Thesis (Ph. D.)--University of Chicago, Dept. of Physics, August 1998. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph.D.)--University of Florida, 2005. / Title from title page of source document. Document formatted into pages; contains 158 pages. Includes vita. Includes bibliographical references.

Thesis (Ph.D.)--Ohio University, March, 2005. / Title from PDF t.p. Includes bibliographical references (p. 152-159)

Thesis (M.Sc.)(Mathematics)--University of Pretoria, 2007. / Includes summary. Includes bibliographical references. Available on the Internet via the World Wide Web.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 48-49). Also available in electronic version.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Balasubramaniam Natarajan / Stochastic hybrid system (SHS) is a class of dynamical systems that experience interaction of both discrete mode and continuous dynamics with uncertainty. State estimation for SHS has attracted research interests for decades with Kalman filter based solutions dominating the area. Mode-based Kalman filter is an extended version of the traditional Kalman filter for SHS. In general, as Kalman filter is unbiased for non-hybrid system estimation, prior research efforts primarily focus on the behavior of error covariance. In SHS state estimate, mode mismatch errors could result in a bias in the mode-based Kalman filter and have impacts on the continuous state estimation quality. The relationship between mode mismatch errors and estimation stability is an open problem that this dissertation attempts to address. Specifically, the probabilistic model of mode mismatch errors can be independent and identically distributed (i.i.d.), correlated across different modes and correlated across time. The proposed approach builds on the idea of modeling the bias evolution as a transformed system. The statistical convergence of the bias dynamics is then mapped to the stability of the transformed system. For each specific model of the mode mismatch error, the system matrix of the transformed system varies which results in challenges for the stability analysis. For the first time, the dissertation derives convergence conditions that provide tolerance regions for the mode mismatch error for three mode mismatch situations. The convergence conditions are derived based on generalized spectral radius theorem, Lyapunov theorem, Schur stability of a matrix polytope and interval matrix method. This research is fundamental in nature and its application is widespread. For example, the spatially and timely correlated mode mismatch errors can effectively capture cyber-attacks and communication link impairments in a cyber-physical system. Therefore, the theory and techniques developed in this dissertation can be used to analyze topology errors in any networked system such as smart grid, smart home, transportation, flight management system etc. The main results provide new insights on the fidelity in discrete state knowledge needed to maintain the performance of a mode-based Kalman filter and provide guidance on design of estimation strategies for SHS.

An attempt is made in this thesis to study some stochastic models of both reliability and inventory systems with reference to the following aspects: (i) the confidence limits with the introduction of common-cause failures. (ii) the Erlangian repair time distributions. (iii) the product interactions and demand interactions. (iv) the products are perishable. This thesis contains six chapters. Chaper 1 is introductory in nature and gives a review of the literature and the techniques used in the analysis of reliability systems. Chapter 2 is a study of component common-cause failure systems. Such failures may greatly reduce the reliability indices. Two models of such systems (series and parallel) have been studied in this chapter. The expressions such as, reliability, availability and expected number of repairs have been obtained. The confidence limits for the steady state availability of these two systems have also been obtained. A numerical example illustrates the results. A 100 (1 - a) % confidence limit for the steady state availability of a two unit hot and warm standby system has been studied, when the failure of an online unit is constant and the repair time of a failed unit is Erlangian. The general introduction of various inventory systems and the techniques used in the analysis of such systems have been explained in chapter 4. Chapter 5 provides two models of two component continuous review inventory systems. Here we assume that demand occurs according to a poisson process and that a demand can be satisfied only if both the components are available in inventory. Back-orders are not permitted. The two components are bought from outside suppliers and are replenished according to (s, S) policy. In model 1 we assume that the lead-time of the components follow an exponential distribution. By identifying the inventory level as a Markov process, a system of difference-differential equations at any time and the steady-state for the state of inventory level are obtained. Tn model 2 we assume that the lead-time distribution of one product is arbitrary and the other is exponential. Identifying the underlying process as a semi-regenerative process we find the stationary distribution of the inventory level. For both these models, we find out the performance measures such as the mean stationary rate of the number of lost demands, the demands satisfied and the reorders made. Numerical examples for the two models are also considered. Chaper 6 is devoted to the study of a two perishable product inventory model in which the products are substitutable. The perishable rates of product 1 and product 2 are two different constants. Demand for product 1 and product 2 follow two independent Poisson processes. For replenishment of product 1 (s, S) ordering policy is followed and the associated lead-time is arbitrary. Replenishment of product 2 is instantaneous. A demand for product 1 which occurs during its stock-out period can be substituted by product 2 with some probability. Expressions are derived for the stationary distribution of the inventor}' level by identifying the underlying stochastic process as a semi-regenerative process. An expression for the expected profit rate is obtained. A numerical illustration is provided and an optimal reordering level maximising the profit rate is also studied. To sum up, this thesis is an effort to improve the state the of art of (i) complex reliability systems and their estimation study (ii) muitiproduct inventory systems. The salient features of the thesis are: (i) Analysis of a two-component reliability system with common-cause failures. (ii) Estimation study of a complex system in which the repair time for both hot standby and warm standby systems are assumed to be Eriangian. (iii) A multi-product continuous review inventory system with product interaction, with a (s, S) policy. (iv) Introduction of the concept of substitutability for products. (v) Derivation of expressions for various statistical measures. (vi) Effective use of the regeneration point technique in deriving various measures for both reliability and inventory systems. (vii) Illustration of the various results by extensive numerical work. (vii) Consideration of relevant optimization problems. / Mathematical Sciences / PhD (Statistics)

Among all stochastic processes with independent increments, essentially only Brownian motion and Poisson process have a chaotic representation property. In the case of a Levy process, several approaches have been proposed in order to construct an orthogonal decomposition of the corresponding L2-space. In this dissertation, we deal with orthogonal (chaotic) decompositions for generalized processes with independent values. We do not suppose stationarity of the process, as a result the Levy measure of the process depends on points of the space. We first construct, in Chapter 3, a unitary isomorphism between a certain symmetric Fock space and the space L2 (D',mu). Here D' is a co-nuclear space of generalized functions (distributions), and mu is a generalized stochastic process with independent values. This isomorphism is constructed by employing the projection spectral theorem for an (uncountable) family of commuting self-adjoint operators. We next derive, in Chapter 4, a counterpart of the Nualart Schoutens decomposition for generalized stochastic process with independent values. Our results here extend those in the papers of Nualart Schoutens and Lytvynov. In Chapter 5, we construct an orthogonal decomposition of the space L2 (D',mu) in terms of orthogonal polynomials on D'. We observe a deep relation between this decomposition and the results of the two previous chapters. Finally, in Chapter 6, we briefly discuss the situation of the generalized stochastic processes of Meixner's type.

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The theory of ordinary factor analysis and its application by means of software packages do not make provision for data sampled from populations with hierarchical structures. Since data are often obtained from such populations - educational data for example ¬the lack of procedures to analyse data of this kind needs to be addressed. A review of the ordinary factor analysis model and maximum likelihood estimation of the parameters in exploratory and confirmatory models is provided, together with practical applications. Subsequently, the concept of hierarchically structured populations and their models, called multilevel models, are introduced. A general framework for the estimation of the unknown parameters in these models is presented. It contains two estimation procedures. The first is the marginal maximum likelihood method in which an iterative expected maximisation approach is used to obtain the maximum likelihood estimates. The second is the Fisher scoring method which also provides estimated standard errors for the maximum likelihood parameter estimates. For both methods, the theory is presented for unconstrained as well as for constrained estimation. A two-stage procedure - combining the mentioned procedures - is proposed for parameter estimation in practice. Multilevel factor analysis models are introduced next, and subsequently a particular two-level factor analysis model is presented. The general estimation theory that was presented earlier is applied to this model - in exploratory and confirmatory analysis. First, the marginal maximum likelihood method is used to obtain the equations for determining the parameter estimates. It is then shown how an iterative expected max¬imisation algorithm is used to obtain these estimates in unconstrained and constrained optimisation. This method is applied to real life data using a FORTRAN program. Secondly, equations are derived by means of the Fisher scoring method to obtain the maximum likelihood estimates of the parameters in the two-level factor analysis model for exploratory and confirmatory analysis. A FORTRAN program was written to apply this method in practice. Real life data are used to illustrate the method. Finally, flowing from this research, some areas for possible further research are pro¬posed. / Thesis (PhD (Applied Statistics))--University of Pretoria, 2007. / Statistics / unrestricted