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Statistica Sinica 30 (2020), 2105-2130

DETECTION AND REPLENISHMENT OF MISSING DATA
IN MARKED POINT PROCESSES

Jiancang Zhuang, Ting Wang and Koji Kiyosugi

Institute of Statistical Mathematics, University of Otago and Kobe University

Abstract: Records of geophysical events, such as earthquakes and volcanic eruptions, are usually modeled as marked point processes. These records often suffer from missing data, resulting in underestimations of the corresponding hazards. We propose a computational approach for replenishing data missing from the records of temporal point processes with time-separable marks. The proposed method is based on the notion that if such a point process is completely observed, it can be transformed into a homogeneous Poisson process, approximately on the unit square [0, 1]², by a biscale empirical transformation (BEPIT). This approach includes three key steps: (1) transforming the process onto [0, 1]² using the BEPIT, and finding a time-mark range that likely contains missing events; (2) estimating a new empirical distribution function based on the data in the time-mark range in which the events are supposed to be completely observed; and (3) generating events in the missing region. We test this method on a synthetic data set, and apply it to records of the volcanic eruptions of the Hakone Volcano in Japan and the aftershock sequence following the 2008 Wenchuan Mw7.9 earthquake in Southwest China. The results show that this algorithm provides a useful way to estimate missing data and to replenish incomplete records of marked point processes. In addition, the replenished data provide estimates of the hazard function that are more robust.

Key words and phrases: Biscale empirical probability, Hakone volcano, integral transformation, marked point process, missing data, Monte Carlo simulation, volcanic eruption, 2008 Wenchuan earthquake.

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