In this thesis, we investigate how well we can reconstruct the best rank-? approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank matrix, its best low-rank approximations may still be reliably computed or estimated from a small number of its entries. This is especially relevant from a statistical viewpoint: the best low-rank approximations to a data matrix are often of more interest than itself because they capture the more stable and oftentimes more reproducible properties of an otherwise complicated data-generating model. In particular, we investigate two agnostic approaches: the first is based on spectral truncation; and the second is a projected gradient descent based optimization procedure. We argue that, while the first approach is intuitive and reasonably effective, the latter has far superior performance in general. We show that the error depends on how close the matrix is to being of low rank. Our results can be generalized to the spectral and entrywise error and provide flexible tools for the error analysis of the follow-up computation. Moreover, we derive a high-order decomposition of the error. With an explicit expression of the main error source, we obtain an improved estimate of the linear form. Both theoretical and numerical evidence is presented to demonstrate the effectiveness of the proposed approaches.

<p>Selecting the most probable multinomial or multivariate hypergeometric category isa multiple-decision selection problem. In this package, xed sampling and inversesampling are used for selecting the most probable category. This package aims atproviding functionality to calculate, display and plot the probabilities of correctlyselecting the most probable category under the least favorable configuration for thesetwo sampling types. A function for finding the specified smallest acceptable samplesize (or cell quota and expected sample size) is included as well.</p> / Master of Science (MSc)

Consider the context of selecting an optimal system from among a finite set of competing systems, based on a "stochastic" objective function and subject to multiple "stochastic" constraints. In this context, we characterize the asymptotically optimal sample allocation that maximizes the rate at which the probability of false selection tends to zero in two scenarios: first in the context of general light-tailed distributions, and second in the specific context in which the objective function and constraints may be observed together as multivariate normal random variates. In the context of general light-tailed distributions, we present the optimal allocation as the result of a concave maximization problem for which the optimal solution is the result of solving one of two nonlinear systems of equations. The first result of its kind, the optimal allocation is particularly easy to obtain in contexts where the underlying distributions are known or can be assumed, e.g., normal, Bernoulli. A consistent estimator for the optimal allocation and a corresponding sequential algorithm for implementation are provided. Various numerical examples demonstrate where and to what extent the proposed allocation differs from competing algorithms. In the context of multivariate normal distributions, we present an exact, asymptotically optimal allocation. This allocation is the result of a concave maximization problem in which there are at least as many constraints as there are suboptimal systems. Each constraint corresponding to a suboptimal system is a convex optimization problem. Thus the optimal allocation may easily be obtained in the context of a "small" number of systems, where the quantifier "small" depends on the available computing resources. A consistent estimator for the optimal allocation and a fully sequential algorithm, fit for implementation, are provided. The sequential algorithm performs significantly better than equal allocation in finite time across a variety of randomly generated problems. The results presented in the general and multivariate normal context provide the first foundation of exact asymptotically optimal sampling methods in the context of "stochastically" constrained simulation optimization on finite sets. Particularly, the general optimal allocation model is likely to be most useful when correlation between the objective and constraint estimators is low, but the data are non-normal. The multivariate normal optimal allocation model is likely to be useful when the multivariate normal assumption is reasonable or the correlation is high. / Ph. D.

<div>The simulation optimization problems refer to the nonlinear optimization problems whose objective function can be evaluated through stochastic simulations. We study two significant discrete simulation optimization problems in this thesis: Ranking and Selection (R&S) and Factor Screening (FS). Both R&S and FS are the "selection" problems defined upon a finite set of candidate systems or factors. They vary mainly in their objectives: the R&S problems is to find the "best" system(s) among all alternatives; whereas the FS is to select important factors that are critical to the stochastic systems. </div><div><br></div><div>In this thesis, we develop efficient sequential procedures for these two problems. For the R&S problem, we propose fully-sequential procedures for selecting the "best" systems with a guaranteed probability of correct selection (PCS). The main features of the stated methods are: (1) a Bonferroni-free model, these procedures overcome the conservativeness of the Bonferroni correction and deliver the exact probabilistic guarantee without overshooting; (2) asymptotic optimality, these procedures achieve the lower bound of average sample size asymptotically; (3) an indifference-zone-flexible formulation, these procedures bridge the gap between the indifference-zone formulation and the indifference-zone-free formulation so that the indifference-zone parameter is not indispensable but could be helpful if provided. We establish the validity and asymptotic efficiency for the proposed procedure and conduct numerical studies to investigates the performance under multiple configurations.</div><div><br></div><div>We also consider the multi-objective R&S (MOR&S) problem. To the best of our knowledge, the procedure proposed is the first frequentist approach for MOR&S. These procedures identify the Pareto front with a guaranteed probability of correct selection (PCS). In particular, these procedures are fully sequential using the test statistics built upon the Generalized Sequential Probability Ratio Test (GSPRT). The main features are: 1) an objective-dimension-free model, the performance of these procedures do not deteriorate as the number of objectives increases, and achieve the same efficiency as KN family procedures for single-objective ranking and selection problem; 2) an indifference-zone-flexible formulation, the new methods eliminate the necessity of indifference-zone parameter while makes use of the indifference-zone information if provided. A numerical evaluation demonstrates the validity efficiency of the new procedure.</div><div><br></div><div>For the FS problem, our objective is to identify important factors for simulation experiments with controlled Family-Wise Error Rate. We assume a Multi-Objective first-order linear model where the responses follow a multivariate normal distribution. We offer three fully-sequential procedures: Sum Intersection Procedure (SUMIP), Sort Intersection Procedure (SORTIP), and Mixed Intersection procedure (MIP). SUMIP uses the Bonferroni correction to adjust for multiple comparisons; SORTIP uses the Holms procedure to overcome the conservative of the Bonferroni method, and MIP combines both SUMIP and SORTIP to work efficiently in the parallel computing environment. Numerical studies are provided to demonstrate the validity and efficiency, and a case study is presented.</div>

This thesis develops adaptive simulation control techniques that differentiate between competing system configurations. Here, a system is a real world environment under analysis. In this context, proposed modifications to a system denoted by different configurations are evaluated using large-scale hybrid simulation. Adaptive control techniques, using ranking and selection methods, compare the relative worth of competing configurations and use these comparisons to control the number of required simulation observations. Adaptive techniques necessitate embedded statistical computations suitable for the variety of data found in detailed simulations, including hybrid and agent-based simulations. These embedded statistical computations apply efficient sampling methods to collect data from simulations running on a network of workstations. The National Airspace System provides a test case for the application of these techniques to the analysis and design of complex systems, implemented here in the Reconfigurable Flight Simulator, a large-scale hybrid simulation. Implications of these techniques for the use of simulation as a design activity are also presented.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2004. / Includes bibliographical references (p. 65).

Thesis (M.Sc.)--Memorial University of Newfoundland, 2001. / Restricted until October 2004. Bibliography: leaves 99-101.

published_or_final_version / Statistics / Master / Master of Philosophy

Thesis (M.Sc. (Applied Statistics)) --University of Pretoria, 2007. / Abstract in English. Includes bibliographical references.

Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 98-103).