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Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Further, most structural systems are likely to exhibit nonlinear and time-varying behavior when subjected to extreme events such as severe earthquake, wind and sea wave excitations. In such cases, a reliable identification approach is behavior and for assessing its reliability. Current work addresses two research themes in the field of stochastic engineering dynamics related to the above challenges. In the first part of the dissertation, the recently developedWiener Path Integral (WPI) technique for determining the joint response probability density function (PDF) of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as non-Gaussian and non-homogeneous stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a beam with a non-Gaussian and non-homogeneous Young’s modulus. Comparisons with MCS data demonstrate the accuracy of the technique. In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear time-variant multi-degree-of-freedom oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear subsystems. In this regard, the oscillator is transformed into an equivalent MISO system in the wavelet domain. Next, a recently developed L1-norm minimization procedure based on compressive sampling theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Finally, these wavelet coefficients are utilized to determine appropriately defined time- and frequency-dependent wavelet based frequency response functions and related oscillator parameters. A nonlinear time-variant system with fractional derivative elements is used as a numerical example to demonstrate the reliability of the technique even in cases of noise corrupted and incomplete data.

The first part of this thesis considers the residual empirical process of a nearly unstable long-memory time series. Chan and Ling [8] showed that the usual limit distribution of the Kolmogorov-Smirnov test statistics does not hold when the characteristic polynomial of the unstable autoregressive model has a unit root. A key question of interest is what happens when this model has a near unit root, that is, when it is nearly non-stationary. In this thesis, it is established that the statistics proposed by Chan and Ling can be extended. The limit distribution is expressed as a functional of an Orenstein-Uhlenbeck process that is driven by a fractional Brownian motion. This result extends and generalizes Chan and Ling's results to a nearly non-stationary long-memory time series. / The second part of the thesis investigates the weak convergence of weighted sums of random variables that are functionals of moving aver- age processes. A non-central limit theorem is established in which the Wiener integrals with respect to the Hermite processes appear as the limit. As an application of the non-central limit theorem, we examine the asymptotic theory of least squares estimators (LSE) for a nearly unstable AR(1) model when the innovation sequences are functionals of moving average processes. It is shown that the limit distribution of the LSE appears as functionals of the Ornstein-Uhlenbeck processes driven by Hermite processes. / Liu, Weiwei. / Adviser: Chan Ngai Hang. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 60-67). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

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Thesis (M. Sc. (Mathematical Statistics)) -- University of the North, 1978 / Refer to the document

Zero-inflated outcomes are common in biomedical studies, where the excessive zeros indicate some special but undetectable events. Quantile regression is potentially advantageous in analyzing zero-inflated outcomes due to two reasons. First, compared to parametric models such as the zero-inflated Poisson and two-part model, quantile regression gives robust and accurate estimation by avoiding likelihood specification and can capture the tail events and heterogeneity over the outcome distribution. Second, while the mean-based regression may be misinterpreted for a zero-inflated outcome, the interpretation of quantiles is naturally compatible with the underlying process that such an outcome intends to measure. Unfortunately, uncorrected linear quantile regression is not directly applicable because of two reasons. First, the feasibility of estimation and validity of inference of quantile regression require the conditional distribution of outcomes to be absolutely continuous, which is violated due to zero-inflation. Second, direct quantile regression implicitly assumes a constant chance to observe a positive outcome, but the degree of zero-inflation varies with the covariates in most cases. Thus the conditional quantile function of the outcome depends on the covariates in a nonlinear fashion. To analyze the zero-inflated outcomes by taking advantage of the merits of quantile regression, we propose a novel quantile regression framework that can address all the issues above. In the first part of this dissertation, we propose a two-part model that comprises a logistic regression for the probability of being positive, and a linear quantile regression for the positive part with subject-specific zero-inflation adjusted. Inference on the estimated conditional quantile and covariate effect are not trivial based on such a two-part model. We then develop an algorithm to achieve a consistent estimation of the conditional quantiles, while circumventing the unbounded variance at the quantile level where the conditional quantile changes from zero to positive. Furthermore, we develop an inference tool to determine the quantile treatment effect associated with a covariate at a given quantile level. We evaluate the proposed method and compare it with existing approaches by simulation studies and a real data analysis aimed at studying the risk factors for carotid atherosclerosis. In the second part, based on the proposed two-part model mentioned above, we develop ZIQRank, a zero-inflated quantile rank-score based test to detect the difference in distributions. The proposed test extends the local inference in the first part to a simultaneous one. It is powerful to handle zero-inflation and heterogeneity simultaneously. It comprises a valid test of logistic regression for the zero-inflation and rank-score based tests on multiple quantiles for the positive part with zero-inflation adjusted. The p-values are combined with a procedure selected according to the extent of zero-inflation and heterogeneity of the data. Simulation studies show that compared to existing tests, the proposed test has a higher power in detecting differential distributions. Finally, we apply the ZIQRank test to a human scRNA-seq data to study differentially expressed genes in Neoplastic and Regular cells. It successfully discovers a group of crucial genes associated with glioma, while the other methods fail to do so. In the third part, we extend the proposed two-part quantile regression model for zero-inflated outcomes and the ZIQRank test to analyze longitudinal data. Each part of the proposed two-part model is modified as a marginal longitudinal model (GEE), conditioning on the outcome at the previous time point and its zero/positive status. We apply the model and the test to study the effect of a recommender system aimed at boosting user engagement of a suite of smartphone apps designed for depressed patients. Our novel model framework demonstrates a dominating performance in model fitting, prediction, and critical feature detection, compared to the existing methods.

Graduation date: 1994

"August 25, 1950." / Bibliography: p. 75-76. / Army Signal Corps Contract No. W-36-039-sc-32037 Project No. 102B. Dept. of the Army Project No. 3-99-10-022.

by David C. Hoaglin, Stephen C. Peters. / Bibliography: leaf [5] / Caption title. "May, 1979." / National Science Foundation Grant SOC75-15702 National Science Foundation Grant MCS77-26902 National Science Foundation Grant MCS78-17697