Nonparametric Approach to Analysis of Space-Time Data on Earthquake Occurrences

Sequential Kernel Estimation of the Conditional Intensity of Nonstationary Point Processes

“…As discussed by Choi and Hall (1999) and others, the parametric approach suffers by many drawbacks. First of all, there is an intrinsic difficulty in deriving reliable mathematical models from the geophysical theory.…”

“…This approach has been pursued in seismology by several authors: the seminal paper by Vere-Jones (1992) compared parametric and nonparametric approaches; the books by Bailey and Gatrell (1995) and Simonoff (1996) provided many numerical applications. Recently, Choi and Hall (1999) have included time in Kernel smoothers and Stock and Smith (2002a,b) apply adaptive estimation.…”

“…As a result they have produced maps of seismic risk which are fundamentally static, and may be useful only if events occur at a constant rate. On the other hand, Choi and Hall (1999) treat time as an additional spatial axis, and estimate the present value of the intensity with a two-sided smoother which also includes future events. This approach poses practical and conceptual problems because time, unlike space, has a causal (unidirectional) nature; moreover, the intensity function is usually defined in conditional form, with respect to the past history of the process.…”

Robust nonparametric estimation of the intensity function of point data

“…Zhuang et al 2002), the estimation of the conditional intensity on real data is usually performed with nonparametric methods. Omitting the depth coordinate z and assuming multiplicative kernels, Vere-Jones (1992), Bailey and Gatrell (1995), Choi and Hall (1999), and Smith (2002a, 2002b) focused on the kernel densitŷ…”

Hybrid kernel estimates of space–time earthquake occurrence rates using the epidemic-type aftershock sequence model

“…Direct kernel smoothing can then be used to estimate the intensity of this presumed Poisson process, and has the advantages of simplicity in computation and adaptability to the data. However, in fact, it is difficult for such intensity rate changes estimated by kernels to express very asymmetric, sharp or highly variable features as seen in Choi and Hall, (1999) and Grillenzoni (2005). These features mainly arise from the fact that earthquake occurrences are history dependent, as for example when a large earthquake occurs that is followed by many aftershocks.…”