Mathematics (Faculty of) http://hdl.handle.net/10012/9924 2021-06-13T00:02:32Z 2021-06-13T00:02:32Z Comparing Distributions with the Probability of Agreement Dadbin, Maziar http://hdl.handle.net/10012/17087 2021-06-04T02:34:05Z 2021-06-03T00:00:00Z Comparing Distributions with the Probability of Agreement Dadbin, Maziar In this thesis we adapt the probability of agreement (PoA) methodology for the comparison of distributions. Most of the commonly used methods for comparing distributions are rooted in hypothesis testing where decisions are made using p-values. The proposed methodology, however, provides a more context-driven comparison by accounting for practically important differences. Two situations are considered: first, the one-sample comparison problem in which we have observed one sample and interest lies in determining whether the sample comes from a given known distribution. Second, we consider the two-sample comparison of distributions in which we have observed two independent samples and interest lies in determining whether these samples have the same distribution. The Horvitz-Thompson estimator is used to estimate the cumulative distribution function(s) corresponding the sample(s) under comparison and the asymptotic normality of the Horvitz-Thompson estimator is used to estimate the PoA. Confidence intervals (CIs) are also determined for the estimated PoA so as to quantify estimation uncertainty. We develop two methods for calculating CIs: one based on asymptotic normality and the delta-method and the other based on the bootstrap. To illustrate the application and interpretation of the methodology, we consider both real world and simulated examples. We also conduct a simulation study that evaluates the bias and variance of the PoA estimator as well as the coverage of the associated CIs. Finally we propose the relative density methodology as a graphical supplement that provides further information about the similarities and differences between the distributions under comparison. In summary, the contributions of this thesis are (1), the generalization of the PoA methodology to the one- and two-sample comparison of distributions, and (2), the suggestion of using the relative density and the PoA methodologies in tandem to gain more thorough information about the similarities and differences between the distributions under comparison. 2021-06-03T00:00:00Z A Floating Ball and Two Asymptotic Problems in Capillarity Chen, Hanzhe http://hdl.handle.net/10012/17085 2021-06-03T02:35:07Z 2021-06-02T00:00:00Z A Floating Ball and Two Asymptotic Problems in Capillarity Chen, Hanzhe The study of capillary phenomena can be traced back to the age of Aristotle. In this thesis, a floating ball and two asymptotic problems in capillarity are considered, all of which include surface tension and gravity. The first problem, the ascent of fluid surface outside a narrow vertical circular tube has been studied for decades. Lo obtained a five-term asymptotic expansion of the fluid height near the boundary for the small Bond number using the method of matched asymptotic expansions. Miersemann gave a rigorous proof of a two-term asymptotic expansion but the error bound he obtained is inferior to Lo's. We reconsider this problem and our goal is to establish a rigorous approach to improve Miersemann's error bound. We construct two piecewise smooth approximate solutions. Each approximate solution consists of an inner solution and an outer solution. The first approximate solution with its inner solution having zero mean curvature is shown as an upper bound. The second approximate solution with its inner solution having constant mean curvature is shown as a lower bound. The approximations are optimized in terms of the transition radius $q$ by use of Olver's theorem. This establishes the two-term asymptotic expansion. Its error bound is an improvement of Miersemann's but is inferior to Lo's. A modification of the outer solution helps improve our error bound. However, we did not achieve Lo's error bound. Interest in floating objects goes back to antiquity. In recent research, many examples of multiple equilibrium configurations have been found with or without surface tension and gravity. We consider a ball floating on an unbounded reservoir. The floating configuration is assumed to be radially symmetric. By a result of Elcrat, Neel and Siegel, the fluid interface is determined by the attachment radius $r_0$ and inclination angle $\psi_0$. Both of these are given in terms of the attachment angle $\phi_0$. However, the zero solution is not included in the parametric solution. So, the graph description of the fluid height is considered, as well. We develop $C^1$ smoothness of the attachment height $u_0$ with respect to $\phi_0$. This requires an extension of Vogel's description of solutions and monotonicity results. As a by-product, Vogel's conjecture on the smoothness of the envelope of exterior solutions is shown. In the study of the number of equilibria and their stability, both force and energy approaches are considered. We classify forces and energies and establish a relation between the total force and the total energy. This requires determining the asymptotic expansion of the interface as the inclination angle tends to zero, which is achieved through the use of Levinson's theorem. A critical point of the total energy can be either a force balanced point or a critical point of the height of the center. Both the total force and the center height contain $u_0$, which has to be found numerically by the shooting method. In order to understand the behavior of the total force and the height curves, both asymptotic analysis and numerical tests are employed. We investigate the limiting behavior of the total force and the height curves for small and large Bond number or when the attachment angle tends to its end points. We perform thousands of numerical tests with different values of Bond numbers and contact angles. Combined with the numerical observations and the results from the asymptotic analysis, we conjecture that there are at most two force balanced points. If there is only one force balanced point, it must be stable. If there are two force balanced points, the one with smaller attachment angle must be stable and the other one with larger attachment angle can be either stable or unstable. For a given contact angle, the information on the number of equilibria and their stability for the floating ball system are illustrated in Bond number versus density ratio figures. We give several such figures with typical contact angles. Finally, two examples are presented. One admits two stable equilibrium configurations. Another example shows a case with no force balanced point where there is an energy minimizer. This prompts discussion of the necessary condition for the floating configuration and a modification of changing topological structure for the floating configurations in this example. 2021-06-02T00:00:00Z Efficient Nested Simulation of Tail Risk Measures for Variable Annuities Dang, Ou http://hdl.handle.net/10012/17084 2021-06-03T02:35:25Z 2021-06-02T00:00:00Z Efficient Nested Simulation of Tail Risk Measures for Variable Annuities Dang, Ou Estimating tail risk measures for portfolios of complex Variable Annuities (VA) is an important enterprise risk management task which usually requires nested simulation. In the nested simulation, the outer simulation stage involves projecting scenarios of key risk factors under the real world measure, while the inner stage is used to value payoffs under guarantees of varying complexity, under a risk neutral measure. In this thesis we propose and analyze three different two-stage simulation procedures that improve the computation efficiency of nested simulation. All three proposals allocate the inner simulations to the specific outer scenarios that are most likely to generate larger losses. These scenarios are identified using a proxy evaluation in the Stage 1 simulation. The proxy evaluation is used only to rank the outer scenarios, not to estimate the tail risk measure directly. The proxy evaluation can be based on a closed-form calculation which works very efficiently for simpler contracts, or a pilot nested simulation using likelihood ratio estimators which accommodates very complex path-dependent contracts. Then in the Stage 2 simulation we allocate the remaining computational budget to the scenarios identified in Stage 1. Our numerical experiments show that, in the VA context, our proposals are significantly more efficient than a standard Monte Carlo experiment, measured by relative mean squared errors (RMSE), when both are given the same computational budget. 2021-06-02T00:00:00Z Continuous Spatial and Temporal Representations in Machine Vision Lu, Thomas http://hdl.handle.net/10012/17081 2021-06-03T02:35:41Z 2021-06-02T00:00:00Z Continuous Spatial and Temporal Representations in Machine Vision Lu, Thomas This thesis explores continuous spatial and temporal representations in machine vision. For spatial representations, we explore the Spatial Semantic Pointer as a biologically plausible representation of continuous space its use in performing spatial memory and reasoning tasks. We show that SSPs can be used to encode visual images into high dimensional memory vectors. These vectors can be used to store, retrieve, and manipulate spatial information, as well as perform search and scanning tasks within the vector algebra space. We also demonstrate the psychological plausibility of these representations by qualitatively reproducing Kosslyn's famous map scanning experiment. For temporal representations, we extend the original 1D Legendre Memory Unit to take multi-dimensional input signals and compare its ability to store temporal information against the Long Short-Term Memory Unit on the task of video action recognition. We show that the multi-dimensional LMU is able to match the LSTM in representing visual data over time. In particular, we demonstrate that the LMU is able to achieve much better performance when the total number of parameters is limited and that the LMU architecture allows it to continue operating at with fewer parameters than the LSTM. 2021-06-02T00:00:00Z