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We develop a methodology to efficiently implement the reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithms of Green, applicable for example to model selection inference in a Bayesian framework, which builds on the \"dragging fast variables\" ideas of Neal. We call such algorithms annealed importance sampling reversible jump (aisRJ). The proposed procedures can be thought of as being exact approximations of idealized RJ algorithms which in a model selection problem would sample the model labels only, but cannot be implemented. Central to the methodology is the idea of bridging different models with fictitious intermediate models, whose role is to introduce smooth intermodel transitions and, as we shall see, improve performance. Efficiency of the resulting algorithms is demonstrated on two standard model selection problems and we show that despite the additional computational effort incurred, the approach can be highly competitive computationally. Supplementary materials for the article are available online.

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

We present a detection problem where several spatially distributed sensors observe Poisson signals emitted from a single radioactive source of unknown position. The measurements at each sensor are modeled by independent inhomogeneous Poisson processes. A method based on Bayesian change-point estimation is proposed to identify the location of the source’s coordinates. The asymptotic behavior of the Bayesian estimator is studied. In particular, the consistency and the asymptotic efficiency of the estimator are analyzed. The limit distribution and the convergence of the moments are also described. The similar statistical model could be used in GPS localization problems.

As an application of Stein's method for Poisson approximation, we prove rates of convergence for the tail probabilities of two scan statistics that have been suggested for detecting local signals in sequences of independent random variables subject to possible change-points. Our formulation deals simultaneously with ordinary and with large deviations.

Complex repairable systems with bathtub-shaped failure intensity will normally go through three periods in the lifecycle, which requires maintenance policies and management decisions accordingly. Therefore, the accurate estimation of change points of different periods has great significance. This paper addresses the challenge of change-point estimation in failure processes for repairable systems, especially for sustained and gradual processes of change. The paper proposes a sectional model composed of two non-homogeneous Poisson processes (NHPPs) to describe the bathtub-shaped failure intensity. In order to obtain the accurate change-point estimator, a novel hybrid method is developed combining bootstrap control charts with the sequential clustering approach. Through Monte Carlo simulations, the proposed change-point estimation method is compared with two powerful estimation procedures in various conditions. The results suggest that the proposed method performs effective and satisfactory for failure processes with no limits of distributions, changing ranges and sampling schemes. It especially provides higher precision and lower uncertainty in detecting small shifts of change. Finally, a case study analysing real failure data from a heavy-duty CNC machine tool is presented. The parameters of the proposed NHPP model are estimated. The change point of the early failure period and the random failure period is also calculated. These findings can contribute to determining the burn-in time in order to improve the reliability of the machine tool.

We generalize the classic change-point problem to a \"change-set\" framework: a spatial Poisson process changes its intensity on an unobservable random set. Optimal detection of the set is defined by maximizing the expected value of a gain function. In the case that the unknown change-set is defined by a locally finite set of incomparable points, we present a sufficient condition for optimal detection of the set using multiparameter martingale techniques. Two examples are discussed.

Change point problems are considered where at some unobservable time the intensity of a point process$(T_{n}),n\\in {\\Bbb N}$, has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of (Tn), ex post decision after observing the point process up to a fixed time$t_{\\ast}$and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.

The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool. The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.

Rotondi and Pagliano considered the problem of identifying periods with different seismicity levels as a multiple-changepoint problem: after having estimated the complete part of the historical catalog related to a seismic source zone, inference was made about the number and the location of k ≥ 1 changepoints of the occurrence rate in a sequence of n observations from a generalized Poisson process through a stepwise procedure. In this work, we extend such a method estimating the number of changepoints, their location, and other model parameters simultaneously. A fundamental role is played by new computational methods based on the simulation of Markov chains. In particular, we have applied a new version of the reversible jump Metropolis-Hastings algorithm to the data of some Italian seismic sources and we have identified periods with alternately higher and lower seismic activity levels.[PUBLICATION ABSTRACT]