Analysis of Time of Occurrence of Earthquakes: A Functional Data Approach
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Analysis of Time of Occurrence of Earthquakes: A Functional Data Approach A. Quintela-del-Río · F. Ferraty · P. Vieu
Received: 5 October 2009 / Accepted: 22 August 2010 / Published online: 1 July 2011 © International Association for Mathematical Geosciences 2011
Abstract There is no single method available for estimating the seismic risk in a given area, and as a result most studies are based on some statistical model. If we denote by Z the random variable that measures the maximum magnitude of earthquakes per unit time, the seismic risk of a value m is the probability that this value will be exceeded in the next time units, that is, R(m) = P (Z > m). Several approximations can be made by adjusting different theoretical distributions to the function R, assuming different distributions for the magnitude of earthquakes. A related method used to treat this problem is to consider the difference between the times of occurrence of consecutive earthquakes, or inter-event times. The hazard function, or failure rate function, of this variable measures the instantaneous risk of occurrence of a new earthquake, supposing that the last earthquake happened at time 0. In this paper, we will consider the estimation of the variable that measures the inter-event time and apply nonparametric techniques; that is, we do not consider any theoretical distribution. Moreover, because the stochastic process associated with this variable can sometimes be non-stationary, we condition each time by the previous ones. We then work with a multidimensional estimation, and consider each multidimensional variable as a functional datum. Functional data analysis deals with data consisting of curves or multidimensional variables. Nonparametric estimation can be applied to functional data, to describe the behavior of seismic zones and their associated instantaneous risk. The applications of estimation techniques are shown by applying them to two different regions and data catalogues: California and southern Spain.
A. Quintela-del-Río () Departamento de Matemáticas, Universidad de A Coruña, A Coruña, Spain e-mail: [email protected] F. Ferraty · P. Vieu Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
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Math Geosci (2011) 43:695–719
Keywords Conditional density · Conditional hazard · Earthquakes · Functional data · Inter-event times · Nonparametric estimation
1 Introduction This paper considers a seismic series as a group of earthquakes that have been occurring over a specific period of time, in a given geographical area. The earthquakes belonging to a seismic series are characterized as random variables, generally with four or five dimensions: latitude, longitude, depth of the epicenter (if measured), magnitude and time of occurrence. The advantage of using stochastic methods to analyze the behavior of seismic series is to understand the mathematical structure that governs the latter; the principal objective is to conduct an exhaustive descriptive study of the time and area where seism
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