Hidden Markov Models for Time-Inhomogeneous and Incompletely Observed Point Processes

Abstract

Many point processes such as earthquakes or volcanic eruptions usually have incomplete records with the degree of incompleteness varying over time. Consequently, hazard estimation from such time-inhomogeneous incomplete records is complicated and potentially biased. Since the number of missing events is unknown, two distinct HMM-type methodologies are proposed: one with the observed process having a fixed number of missing events between each pair of consecutively observed events, and the other with the observed process having a variable number of missing events between each pair of consecutively observed events in an incomplete point process record.

In the first approach, a general class of inhomogeneous hidden semi-Markov models (IHSMMs) is proposed for modelling incompletely observed point processes when incompleteness does not necessarily behave in a stationary and memoryless manner. The key feature of the proposed model is that the sojourn times of the hidden states in the semi-Markov chain depend on time, making it an inhomogeneous semi-Markov chain. We check a conjecture of consistency of the parameter estimators of the proposed model by simulation study using direct numerical optimization of the log-likelihood function. We apply this class of models to a global volcanic eruption catalogue to investigate the time-dependent incompleteness of the record by proposing a particular IHSMM with time-dependent shifted Poisson distributed state durations and a renewal process as the observed process with a fixed number of missing events between each pair of consecutively observed events in the record. A combination of the Akaike Information Criterion and residual analysis is used to choose the best model. The selected inhomogeneous hidden semi-Markov model provides useful insights into the completeness of a global record of volcanic eruptions during the last 2000 years, demonstrating the effectiveness of this method.

In the second approach, shifted compound Poisson-gamma (SCPG) and time-dependent SCPG (TSCPG) renewal processes are introduced in order to model the unknown and time-dependent random variable number of missing events between each pair of consecutively observed events in incompletely observed point processes. The SCPG renewal process models the shifted Poisson distributed number of missing events, and the TSCPG renewal process models the time-dependent shifted Poisson distributed number of missing events between each pair of consecutively observed events in the gamma renewal process. In addition to IHSMMs and SCPG renewal processes, a special case of inhomogeneous hidden Markov models (IHMMs) is developed to examine nonstationary incompleteness of point processes. The multinomial logistic functions are adopted to formulate the time-varying transition probabilities in the proposed IHMM in the way that characterizes the temporal structure of the missingness of events in records. The SCPG and TSCPG renewal processes are used as the observed processes in HMMs, HSMMs, IHMMs and IHSMMs to model the time-dependent incomplete point process records. Simulation experiments are employed to check the performance of proposed renewal processes with different types of HMMs. We apply these models to a global volcanic eruption record during the last 10000 years to analyze and demonstrate how we estimate the completeness of the record and the future hazard rate. All proposed models can be utilized to model other types of inhomogeneous processes with or without missing data.