335 research outputs found
Large deviations of stochastic systems and applications
This dissertation focuses on large deviations of stochastic systems with applications to optimal control and system identification. It encompasses analysis of two-time-scale Markov processes and system identification with regular and quantized data. First, we develops large deviations principles for systems driven by continuous-time Markov chains with twotime scales and related optimal control problems. A distinct feature of our setup is that the Markov chain under consideration is time dependent or inhomogeneous. The use of two time-scale formulation stems from the effort of reducing computational complexity in a wide variety of applications in control, optimization, and systems theory. Starting with a rapidly fluctuating Markovian system, under irreducibility conditions, both large deviations upper and lower bounds are established first for a fixed terminal time and then for time-varying dynamic systems. Then the results are applied to certain dynamic systems and LQ control problems.
Second, we study large deviations for identifications systems. Traditional system identification concentrates on convergence and convergence rates of estimates in mean squares, in distribution, or in a strong sense. For system diagnosis and complexity analysis, however, it is essential to understand the probabilities of identification errors over a finite data window. This paper investigates identification errors in a large deviations framework. By considering both space complexity in terms of quantization levels and time complexity with respect to data window sizes, this study provides a new perspective to understand the fundamental relationship between probabilistic errors and resources that represent data sizes in computer algorithms, sample sizes in statistical analysis, channel bandwidths in communications, etc.
This relationship is derived by establishing the large deviations principle for quantized identification that links binary-valued data at one end and regular sensors at the other. Under some mild conditions, we obtain large deviations upper and lower bounds. Our results accommodate
independent and identically distributed noise sequences, as well as more general classes of mixing-type noise sequences. Numerical examples are provided to illustrate the theoretical results
Large Deviations for Small Noise Diffusions in a Fast Markovian Environment
A large deviation principle is established for a two-scale stochastic system
in which the slow component is a continuous process given by a small noise
finite dimensional It\^{o} stochastic differential equation, and the fast
component is a finite state pure jump process. Previous works have considered
settings where the coupling between the components is weak in a certain sense.
In the current work we study a fully coupled system in which the drift and
diffusion coefficient of the slow component and the jump intensity function and
jump distribution of the fast process depend on the states of both components.
In addition, the diffusion can be degenerate. Our proofs use certain stochastic
control representations for expectations of exponential functionals of finite
dimensional Brownian motions and Poisson random measures together with weak
convergence arguments. A key challenge is in the proof of the large deviation
lower bound where, due to the interplay between the degeneracy of the diffusion
and the full dependence of the coefficients on the two components, the
associated local rate function has poor regularity properties.Comment: 42 page
Engineering applications of Bayesian statistical methods
This dissertation makes Bayesian contributions to engineering statistics in three basic areas. These are methods for combining information, modeling repairable system reliability, and designing experiments.;A recursive Bayesian hierarchical model (RBHM) is presented. An RBHM can be used to combine information from physical data, data from a computer model of a process, and experts. In an example involving a fluidized bed process, an RBHM is used to estimate location and scale biases of one source of information for another.;The need to document the reliability of the Blue Mountain supercomputer motivates the work on system reliability. A detailed reliability analysis of this supercomputer is presented, using a Bayesian hierarchical nonhomogeneous Poisson process model. Further, some flexible new families of intensities for nonhomogeneous Poisson processes are defined and Bayes inference for them is discussed.;Finally, the problem of estimating expected information gain for planned data collection is considered. Two methods of estimation are applied to the so called random fatigue-limit model, a 5 parameter model important in some materials engineering applications
Genealogical particle analysis of rare events
In this paper an original interacting particle system approach is developed
for studying Markov chains in rare event regimes. The proposed particle system
is theoretically studied through a genealogical tree interpretation of
Feynman--Kac path measures. The algorithmic implementation of the particle
system is presented. An estimator for the probability of occurrence of a rare
event is proposed and its variance is computed, which allows to compare and to
optimize different versions of the algorithm. Applications and numerical
implementations are discussed. First, we apply the particle system technique to
a toy model (a Gaussian random walk), which permits to illustrate the
theoretical predictions. Second, we address a physically relevant problem
consisting in the estimation of the outage probability due to polarization-mode
dispersion in optical fibers.Comment: Published at http://dx.doi.org/10.1214/105051605000000566 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Nonasymptotic analysis of adaptive and annealed Feynman-Kac particle models
Sequential and quantum Monte Carlo methods, as well as genetic type search
algorithms can be interpreted as a mean field and interacting particle
approximations of Feynman-Kac models in distribution spaces. The performance of
these population Monte Carlo algorithms is strongly related to the stability
properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these
models in terms of Dobrushin ergodic coefficients of the reference Markov
transitions and the oscillations of the potential functions. Sufficient
conditions for uniform concentration inequalities w.r.t. time are expressed
explicitly in terms of these two quantities. We provide an original
perturbation analysis that applies to annealed and adaptive Feynman-Kac models,
yielding what seems to be the first results of this kind for these types of
models. Special attention is devoted to the particular case of Boltzmann-Gibbs
measures' sampling. In this context, we design an explicit way of tuning the
number of Markov chain Monte Carlo iterations with temperature schedule. We
also design an alternative interacting particle method based on an adaptive
strategy to define the temperature increments. The theoretical analysis of the
performance of this adaptive model is much more involved as both the potential
functions and the reference Markov transitions now depend on the random
evolution on the particle model. The nonasymptotic analysis of these complex
adaptive models is an open research problem. We initiate this study with the
concentration analysis of a simplified adaptive models based on reference
Markov transitions that coincide with the limiting quantities, as the number of
particles tends to infinity.Comment: Published at http://dx.doi.org/10.3150/14-BEJ680 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Essays in Applied Bayesian Analysis
With continuing rapid developments in computational power, Bayesian statistical methods, because of their user-friendliness and estimation capabilities, have become increasingly popular in a considerable variety of application fields. In this thesis, applied Bayesian methodological topics and empirical examples focusing on nonhomogeneous hidden Markov models (NHMMs) and measurement error models are explored in three chapters. In the first chapter, a subsequence-based variational Bayesian inference framework for NHMMs is proposed in order to address the computational problems encountered when analyzing datasets containing long sequences. The second chapter concentrates on measurement error models, where a Bayesian estimation procedure is proposed for the partial potential impact fraction (pPIF) with the presence of measurement error. The third chapter focuses on an empirical application in marketing, where a coupled nonhomogeneous hidden Markov model (CNHMM) is introduced to provide a novel framework for customer relationship management
Applications of stochastic modeling in air traffic management : Methods, challenges and opportunities for solving air traffic problems under uncertainty
In this paper we provide a wide-ranging review of the literature on stochastic modeling applications within aviation, with a particular focus on problems involving demand and capacity management and the mitigation of air traffic congestion. From an operations research perspective, the main techniques of interest include analytical queueing theory, stochastic optimal control, robust optimization and stochastic integer programming. Applications of these techniques include the prediction of operational delays at airports, pre-tactical control of aircraft departure times, dynamic control and allocation of scarce airport resources and various others. We provide a critical review of recent developments in the literature and identify promising research opportunities for stochastic modelers within air traffic management
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