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\begin{document}

\title{Bayesian Updating for Time-Intervals of Different Magnitude Thresholds
in Marked Point Process and Its Application to Time-Series of ETAS Model}



\author[1]{Hiroki Tanaka}%
\author[1]{Ken Umeno}%
\affil[1]{Kyoto University}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\begin{abstract}
In this presentation, we introduce a Bayesian updating method for
inter-event times of different magnitude thresholds in marked point
process, apply it to time-series of the ETAS model {[}1{]}, and discuss
the effectiveness in probabilistic forecasting of forthcoming large
event considering the information on smaller events. To investigate
magnitude threshold dependence of the inter-event time distribution of
earthquakes, the conditional probability between inter-event times of
different magnitude thresholds is proposed {[}2{]}. This gives the
one-to-one statistical relationship between inter-event times of
different magnitude thresholds. Firstly, we show the Bayes' theorem on
this conditional probability and derive the representation of the
inverse probability density function. Secondly, we extend it to the
Bayesian updating that gives the relationship between multiple intervals
for lower threshold and an interval for upper threshold. We show the
derivation of the inverse probability density function and its
approximation function for uncorrelated marked point process (background
seismicity in the ETAS model). The condition for the inverse probability
density function to have a peak is also shown. The approximation
function consists of two parts, a kernel-part that determines its
outline and a correction term. The former has an easy form to handle
numerically and is applicable to the time-series with correlations among
events. Thirdly, based on the results for uncorrelated time-series, we
apply the Bayesian updating method to time-series of the ETAS model. The
mode of the approximation function is numerically shown to be nearly the
same as that of the kernel-part. Therefore, the mode of the kernel part
is used as the estimate of the occurrence time of forthcoming large
event. By using the relative error between the estimate and the actual
occurrence time of large event, effectiveness of the estimation with the
approximation function is statistically evaluated. As a result, it is
shown that if the time-series is dominated by stationary part,
immediately or long after the large event, the forecasting is
effectively conducted. {[}1{]} Y. Ogata, Ann. Inst. Statist. Math.
50(2), 379 (1998). {[}2{]} H. Tanaka and Y. Aizawa, J. Phys. Soc. Jpn.
86, 024004 (2017).%
\end{abstract}%



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