See also self-exciting point processes and mutually exciting point processes.

For a review, see: González, J. A., Rodrı́guez-Cortés, F. J., Cronie, O., & Mateu, J. (2016). Spatio-temporal point process statistics: A review. *Spatial Statistics*, *18*, 505–544. doi:10.1016/j.spasta.2016.10.002

I’ve written a full review of *self-exciting* spatiotemporal point processes; see Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications. *Statistical Science*, *33*(3), 299–318. doi:10.1214/17-STS629. https://arxiv.org/abs/1708.02647

Self-exciting spatiotemporal point process parameter estimates are asymptotically normal, under regularity conditions: Rathbun, S. L. (1996). Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes. *Journal of Statistical Planning and Inference*, *51*(1), 55–74. doi:10.1016/0378-3758(95)00070-4

Log-Gaussian Cox processes model the conditional intensity as the exponential of a Gaussian process with specific covariance structure. These can be tweaked to include covariates. The Gaussian process makes these models flexible, but seems to make them difficult to interpret.

For an introduction, see Diggle, P. J., Moraga, P., Rowlingson, B., & Taylor, B. M. (2013). Spatial and spatio-temporal log-gaussian cox processes: Extending the geostatistical paradigm. *Statistical Science*, *28*(4), 542–563. doi:10.1214/13-sts441

Baddeley, A., Chang, Y.-M., Song, Y., & Turner, R. (2012). Nonparametric estimation of the dependence of a spatial point process on spatial covariates.

*Statistics and Its Interface*,*5*(2), 221–236. doi:10.4310/SII.2012.v5.n2.a7A nonparametric method to estimate the effect of a single covariate on a purely spatial point process. Further extended by:

M.I. Borrajo, W. González-Manteiga, M.D. Martínez-Miranda (2017). Kernel intensity estimation, bootstrapping and bandwidth selection for inhomogeneous point processes depending on spatial covariates. https://arxiv.org/abs/1703.03213

Develops mathematical theory of the method, a bootstrapping procedure for bandwidth selection, and suggests extensions to spatio-temporal processes and multiple covariates.

M.I. Borrajo, W. González-Manteiga, M.D. Martínez-Miranda (2017). Testing covariate significance in spatial point process first-order intensity. https://arxiv.org/abs/1709.07716

A hypothesis testing procedure for the nonparametric estimator.

A good strategy is to integrate the conditional intensity over cells and compare the integral to the true count.

- Bray, A., & Schoenberg, F. P. (2013). Assessment of point process models for earthquake forecasting.
*Statistical Science*,*28*(4), 510–520. doi:10.1214/13-sts440 - Integrating over Voronoi cells proves to produce better residuals: Bray, A., Wong, K., Barr, C. D., & Schoenberg, F. P. (2014). Voronoi residual analysis of spatial point process models with applications to california earthquake forecasts.
*The Annals of Applied Statistics*,*8*(4), 2247–2267. doi:10.1214/14-aoas767

Other diagnostic statistics are available, via various schemes to transform the data into a Poisson process:

- Clements, R. A., Schoenberg, F. P., & Schorlemmer, D. (2011). Residual analysis methods for space–time point processes with applications to earthquake forecast models in california.
*The Annals of Applied Statistics*,*5*(4), 2549–2571. doi:10.1214/11-aoas487

Given a conditional intensity function, we can do a time rescaling to get a unit-rate Poisson process. This can be the basis of some goodness-of-fit tests.

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. doi:10.1162/08997660252741149Schoenberg, F. P. (2002). On rescaled Poission processes and the Brownian bridge.

*Annals of the Institute of Statistical Mathematics*,*54*(2), 445–457. doi:10.1023/A:1022494523519Rescaling with the true conditional intensity function produces a Poisson process, yes, but rescaling with the maximum likelihood estimate of the intensity does

*not*, particularly in small samples. The rescaled process converges asymptotically to a Brownian bridge, and is self-correcting, meaning the numbers of events in subsequent time periods have negative covariance.Vere-Jones, D., & Schoenberg, F. P. (2004). Rescaling marked point processes.

*Australian & New Zealand Journal of Statistics*,*46*(1), 133–143. doi:10.1111/j.1467-842X.2004.00319.xExtension to marked processes, of which spatiotemporal processes are a special case.

Via weighting, we can produce *K* functions and other descriptive statistics which behave like they’re from a homogeneous Poisson process, to test model fit:

- [To read] Adelfio, G., & Schoenberg, F. P. (2008). Point process diagnostics based on weighted second-order statistics and their asymptotic properties.
*Annals of the Institute of Statistical Mathematics*,*61*(4), 929–948. doi:10.1007/s10463-008-0177-1