See also self-exciting point processes and mutually exciting point processes.
For a review, see: González, J. A., Rodríguez-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. 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.a7
A 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.
Other diagnostic statistics are available, via various schemes to transform the data into a Poisson process:
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:10.1162/08997660252741149
Schoenberg, 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:1022494523519
Rescaling 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.x
Extension 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: