Reinhart 2018 - Crime, Epidemic and Network Applications

Summary

The ETAS template transfers to crime (near-repeat victimisation), infectious disease (transmission as excitation) and events on networks (email replies, contagion of violence). Conclusion of the review: the background/triggered split must be interpreted with care when the true mechanism is fully contagious.

Overview

Different fields choose different background and kernel structures; the shared pieces are the conditional intensity and the branching-structure toolbox.

Main Content

Crime (Sec. 4.2)

Mohler et al. (2011): $\lambda (s,t)=\nu (t)\mu (s)+{\sum }_{t_i<t}g(s-s_i,t-t_i)$, nonparametric via Algorithm 1 (subsampled for 5,376 residential burglaries). Mohler (2014) parametric version with leading indicators $M_i$: $\mu (s)={\sum }_i\frac{{\alpha }_{M_i}}{2\pi {\eta }^{2}T}\exp (-\Vert s-s_i{\Vert }^2/2{\eta }^2)$, $g(s,t,M)=\frac{{\theta }_M}{2\pi \omega {\sigma }^2}{e}^{-t/\omega }{e}^{-\Vert s{\Vert }^2/2{\sigma }^2}$. Chicago, 78,852 violent crimes 2007-12, target = homicide; the self-exciting model outperformed fixed hot-spots ($\theta =0$) and near-repeat-only ($\alpha =0$) models. Theory background: routine activities, rational choice, law of crime concentration (Weisburd 2015).

Epidemics (Sec. 4.3)

Meyer, Elias & Hohle (2012): invasive meningococcal disease, 636 cases in Germany 2002-08: $\lambda (s,t)=\rho (s,t)\exp ({\beta }^{\top }z(s,t))+{\sum }_{j\in I^{\ast }(s,t)}{e}^{{\eta }_j}g(t-t_j)f(\Vert s-s_j\Vert )$, ${\eta }_j={\gamma }_0+{\gamma }^{\top }{m}_j$, Gaussian spatial $f$, constant temporal $g$; Meyer & Held (2014) power-law $f$. Schoenberg, Hoffman & Harrigan (2017): recursive model $\lambda =\mu +\iint H(\lambda (s^{\prime },t^{\prime }))gdN$ with $H(x)=\kappa x^{-\alpha }$ (decreasing productivity), fitted to LA measles.

Networks (Sec. 4.4)

Fox et al. (2016): email at West Point, ${\lambda }_i(t)={\nu }_i\mu (t)+{\sum }_j{\sum }_{r_k^{ij}<t}{\theta }_{ij}{\omega }_i{e}^{-{\omega }_i(t-r_k^{ij})}$ (EM + bootstrap). Zipkin et al. (2015): a self-exciting process per edge to infer missing sender/recipient. Green, Horel & Papachristos (2017): contagion of gun violence, ${\lambda }_k(t)=\mu (t)+{\sum }_{t_i<t}{\phi }_{k_i,k}(t-t_i)$ with ${\phi }_{u,v}=\beta e^{-\beta t}\alpha \text{dist}(u,v{)}^{-2}$ for dist $\le 3$.

Interpretation caveat (Sec. 5)

If an infection has no unobserved carriers, “background” cases only reflect unobserved infections; better reporting lowers apparent background and the estimated $m$ is underestimated (some triggered infections are missing). Self-exciting models are most meaningful when the generative process is truly clustered.

Examples

PyMC sketch for a marketing analogue (untested; conceptual): counts of daily conversions $y_t$ with discrete-time Hawkes intensity ${\lambda }_t={\mu }_t+\alpha {\sum }_{k\ge 1}{w}_k{y}_{t-k}$, $\sum w_k=1$, $y_t\sim \text{Poisson}({\lambda }_t)$, ${\mu }_t=\exp (\text{media effect})$. Report $\alpha$ (endogenous share) alongside media lift. This is an analogy beyond the paper.

Connections

• Social Network Models and Hoff 2002 - Latent Space Models Overview for networks as outcomes instead of event streams.
• Marketing measurement: contagion vs. exogenous demand; see MMM Model Selection and Application.

See Also

Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes - Overview