exponentially distributed with mean 1. (This reduces to the previous result when the intensity function is homogeneous.)

Now suppose that the intensity function is not fixed, but depends on some random inputs, including possibly the history of the process. We write this as $\lambda(t|\mathcal{H}_t)$ , where $\mathcal{H}_t$ is supposed to summarize everything that goes into setting the intensity. (Even more technically, $\mathcal{H}_t$ is a sigma-field, and the collection of them over various t forms a filtration; the intensity is a random process adapted to this filtration. I could get even more technical if you make me.) If one now defines the rescaled time $\tau(t) = \int_{0}^{t}{\lambda(t|\mathcal{H}_t) dt}$ it still the case that $\tau(t_1), \tau(t_2), \ldots$ looks exactly like a realization of a standard Poisson process. But notice that the way we have re-scaled time is random, and possibly different from one realization of the original point process to another, because the times at which events happened can be included in the information represented by $\mathcal{H}_t$ .

This idea is not limited to point processes; one can transform many other sorts of stochastic processes to standardized versions by the appropriate random time changes. (A trivial example: If $ W(t) $ is a standard Wiener process, then $X(t) = \sigma W(t)$ is a non-standard Wiener process, where $X(t) - X(s) \sim \mathcal{N}(0,\sigma^2 |t-s|)$ . But $X(t/\sigma^2)$ is then a standard Wiener process again.) Generally speaking, to know how to do the transformation requires that one know something about the structure and parameters of the process.

This leads to what I can only call a brilliant little trick for doing statistical inference on stochastic processes, first made explicit (so far as I know) by Brown et al. If one is trying to fit a model to a point process, for example, what's going on is that you have a set of possible guesses about the conditional intensity function, say $\phi(t|\mathcal{H}_t;\theta)$ , where $\theta$ is a parameter indexing the different functions. For one parameter value, call it $\theta_0$ , this is actually right: $\phi(t|\mathcal{H}_t;\theta_0) = \lambda(t|\mathcal{H}_t)$ If your guess at $\theta_0$ , call it $\hat{\theta}$ , is right, then you can use $\phi(t|\mathcal{H}_t;\hat{\theta})$ to transform the original point process into a realization of a standard Poisson process. And it is really easy to test whether something is a realization of such a process (are the inter-event times exponentially distributed with mean 1? are the independent?). Notice that there are no free parameters in the hypothesis one ends up testing. So you can estimate the parameter however you like, and then do a back-end test on the goodness of fit.

I would really like to know more about how this can be done for other processes, and its strengths and limitations as e.g. a means of model selection.

Emery N. Brown Riccardo Barbieri Valérie Ventura, Robert E. Kass and Loren M. Frank, "The Time-Rescaling Theorem and Its Applications to Neural Spike Train Data Analysis", Neural Computation 14 (2002): 325--346 [PDF reprint]
Olav Kallenberg, Foundations of Modern Probability

Uwe Küchler and Michael Sorensen, Exponential Families of Stochastic Processes