Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models

Irene Votsi, Nikolaos Limnios, George Tsaklidis and Eleftheria Papadimitriou have a new paper in Methodology and Computing in Applied Probability. In the paper major earthquakes in the Northern Aegean Sea are modelled by a semi-Markov process. Earthquakes of above 5.5 in magnitude are categorized into three groups, [5.5,5.6], [5.7,6.0] and [6.1,7.2] (7.2 being the highest record event in the dataset). The model assumes that probability the next earthquake is of a particular category depends only on the category of the current earthquake (embedded Markov chain), and given a particular transition i->j is to occur, the waiting times between events has some distribution f_{ij}(t). Hence the inter-event times will follow a mixture distribution dependent on i. Note that the process can renew in the same state (i.e. two earthquakes of similar magnitude can occur consecutively).
The dataset used is quite small, involving only 33 events.
The observed matrix of transitions is

Clearly something of basic interest would be whether there is any evidence of dependence on the previous earthquake type. A simple test gives 2.55 on 2 df suggesting no evidence against independence.
The authors instead proceed with a fully non-parametric method which uses the limited number of observations per transition type (i.e. between 1 & 6) to estimate the conditional staying time distributions. While a semi-Markov process of this type may well be appropriate for modelling earthquakes, the limited dataset available is insufficient to get usable estimates.

There are some odd features to the paper. Figure 3 in particular is puzzling as the 95% confidence intervals for the expected number of magnitude [6.1,7.2] earthquakes are clearly far too narrow. It also seems strange that there is essentially no mention of the flowgraph/saddlepoint approximation approach to analyzing semi-Markov processes of this nature.

Experimenting with the Coxian Phase-Type distribution to uncover suitable fits

Marshall and Zenga have a new paper in Methodology and Computing in Applied Probability. This considers the best approach to fitting Coxian phase-type distributions to fully observed survival or duration data. Several previous studies have shown that the EM algorithm, Nelder-Mead and quasi-Newton approaches give estimates that are highly dependent on initial values. In a series of simulations it is shown that the quasi-Newton algorithm is preferable to Nelder-Mead in converging to values closer to the true parameters. However, in the absence of a criterion for global convergence it still seems wise to consider several starting values to try to ensure a global rather than local optimum is obtained.

About Earthquake Forecasting by Markov Renewal Processes

There is a new paper in Methodology and Computing in Applied Probability by Garavaglia and Pavani. This concerns the forecasting of earthquakes. They analyse data on severe earthquakes in Turkey during the 20th century. Two approaches were considered. Firstly a two-state semi-Markov model is proposed to model the process. The states represent the occurrence of earthquakes of magnitude 5.5 - 6.3 and greater than 6.3 respectively. However, the probability that the next earthquake is of magnitude greater than 6.3 is seen to not depend on the magnitude of the last earthquake. Hence, instead a model based only on the inter-event times is considered using an exponential-Weibull mixture distribution.