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

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 λ 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. The weighted K-function is powerful for evaluating the degree of clustering or inhibition in a model. Competing models are also compared using pixel-based approaches, such as Pearson residuals and deviance residuals. The different residual analysis techniques are demonstrated using the CSEP models and are used to highlight certain deficiencies in the models, such as the overprediction of seismicity in inter-fault zones for the model proposed by Helmstetter, Kagan and Jackson [Seismological Research Letters 78 (2007) 78–86], the underprediction of the model proposed by Kagan, Jackson and Rong [Seismological Research Letters 78 (2007) 94–98] in forecasting seismicity around the Imperial, Laguna Salada, and Panamint clusters, and the underprediction of the model proposed by Shen, Jackson and Kagan [Seismological Research Letters 78 (2007) 116–120] in forecasting seismicity around the Laguna Salada, Baja, and Panamint clusters.