Bayesian Updating on Time Intervals at Different Magnitude Thresholds in a Marked Point Process and Its Application to Synthetic Seismic Activity

Abstract

We present a Bayesian updating method on the inter-event time intervals at different magnitude thresholds in a marked point process toward probabilistic forecasting of an upcoming large event using temporal information on smaller events. Bayes’ theorem 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. We derive the inverse probability density function and the condition under which it has a peak. Further, we derive its approximation function that consists of the kernel part that includes the product of conditional probabilities and a correction term. Bayesian updating is applied numerically to the time series of a seismic activity model. We estimate the time of an upcoming large event using the maximum point of the kernel part and evaluate its accuracy by the relative error with the actual occurrence time. We further evaluate the forecasting effectiveness by the continuity of updates with acceptable accuracy before the large event. Statistical results indicate that forecasting is relatively effective immediately or long after the last major event in which stationarity dominates in time series.