Modern, powerful techniques for the residual analysis of spatial-temporal point process models are reviewed and their power under various null and alternative hypotheses is compared. Residual methods can be divided into two schemes: transformation-based and pixel-based methods. Rescaling, thinning and superposition are useful transformation-based methods for the residual analysis of spatial- temporal point processes. These techniques involve transforming the original point process into a new process that should be a homogeneous Poisson process if and only if the fitted model is correct, so that one may inspect the residual process for homogeneity using standard tests for homogeneity as a means of assessing the goodness-of-fit of the model. Unfortunately, tests of homogeneity performed on residuals based on these three residual methods tend to have low power when the modeled conditional intensity of the original process is volatile. For such purposes, we propose the method of super-thinning, which combines thinned residuals and superposition. This technique involves the use of a tuning parameter, k, which controls how much thinning and superposition are performed to homogenize the process. The method is applied to the assessment of a parametric space-time point process model for the origin times and epicentral locations of recent major California earthquakes.

These residual methods are then applied to California earthquake forecast models used in the Collaboratory for the Study of Earthquake Predictability (CSEP). Assessments of these earthquake forecasting models have previously been performed using simple, low-power means such as the L-test and N-test. We instead propose using the transformation residual methods for model assessment, and the pixel-based methods, such as Pearson and deviance residuals, to compare competing models. The different residual analysis techniques are demonstrated using the CSEP models and are used to highlight certain deficiencies in the models.
Both pixel-based and transformation methods are evaluated through a simulation study by applying each method to a group of Hawkes processes that contain different degrees of clustering and inhibition. Pixel-based methods, such as raw, Pearson, inverse, deviance, and tessellation residuals appear to be generally weaker than transformed residuals at detecting insufficient or excessive local clustering in the model. The transformation method of super-thinning is shown to have relatively high power when the value of the tuning parameter, k, is carefully chosen.

Finally, we introduce the R package stppResid, which implements both transformation and pixel-based residual analysis for space-time point process models. We illustrate the use of each of these residual tools by applying them to a well known space-time point process model fitted to a red banana data set.