The Magnitude Threshold and Missing and Pseudo Links in Markov Chains

A crisp step function is not an adequate threshold for studies of Markovian occurrence of large earthquakes, because it can lead to missing or pseudo links in an observed sequence that should be a Markov chain. A more realistic threshold is a fuzzy one where there is a transition magnitude band, located between those magnitudes that are too small for the earthquakes to be part of a Markovian process and those who are certainly large enough for the earthquakes to be part of it, where earthquakes may or may not be part of the process. This fuzzy threshold is described by a membership function that gives the probability of an earthquake with a given magnitude belonging to the process. We propose a membership function with probabilities in the transition band proportional to the seismic moment. To estimate empirical transition probabilities when considering a fuzzy magnitude threshold, we propose a counting strategy for the observed transitions and justify it through Monte Carlo simulations. The counting strategy is illustrated by application to the model from a previous seismic study of the Japan area by testing, through Monte Carlo simulations, how well the counting strategy results resemble optimum estimations of the transition probabilities. The simulations are also used to study the behavior of three Markovianity measures, and it is found that the peak values of these measures are not useful in identifying the true transition band, but that this band may be better identified by using the whole set of values taken by each measure for different transition band models. As an illustration, the measures were applied to real data from the previous study, a short set corresponding to a single realization, and found that the behavior of the measures does not agree with those expected from a crisp threshold, but agree, within the limitations of the data, with either a fuzzy threshold going from zero probability for magnitudes \(M\le 6.9\) to probability one for \(M\ge 7.2\) or from zero probability for magnitudes \(M\le 7.0\) to probability one for \(M\ge 7.2\).