QSpace Collection:http://hdl.handle.net/1974/7582015-08-30T12:04:30Z2015-08-30T12:04:30ZOptimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov SourcesWood, RICHARDhttp://hdl.handle.net/1974/134572015-07-29T05:14:11Z2015-07-28T04:00:00ZTitle: Optimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
Authors: Wood, RICHARD
Abstract: Optimal zero-delay coding of a finite state Markov source through quantization is considered. Building on previous literature, the existence and structure of optimal policies are studied using a stochastic control problem formulation. In the literature, the optimality of deterministic Markov coding policies (or Walrand-Varaiya type policies) for infinite horizon problems has been established. This work expands on this result for systems with finite source alphabets, proving the optimality of de- terministic and stationary Markov coding policies for the infinite horizon setup. In addition, the ε-optimality of finite memory quantizers is established and the depen- dence between the memory length and ε is quantified. An algorithm to find the optimal policy for the finite time horizon problem is presented. Numerical results produced using this algorithm are shown.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2015-07-27 15:57:18.6672015-07-28T04:00:00ZOptimal Quantization and Approximation in Source Coding and Stochastic ControlSaldi, NACIhttp://hdl.handle.net/1974/131472015-06-28T08:00:12Z2015-06-23T04:00:00ZTitle: Optimal Quantization and Approximation in Source Coding and Stochastic Control
Authors: Saldi, NACI
Abstract: This thesis deals with non-standard optimal quantization and approximation problems in source coding and stochastic control.
The first part of the thesis considers randomized quantization. Adapted from stochastic control, a general representation of randomized quantizers that is probabilistically equivalent to common models in the literature is proposed via mixtures of joint probability measures induced by deterministic quantizers. Using this general model, we prove the existence of an optimal randomized quantizer for the generalized distribution preserving quantization problem. A Shannon theoretic version of this source coding problem is also considered, in which an optimal (minimum distortion) coding of stationary and memoryless source is studied under the requirement that the quantizer's output distribution also be stationary and memoryless possibly different than source distribution. We provide a characterization of the achievable rate region where the rate region includes both the coding rate and the rate of common randomness shared between the encoder and the decoder.
In the second part of the thesis, we consider the quantization problems in stochastic control from viewpoints of information transmission and computation. The first problem studies the finite-action approximation (via quantization of the action space) of deterministic stationary policies of a discrete time Markov decision process (MDP), while the second problem considers finite-state approximations (via quantization of the state space) of discrete time Markov decision process. Under certain continuity conditions on the components of the MDP, we establish that optimal policies for the finite models can approximate with arbitrary precision optimal deterministic stationary policies for the original MDP. Combining these results leads to a constructive scheme for obtaining near optimal solutions via well known algorithms developed for finite state/action MDPs. For both problems, we also obtain explicit bounds on the approximation error in terms of the number of representation points in the quantizer, under further conditions.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-06-19 12:20:57.0862015-06-23T04:00:00ZMathematics Problems and Thinking Mathematically in Undergraduate MathematicsMatthews, Asia Rhttp://hdl.handle.net/1974/130452015-05-01T15:20:53Z2015-05-01T04:00:00ZTitle: Mathematics Problems and Thinking Mathematically in Undergraduate Mathematics
Authors: Matthews, Asia R
Abstract: Mathematics is much more than a formal system of procedures and formulae; it is also a way of thinking built on creativity, precision, reasoning, and representation. I present a model for framing the process of doing mathematics within a constructivist ideology, and I discuss two fundamental parts to this process: mathematical thinking and the design of undergraduate mathematics problems. I highlight the mathematical content and the structuredness of the problem statement and I explain why the initial work of re-formulating an ill-structured problem is especially important in learning mathematics as a mental activity. Furthermore, I propose three fundamental processes of mathematical thinking: Discovery (acts of creation), Structuring (acts of arranging), and Justification (acts of reflection). In the empirical portion of the study, pairs of university students, initially characterized by certain affective variables, were observed working on carefully constructed problems. Their physical and verbal actions, considered as proxies of their mental processes, were recorded and analyzed using a combination of qualitative and quantitative measurement. The results of this research indicate that ill-structured problems provide opportunities for a concentration of Discovery and Structuring. Though all of the identified processes of mathematical thinking were observed, students who are highly metacognitive appear to engage in more frequent and advanced mathematical thinking than their less metacognitive peers. This study highlights pedagogical opportunities, for both highly metacognitive students as well as for those who demonstrate fewer metacognitive actions, arising from the activity of doing ill-structured problems. The implications of this work are both theoretical, providing insight into the relationship between metacognition and student “performance,” and practical, by providing a simple tool for identifying processes of mathematical thinking.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-04-30 11:28:32.4162015-05-01T04:00:00ZApplications of Multitaper Spectral Analysis to Nonstationary DataRahim, KARIMhttp://hdl.handle.net/1974/125842014-10-18T05:15:42Z2014-10-15T04:00:00ZTitle: Applications of Multitaper Spectral Analysis to Nonstationary Data
Authors: Rahim, KARIM
Abstract: This thesis is concerned with changes in the spectrum over time observed in Holocene climate data as recorded in the Burgundy grape harvest date series. These changes represent nonstationarities, and while spectral estimation techniques are relatively robust in the presence of nonstationarity--that is, they are able to detect significant contributions to power at a given frequency in cases where the contribution to power at that given frequency is not constant over time--estimation and prediction can be improved by considering nonstationarity. We propose improving spectral estimation by considering such changes. Specifically, we propose estimating the level of change in frequency over time, detecting change-point(s) and sectioning the time series into stationary segments. We focus on locating a change in frequency domain in time, and propose a graphical technique to detect spectral changes over time. We test the estimation technique in simulation, and then apply it to the Burgundy grape harvest date series. The Burgundy grape harvest date series was selected to demonstrate the introduced estimator and methodology because the time series is equally spaced, has few missing values, and a multitaper spectral analysis, which the methodology proposed in this thesis is based on, of the grape harvest date series was recently published. In addition, we propose a method using a test for goodness-of-fit of autoregressive estimators to aid in assessment of change in spectral properties over time.
This thesis has four components: (1) introduction and study of a level-of-change estimator for use in the frequency domain change-point detection, (2) spectral analysis of the Burgundy grape harvest date series, (3) goodness-of-fit estimates for autoregressive processes, and (4) introduction of a statistical software package for multitaper spectral analysis. We present four results. (1) We introduce and demonstrate the feasibility of a level-of-change estimator. (2) We present a spectral analysis and coherence study of the Burgundy grape harvest date series that includes locating a change-point. (3) We present a study showing an advantage using multitaper spectral estimates when calculating autocorrelation coefficients. And (4) we introduce an R software package, available on the CRAN, to perform multitaper spectral estimation.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2014-10-15 00:52:05.8422014-10-15T04:00:00Z