deals with Markovian arrival processes for earthquake clustering analysis. The temporal variations of the seismicity rate in certain areas are considered as the manifestation of the physical mechanism behind the seismicity behavior, which exhibits periods of excitation (clustering) and periods of relative quiescence (background seismicity). The method used for this purpose is based on the application of a bivariate stochastic point process, the Markovian arrival process (MAP), (Nt, Jt)t∈R+, whose intensity function, λt, is modulated by the latent Markov process, Jt. The hidden states of the latter correspond to distinct occurrence rates of the counting process, Nt, which enables the modeling of changes in the seismicity rate.
Chapter 10, by Rodi Lykou and George Tsaklidis, deals with change point detection techniques on seismicity models. They present the detection techniques of the Poisson process, ETAS model and G–R law, and suggested approaches for seismicity rate changes, covering parametric and non-parametric tests and approaches lacking statistical hypothesis testing. Change detection in seismicity rates has been combined with the use of probabilistic forecasting models and simultaneous studies in the stress field and space domain. It constitutes a powerful tool for solving non-stationarity problems leading to promising results for the underlying mechanism of seismogenesis.
Chapter 11, by Vlad Stefan Barbu, Alex Karagrigoriou and Andreas Makrides, deals with semi-Markov processes for earthquake forecasts. In particular, the authors provide a different approach of earthquake forecasting using a special type of semi-Markov processes. The seismic zones are considered as the states of a semi-Markov process and a methodology is employed for estimating the transition probabilities of seismic events transit, from one specific seismic zone to another. The sojourn times in a certain state belong to a general class of distributions presenting the advantage that it is closed under minima and includes various continuous distributions like the exponential, Weibull, Pareto, Rayleigh and Erlang truncated exponential distribution.
The authors of this preface are grateful to all authors for their excellent collaboration.
December 2020
1
Kernel Density Estimation in Seismology
Stanisław LASOCKI
Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland
1.1. Introduction
The basis of statistical seismology is the probabilistic distribution of earthquake parameters. Unfortunately, the models of probabilistic distributions of earthquake parameters are mostly unknown, and those which are thought to be known are often disproved by more thorough testing.
On the other hand, increasing quality and decreasing prices of seismic monitoring systems, increasing density of seismic networks and the development of event picking tools cause the current seismic catalogs to be large and rising. This situation opens up the area for model-free approaches to probabilistic estimation and statistical inferences, which to be accurate, require considerable sample sizes.
The kernel density estimation is a model-free estimation of probability functions of continuous random variables. The estimation is carried out solely from sample data.
There is no attempt here to comprehensively present the kernel density estimation method. A detailed description and the discussion of the method can be found in the textbooks by Silverman (1986), Wand and Jones (1995), Scott (2015) and in a multitude of high-level research papers. The method is also implemented in Matlab, R and Python, among others, and in many statistical packages. In this chapter, the author presents how this method has been applied to selected problems of seismology. This presentation begins with a short and simplified theoretical introduction. The kernel density estimation has been fast developing from both theoretical and practical sides; hence, the techniques presented here do not aspire to be optimal. There is plenty of space for future modifications and developments.
In the case of a univariate random variable X, and a constant kernel, the kernel estimator of the actual probability density function (PDF) of X, fX(x), takes the form:
[1.1]
where {xi}, i = 1,.., n is the sample data, K(•) is the kernel function, which is a PDF symmetric about zero, and h is the bandwidth, whose value decides how much smoothing has been applied to the sample data.
In the presented seismological applications of the kernel density estimation, we use the normal kernel function:
For this kernel function, the kernel estimates of PDF and the cumulative distribution function (CDF,
[1.4]
where Ф(·) is the CDF of standard normal distribution.
As it will follow, the kernel method is used, among others, to estimate the magnitude distribution, whose distribution is exponential-like or light-tailed. Because of that, the tail values are sparse in a sample. Such sparsity can result in spurious irregularities in the estimate on tails, if a constant bandwidth is used. We can alleviate this problem by using an adaptive kernel with variable bandwidth. Because the estimates of magnitude distribution functions serve in the probabilistic seismic hazard analysis (PSHA), the quality of the estimate in the tail part is of utmost importance.
The adaptive kernel estimate of PDF used here takes the form:
and the estimate of CDF is:
[1.6]
where ωi are the local weights widening the bandwidth associated with the data points from the range where the data is sparse. The weights take the form:
where
