Abstract Not Provided. / Thesis / Master of Science (MSc)

We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.

Earthquake statistics is a growing field of research with direct application to probabilistic seismic hazard evaluation. The earthquake process is a complex spatio-temporal phenomenon, and has been thought to be an example of the self-organised criticality (SOC) paradigm, in which events occur as cascades on a wide range of sizes, each determined by fine details of the rupture process. As a consequence, deterministic prediction of specific event sizes, locations, and times may well continue to remain elusive. However, probabilistic forecasting, based on statistical patterns of occurrence, is a much more realistic goal at present, and is being actively explored and tested in global initiatives. This thesis focuses on the temporal statistics of earthquake populations, exploring the uncertainties in various commonly-used procedures for characterising seismicity and explaining the origins of these uncertainties. Unlike many other SOC systems, earthquakes cluster in time and space through aftershock triggering. A key point in the thesis is to show that the earthquake inter-event time distribution is fundamentally bimodal: it is a superposition of a gamma component from correlated (co-triggered) events and an exponential component from independent events. Volcano-tectonic earthquakes at Italian and Hawaiian volcanoes exhibit a similar bimodality, which in this case, may arise as the sum of contributions from accelerating and decelerating rates of events preceding and succeeding volcanic activity. Many authors, motivated by universality in the scaling laws of critical point systems, have sought to demonstrate a universal data collapse in the form of a gamma distribution, but I show how this gamma form is instead an emergent property of the crossover between the two components. The relative size of these two components depends on how the data is selected, so there is no universal form. The mean earthquake rate—or, equivalently, inter-event time—for a given region takes time to converge to an accurate value, and it is important to characterise this sampling uncertainty. As a result of temporal clustering and non-independence of events, the convergence is found to be much slower than the Gaussian rate of the central limit theorem. The rate of this convergence varies systematically with the spatial extent of the region under consideration: the larger the region, the closer to Gaussian convergence. This can be understood in terms of the increasing independence of the inter-event times with increasing region size as aftershock sequences overlap in time to a greater extent. On the other hand, within this high-overlap regime, a maximum likelihood inversion of parameters for an epidemic-type statistical model suffers from lower accuracy and a systematic bias; specifically, the background rate is overestimated. This is because the effect of temporal overlapping is to mask the correlations and make the time series look more like a Poisson process of independent events. This is an important result with practical relevance to studies using inversions, for example, to infer temporal variations in background rate for time-dependent hazard estimation.

by Wing-yi, Tam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 74-77). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Cox and Lewis' Model and Weibull Process Model / Chapter Section 1 --- Nonhomogeneous Poisson Process (NHPP) --- p.5 / Chapter Section 2 --- Cox and Lewis' Model --- p.7 / Chapter Section 3 --- Weibull Process Model --- p.11 / Chapter Section 4 --- Test of NHPP --- p.14 / Chapter Chapter 3 --- Inference for Geometric Process with Inverse Gaussian Distribution / Chapter Section 1 --- Geometric Process (GP) --- p.18 / Chapter Section 2 --- Inverse Gaussian Distribution (IG) --- p.22 / Chapter Section 3 --- Simulation --- p.25 / Chapter Section 4 --- Conclusion --- p.33 / Chapter Chapter 4 --- Comparison Geometric Process Model and NHPP model in Fitting a Point Process / Chapter Section 1 --- Introduction --- p.34 / Chapter Section 2 --- Real Data Examples --- p.39 / Chapter Section 3 --- Conclusion --- p.45 / Tables and Graphs --- p.48 / Appendices --- p.71 / References --- p.74

No description available.

Thesis (Ph. D.)--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 76-78).

Thesis (Ph. D.)--Washington State University, August 2006. / Includes bibliographical references (p. 67-68).

This thesis introduces three new model-free tests for isotropy, equal distribution and random superposition in non-rectangular windows respectively. For isotropy, a bootstrap-type test is proposed. The corresponding test statistic assesses the discrepancy between the uniform distribution and the empirical normalised reduced second-order moment measure of a sector of fixed radius with increasing central angle. The null distribution of the discrepancy is then estimated by stochastic reconstruction, which generates bootstrap-type samples of point patterns that resemble the spatial structure of the given pattern. The new test is applicable for small sample sizes and is shown to have more robust powers to different choices of user-chosen parameter when compared with the asymptotic chi-squared test by Guan et al. (2006) in our simulation. For equal distribution, a model-free asymptotic test is introduced. The proposed test statistic compares the discrepancy between the empirical second-order product densities of the observed point patterns at some pre-chosen lag vectors. Under certain mild moment conditions and a weak dependence assumption, the limiting null distribution of the test statistic is the chi-squared distribution. Simulation results show that the new test is more powerful than the permutation test by Hahn (2012) for comparing point patterns with similar structures but different distributions. The new test for random superposition is a modification of the toroidal shift test by Lotwick and Silverman (1982). The idea is to extrapolate the pattern observed in a non-rectangular window to a larger rectangular region by the stochastic reconstruction so that the toroidal shift test can be applied. Simulation results show that the powers of the test applied to patterns with extrapolated points are remarkably higher than those of the test applied to the largest inscribed rectangular windows, with only slightly increased type I error rates. Real data sets are used to illustrate the advantages of the tests developed in this thesis over the existing tests in the literature.

Point processes are mechanisms that beget point patterns. Realisations of point processes are observed in many contexts, for example, locations of stars in the sky, or locations of trees in a forest. Inferring the mechanisms that drive point processes relies on the development of models that appropriately account for the dependencies inherent in the data. Fitting models that adequately capture the complex dependency structures in either space, time, or both is often problematic. This is commonly due to—but not restricted to—the intractability of the likelihood function, or computational burden of the required numerical operations. This thesis primarily focuses on developing point process models with some hierarchical structure, and specifically where this is a latent structure that may be considered as one of the following: (i) some unobserved construct assumed to be generating the observed structure, or (ii) some stochastic process describing the structure of the point pattern. Model fitting procedures utilised in this thesis include either (i) approximate-likelihood techniques to circumvent intractable likelihoods, (ii) stochastic partial differential equations to model continuous spatial latent structures, or (iii) improving computational speed in numerical approximations by exploiting automatic differentiation. Moreover, this thesis extends classic point process models by considering multivariate dependencies. This is achieved through considering a general class of joint point process model, which utilise shared stochastic structures. These structures account for the dependencies inherent in multivariate point process data. These models are applied to data originating from various scientific fields; in particular, applications are considered in ecology, medicine, and geology. In addition, point process models that account for the second order behaviour of these assumed stochastic structures are also considered.

We characterize the equilibrium behavior of a class of stochastic particle systems, where particles (representing customers, jobs, animals, molecules, etc.) enter a space randomly through time, interact, and eventually leave. The results are useful for analyzing the dynamics of randomly evolving systems including spatial service systems, species populations, and chemical reactions. Such models with interactions arise in the study of species competitions and systems where customers compete for service (such as wireless networks). The models we develop are space-time measure-valued Markov processes. Specifically, particles enter a space according to a space-time Poisson process and are assigned independent and identically distributed attributes. The attributes may determine their movement in the space, and whenever a new particle arrives, it randomly deletes particles from the system according to their attributes. Our main result establishes that spatial Poisson processes are natural temporal limits for a large class of particle systems. Other results include the probability distributions of the sojourn times of particles in the systems, and probabilities of numbers of customers in spatial polling systems without Poisson limits.