Information Gains for Stress Release Models

Abstract

The information gain of a point process model quantifies its predictability, relative to a reference model such as the Poisson process. This is bounded above by the entropy gain, or difference between the point process entropy rates. This provides a bound on the utility of the model as a forecasting tool, separate from the usual “goodness of fit” assessment criteria. The stress release model is a point process with an underlying state variable increasing linearly with time, and decreased by events. Assuming the intensity to be an exponential function of this state, we derive an analytic expression for the entropy gain. This is illustrated, using various magnitude distributions, for earthquake data from north China, and extensions to a multivariate linked model outlined. The results measure the effectiveness of the stress release process as a predictive tool. Comparisons are made with a scale derived from the Gamma renewal process and using Molchan's ν-τ diagram.