This thesis introduces three new model-free tests for isotropy, equal distribution and random superposition in non-rectangular windows respectively. For isotropy, a bootstrap-type test is proposed. The corresponding test statistic assesses the discrepancy between the uniform distribution and the empirical normalised reduced second-order moment measure of a sector of fixed radius with increasing central angle. The null distribution of the discrepancy is then estimated by stochastic reconstruction, which generates bootstrap-type samples of point patterns that resemble the spatial structure of the given pattern. The new test is applicable for small sample sizes and is shown to have more robust powers to different choices of user-chosen parameter when compared with the asymptotic chi-squared test by Guan et al. (2006) in our simulation. For equal distribution, a model-free asymptotic test is introduced. The proposed test statistic compares the discrepancy between the empirical second-order product densities of the observed point patterns at some pre-chosen lag vectors. Under certain mild moment conditions and a weak dependence assumption, the limiting null distribution of the test statistic is the chi-squared distribution. Simulation results show that the new test is more powerful than the permutation test by Hahn (2012) for comparing point patterns with similar structures but different distributions. The new test for random superposition is a modification of the toroidal shift test by Lotwick and Silverman (1982). The idea is to extrapolate the pattern observed in a non-rectangular window to a larger rectangular region by the stochastic reconstruction so that the toroidal shift test can be applied. Simulation results show that the powers of the test applied to patterns with extrapolated points are remarkably higher than those of the test applied to the largest inscribed rectangular windows, with only slightly increased type I error rates. Real data sets are used to illustrate the advantages of the tests developed in this thesis over the existing tests in the literature.

We give asymptotics near the boundary for the distribution of the first exit time of the isotropic alpha-stable Levy process on bounded Lipschitz sets in real euclidean space. These asymptotics bear some relation to the existence of limits in the Yaglom sense of alpha-stable processes. Our approach relies on the uniform integrability of the ratio of Green functions on bounded Lipschitz sets. We use bounds for the heat remainder to give the first two terms in the small time asymptotic expansion of the trace of the heat kernel of unimodal Levy processes satisfying some weak scaling conditions on bounded Lipschitz domains.

Point processes are mechanisms that beget point patterns. Realisations of point processes are observed in many contexts, for example, locations of stars in the sky, or locations of trees in a forest. Inferring the mechanisms that drive point processes relies on the development of models that appropriately account for the dependencies inherent in the data. Fitting models that adequately capture the complex dependency structures in either space, time, or both is often problematic. This is commonly due to—but not restricted to—the intractability of the likelihood function, or computational burden of the required numerical operations. This thesis primarily focuses on developing point process models with some hierarchical structure, and specifically where this is a latent structure that may be considered as one of the following: (i) some unobserved construct assumed to be generating the observed structure, or (ii) some stochastic process describing the structure of the point pattern. Model fitting procedures utilised in this thesis include either (i) approximate-likelihood techniques to circumvent intractable likelihoods, (ii) stochastic partial differential equations to model continuous spatial latent structures, or (iii) improving computational speed in numerical approximations by exploiting automatic differentiation. Moreover, this thesis extends classic point process models by considering multivariate dependencies. This is achieved through considering a general class of joint point process model, which utilise shared stochastic structures. These structures account for the dependencies inherent in multivariate point process data. These models are applied to data originating from various scientific fields; in particular, applications are considered in ecology, medicine, and geology. In addition, point process models that account for the second order behaviour of these assumed stochastic structures are also considered.

Please read the abstract in the front section of this document / Dissertation (MSc (Mathematics))--University of Pretoria, 2005. / Mathematics and Applied Mathematics / unrestricted

M.Ing. (Electrical and Electronic Engineering) / This treatise presents an investigation into the application of multivariable frequency domain techniques in the modelling and control of a helicopter aircraft in forward flight. The presentation is structured in the following sectioned format: I Hypotheses are stated which deal with the use of linear, multivariable, frequency domain theory in the modelling and control of helicopter aircraft. II The stated hypotheses are investigated by the application of relevant theories and techniques to a reference case plant - a single rotor helicopter in forward flight. III Conclusions drawn from the results are used to assess the validity of the hypotheses. The subject matter of the presentation may be summarized as follows: The hypotheses are initially placed in perspective by a discussion of the incentives for their formulation. In essence, the hypotheses state that helicopter dynamics, in a multivariable systems characterization, can be modelled and an appropriate flight control system designed by the use of linear frequency domain theory. The plant in reference to which the hypotheses are investigated is a single rotor utility helicopter - the Aerospatiale Alouette III. A single flight condition - a typical cruising condition - is considered. A comprehensive, nonlinear digital computer simulation of the aircraft is used as a substitute for the actual plant in the execution of the modelling and control design processes. The plant is modelled in terms of a linear model structure, in the form of the frequency response function, by linearization of its highly nonlinear dynamics about an operating point (datum flight condition). The frequency response function model parameters are identified by power spectral density analysis procedures. This method, based on random signal excitation of the plant, provides a valuable quantitative measure of the accuracy of the linearization performed in the identification. The measure, the coherence function, is used as a criterion for the robustness required of a control system of which the design is based on a linear model of a nonlinear plant.

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The travel-time flow relationship is not always increasing in nature, it is very difficult to predict precisely. Traditional method fails to replicate this unique conditions. Until millennium, although various researchers and practitioners have given much attention to develop travel-time flow relationships, the advancement to improve travel-time flow relationships was not substantial. The knowledge about the travel-time flow relationship is not commensurate with or parallel to the advancement of new knowledge in other fields. After millennium, most investigators did not devote enough attention to create new knowledge, except for application and performance evaluation of the existing knowledge. Therefore, it is necessary to provide a new theoretical and methodological advancement in travel-time flow relationship. Consequentially, this research proposes a new methodology, which considers stochastic behavior of travel-time flow relationship with probabilistic Bayesian statistics and logistic growth mapping techniques. This research moderately improves the travel-time flow relationship. The unique contribution of this research is that the proposed methods outperforms the existing traditional travel-time flow theory, assumptions, and modeling techniques. The results shows that the proposed model is considerably a good candidate for travel-time predictions. The proposed model performs 36 percent better and accurate travel-time predictions in compared to the existing models. Furthermore, travel-time flow relationship need capacity and free-flow speed estimations. Traditionally, practice of capacity estimation is mostly practical, subjective, and not steady-state capacity. Therefore, a robust and stable capacity-estimation method was developed to eliminate the subjectivity of capacity estimation. The proposed model shows robust and capable of replicating steady-state capacity estimation. The free-flow speed estimation should relate to the traffic-flow speed model while the density is zero. Therefore, this research investigates the existing deterministic speed-density models and recommends a better methodology in free-flow speed estimation. This research presents how the undefined practice of free-flow speed selection can be sensitive. Additionally, finding suitable concurrent travel-time data and traffic volume is crucial and very challenging. To collect concurrent data, this research investigates and develops several technologies such as crowdsource, web app, virtual sensor method, test vehicle, smartphone, global positioning system, and utilized several state and local agencies data collection efforts. Keywords: Travel-Time Flow, Travel-Time Delay, Volume-Delay Function, Travel Time, Origin-Destination Survey, Travel Demand Model, Travel Data Collection, Transportation Survey, Internet Sensor, Crowdsourcing, Virtual Sensor Method, VSM, Transportation Planning, GPS, Smartphone, Loop Detector, Travel -Time Prediction, Travel-Speed Prediction, TDM, Bayesian Inference, Logistic Growth Function.

Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations. We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.

This research considers a single-item two-echelon supply chain facing a sequence of stochastic bulky customer demand with random order inter-arrival time and random demand size. The demand process is a general renewal process and the cost functions for both parties involve the renewal function and its integral. The complexity of the general renewal function causes the computational intractability in deciding the optimal order quantities, so approximations for the renewal function and its integral are introduced to address the computational complexity. Asymptotic expansions are commonly used in the literature to approximate the renewal function and its integral when the optimal decisions are relatively large compared to the mean of the inter-renewal time. However, the optimal policies do not necessarily fall in the asymptotic region. So the use of asymptotic expansions to approximate the renewal function and its integral in the cost functions may cause significant errors in decision making. To overcome the inaccuracy of the asymptotic approximation, this research proposes a modified approximation. The proposed approximation provides closed form functions for the renewal function and its integral which could be applied to various optimization problems such as inventory planning, supply chain management, reliability and maintenance. The proposed approximations are tested with commonly used distributions and applied to an application in the literature, yielding good performance. By applying the proposed approximation method to the supply chain cost functions, this research obtains the optimal policies for the decentralized and the centralized cases. The numerical results provide insights into the cost savings realized by the centralization of the supply chain compared to the decentralized case. Furthermore, this research investigates coordination schemes for the decentralized case to improve the utilities of parties. A cost sharing mechanism in which the vendor offers the retailer a contract as a compensation of implementing vendordesired inventory policy is investigated. The sharing could be realized by bearing part of the retailer’s inventory holding cost or fixed cost. The contract is designed to minimize the vendors cost while satisfying the individual rationality of the retailer. Other forms of coordination mechanisms, such as the side payment and delayed payment, are also discussed.

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