Reinhart 2018 - Hawkes Processes and Conditional Intensity

Summary

A self-exciting process has conditional intensity equal to a background rate plus a sum of triggering kernels centred at past events. The mean number of direct offspring is $m=\int g$; the process is subcritical (finite clusters) if $m<1$. Equivalent Poisson-cluster representation: background events (“immigrants”) each start a branching tree.

Overview

Regression on gridded counts suffers the Modifiable Areal Unit Problem; point-process models avoid aggregation by modelling the intensity of events at $(s,t)$ directly. Reinhart distinguishes: (i) homogeneous/inhomogeneous Poisson; (ii) log-Gaussian Cox processes (log-intensity is a Gaussian process; random but independent of event history); (iii) cluster processes (Neyman-Scott, Matern); (iv) self-exciting processes whose intensity depends on the history ${\mathcal{H}}_t$.

Main Content

Conditional intensity

For a simple temporal point process with event times $t_i\in [0,T)$, history ${\mathcal{H}}_t$ and counting measure $N$,

$$\lambda (t\mid {\mathcal{H}}_t)=\underset{\Delta t\to 0}{\lim }\frac{\mathbb{E}[N[t,t+\Delta t)\mid {\mathcal{H}}_t]}{\Delta t}.$$

Spatio-temporally (eq. 1): $\lambda (s,t\mid {\mathcal{H}}_t)=\lim \frac{\mathbb{E}[N(B(s,\Delta s)\times [t,t+\Delta t))\mid {\mathcal{H}}_t]}{|B(s,\Delta s)|\Delta t}$.

Hawkes (self-exciting) process

Temporal: $\lambda (t\mid {\mathcal{H}}_t)=\nu +{\int }_0^{t}g(t-u)dN(u)=\nu +{\sum }_{i:t_i<t}g(t-t_i)$, with background rate $\nu$ and triggering function $g\ge 0$, $g(u)=0$ for $u<0$ (Hawkes 1971). Spatio-temporal (eq. 2): $\lambda (s,t\mid {\mathcal{H}}_t)=\mu (s)+{\sum }_{i:t_i<t}g(s-s_i,t-t_i)$. Often $g$ is separable, $g=f(s-s_i)h(t-t_i)$.

Poisson-cluster representation (Hawkes & Oakes 1974)

Any stationary self-exciting process with finite intensity is a Poisson cluster process: background events (cluster centres) form a Poisson process of rate $\nu$ (or $\mu (s)$); each event independently produces a Poisson number of direct offspring with mean

$$m={\int }_0^{\infty }g(t)dt(\text{spatio-temporal: }m={\int }_{\mathcal{X}}{\int }_0^{T}g(s,t)dtds)$$

placed according to $g$ normalised to a density. If $m<1$ clusters are a.s. finite, generation sizes form a geometric progression, and expected total cluster size (including the immigrant) is $1/(1-m)$.

Proof sketch. Offspring of an event at $t_i$ form an inhomogeneous Poisson process with intensity $g(\cdot -t_i)$ by the superposition property; recursion gives a Galton-Watson tree with Poisson($m$) offspring; mean total progeny is ${\sum }_k{m}^k=1/(1-m)$.

Marks and ground process

A marked process has events $(s_i,t_i,{\kappa }_i)$, ${\kappa }_i\in K$. With ground-process intensity ${\lambda }_g$ (eq. 4): $\lambda (s,t,\kappa )={\lambda }_g(s,t)f(\kappa \mid s,t)$. Marks are independent if conditionally independent given $(s_i,t_i)$ and unpredictable if ${\kappa }_i$ is independent of the earlier events. A multivariate process has a finite mark set (event types).

Figure 1 (p. 4) shows an observed process decomposed into background plus generations 1 and 2 with parent arrows; the cluster structure is latent. Figure 2 (p. 6) compares an inhomogeneous Poisson process (high intensity in a central square) with a self-exciting process using it as background and a short-bandwidth Gaussian triggering kernel (expected cluster size 4): clumps visibly emerge.

Examples

Exponential-kernel Hawkes, branching ratio

Let $g(u)=\alpha \beta e^{-\beta u}$. Then $m=\alpha$ (dimensionless branching ratio) and the mean intensity under stationarity is $\nu /(1-\alpha )$. With $\alpha =0.75$, expected cluster size $=1/(1-0.75)=4$, the value used in Fig. 2.

import numpy as np
nu, a, b = 0.2, 0.75, 1.5
print("branching ratio m =", a, " cluster size =", 1/(1-a), " mean rate =", nu/(1-a))
rng = np.random.default_rng(0)
def sim_cluster(T):                      # Alg.-5 style: immigrants then generations
    ev = list(rng.uniform(0, T, rng.poisson(nu*T))); gen = ev[:]
    while gen:
        new = []
        for t0 in gen:
            k = rng.poisson(a)
            new += list(t0 + rng.exponential(1/b, k))
        gen = [t for t in new if t < T]; ev += gen
    return np.sort(ev)
x = sim_cluster(2000.); print(len(x)/2000., "events per unit time (theory 0.8)")

Connections

• Reinhart 2018 - Likelihood, EM and Bayesian Estimation - how to fit $(\nu ,g)$.
• Reinhart 2018 - ETAS Earthquake Models and Omori Law - $g$ = modified Omori law with magnitude-dependent productivity.
• Relevance to marketing measurement: the branching ratio $m$ is the “viral coefficient” analogue; $m\to 1$ signals runaway cascades, so attribution models that ignore it overstate the exogenous (paid) share.

See Also

Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes - Overview | Simpson 2016 - Going Off Grid - Overview