Abstract

We present a Bayesian updating method on the inter-event times at different magnitude thresholds in a marked point process, toward the probabilistic forecasting of an upcoming large event using temporal information on smaller events. Bayes' theorem in a marked point process that yields the one-to-one relationship between intervals at lower and upper magnitude thresholds is presented. This theorem is extended to Bayesian updating for an uncorrelated marked point process that yields the relationship between multiple consecutive lower intervals and one upper interval. The inverse probability density function and its approximation function are derived. For the former, the condition for having a peak is shown. The latter is easier to apply to the time series of the ETAS model, and it consists of the kernel part, which includes the product of the conditional probabilities, and the correction term. The maximum point of the kernel part is shown to be not significantly affected by the correction term when applying the Bayesian updating to the ETAS model time series numerically. The occurrence time of the upcoming large event is estimated using this maximum point, and its accuracy is evaluated considering the relative error with the actual occurrence time. Moreover, forecasting is evaluated to be effective by the continuity of the updates with the accuracy within an acceptable range prior to the upcoming large event. Under these conditions, the statistical analysis indicates that forecasting is relatively effective immediately or long after the last major event in which stationarity is dominant in the time series.