Analysis of Time of Occurrence of Earthquakes: A Functional Data Approach |
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Authors: | A Quintela-del-Río F Ferraty P Vieu |
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Institution: | (1) Department of Mathematics, The College of William & Mary, Williamsburg, VA, USA; |
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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. |
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