Spatiotemporal point processes

Alex Reinhart – Updated July 25, 2018 notebooks · refsmmat.com

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

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

Covariates

Residual diagnostics

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:

Time rescaling

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.

Fit tests

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: