Point processes offer a convenient mathematical representation of earthquakes, volcanic eruptions, crimes and many other processes which occur at random times and locations. The data available in these fields has exploded with modern recording technology, and yet many data sets suffer from significant incompleteness. In the case of earthquakes, for example, the recording of aftershocks in the period following a large earthquake is typically biased, as large aftershocks are more likely to be detected than smaller ones. Similar missing data plagues studies of volcanic eruptions, or records of communications in social networks, often leading to biased research conclusions.
In previous work, researchers have explored a variety of means for filling in missing data, often by making strong assumptions about the statistical nature of the process in question. Such methods should make as few assumptions as possible, of course, especially when the temporal structure and the distribution of event sizes are unknown. In previous work, LML External Fellow Jiancang Zhuang introduced a stochastic algorithm able to restore missing aftershocks in an aftershock sequences, and working with one minimal assumption – that the empirical point process is time-separable. That is, the distribution of events over time and magnitude can be written as a product of functions in time and magnitude alone.
In a new study, Zhuang and colleagues analyse in more detail the mathematical foundations of the approach. As they describe, the method rests on the observation that, if such a point process is observed completely, without missing events, then a suitable mathematical transform will turn it into a homogeneous Poisson process. Hence, missing data can be estimated by studying how the transformed data deviates from such a form. They illustrate the method by removing the bias from missing data in the eruption records for both the Hakone volcano in Japan and the earthquake catalogue from Southwest China.
The paper is available at http://bemlar.ism.ac.jp/zhuang/pubs/zhuang2019statsini.pdf