Point Process and Conditional Intensity
Table of Contents
1. Summary
A point process is a random set of times at which events happen. The processes in Stochastic Processes that came before this one all answer the question "what is the level now?". A point process answers a different question: "when does the next event happen?". For a market data feed the event is a tick, and the question is when the next tick arrives.
One quantity describes every point process in this cluster: the conditional intensity \(\lambda(t)\), the rate at which an event is expected at time \(t\), given every event before \(t\). The Poisson process is the case where \(\lambda\) never changes. The Hawkes process is the case where each event raises \(\lambda\) for a while.
2. Layperson's mental model
Picture a gauge that shows how likely a tick is in the next instant. When the gauge is high, ticks come close together. When it is low, they come far apart. A point process is a rule for how the gauge moves. In a Poisson process the gauge stays where it is. In a Hawkes process every tick pushes the gauge up, and the gauge then sinks back by itself.
3. Detail
3.1. The counting process and the intensity
The counting process \(N(t)\) is the number of events in \((0, t]\). It starts at zero and goes up by one at each event. The history \(\mathcal{H}_t\) is the list of event times up to \(t\). The conditional intensity is
\begin{equation} \lambda(t) = \lim_{h \to 0} \frac{1}{h}\, P\big(N(t+h) - N(t) = 1 \mid \mathcal{H}_t\big) \end{equation}A small \(h\) gives \(P \approx \lambda(t)\,h\). So \(\lambda\) has the unit of events for each unit of time. It is a rate, not a probability, and it can be larger than one.
3.2. The compensator and the test of a fit
The compensator \(\Lambda(t) = \int_0^t \lambda(s)\,ds\) is the number of events the model expects up to \(t\). The time-rescaling theorem says that if the model is right, the gaps \(\Lambda(t_k) - \Lambda(t_{k-1})\) between rescaled event times are independent and follow the unit exponential law. That gives a test that works for every point process in this cluster. Compute the rescaled gaps and check that they look like a sample from \(\text{Exp}(1)\). Brown and others (2002) describe the test for spike trains. The same test checks a simulated tick stream against its own parameters.
3.3. Two ways to simulate
- Inversion. Draw \(E \sim \text{Exp}(1)\) and solve \(\Lambda(t_k) - \Lambda(t_{k-1}) = E\) for the next time. This works when \(\Lambda\) has an inverse in closed form, as it does for a constant rate.
- Thinning. Find a bound \(\bar\lambda \ge \lambda(s)\) for the next stretch of time. Draw candidate times at rate \(\bar\lambda\), and keep each candidate with probability \(\lambda(s)/\bar\lambda\). Lewis and Shedler (1979) introduced this for a rate that changes with time. Ogata (1981) extended it to a rate that depends on the history, which is the case here.
3.4. Why ticks are a point process
Each tick is an event with a time stamp. A fixed timer, in which a tick comes every \(T\) seconds, is the degenerate case: the gaps do not vary. Real tick streams show bursts and quiet spells, so the gaps vary and they are not independent of each other. The notes linked below give the two simplest models of that variation. How the system uses them is in the tick arrival note.
4. See also
- Stochastic Processes — the hub; this note is the foundation of its point-process family.
- Poisson Process — the intensity that never changes.
- Hawkes Process — the intensity that each event raises.
- Markov Property — a point process with a constant intensity has no memory; an intensity that depends on the history does.
4.1. Further reading
- Daley, D. J., & Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, vol. 1, 2nd ed. Springer. The standard text; defines the conditional intensity and the compensator.
- Lewis, P. A. W., & Shedler, G. S. (1979). "Simulation of nonhomogeneous Poisson processes by thinning." Naval Research Logistics Quarterly, 26(3), 403-413. https://doi.org/10.1002/nav.3800260304.
- Ogata, Y. (1981). "On Lewis' simulation method for point processes." IEEE Transactions on Information Theory, 27(1), 23-31. Thinning for an intensity that depends on the history.
- Brown, E. N., Barbieri, R., Ventura, V., Kass, R. E., & Frank, L. M. (2002). "The time-rescaling theorem and its application to neural spike train data analysis." Neural Computation, 14(2), 325-346. https://doi.org/10.1162/08997660252741149.
- Wikipedia: Point process.