Residual analysis methods for space--time point processes with applications to earthquake forecast models in California

TL;DR

This paper reviews advanced residual analysis techniques for space-time point process models, applies them to California earthquake forecasts, and compares their effectiveness to traditional methods, highlighting super-thinning as a promising approach.

Contribution

It introduces and evaluates modern residual analysis methods for earthquake forecast models, emphasizing the advantages of super-thinning over traditional techniques.

Findings

Rescaled residuals assess overall model fit but are impractical with volatile intensities.

Residual thinning and superposition identify poorly fitted spatial regions but have limited power with extreme intensities.

Super-thinning offers a more powerful alternative for residual analysis in earthquake forecasting models.

Abstract

Modern, powerful techniques for the residual analysis of spatial-temporal point process models are reviewed and compared. These methods are 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 residual methods based on rescaling, thinning, superposition, weighted K-functions and deviance residuals. Rescaled residuals can be useful for assessing the overall fit of a model, but as with thinning and superposition, rescaling is generally impractical when the conditional intensity $\lambda$ is volatile. While residual thinning and superposition may be useful for identifying spatial locations where a model fits poorly, these methods have limited power when the modeled conditional intensity assumes extremely low or high values somewhere in the observation region, and this is commonly the case for earthquake forecasting models. A recently proposed hybrid method of thinning and superposition, called super-thinning, is a more powerful alternative.