QSpace Community: Queen's UniversityInformation
http://hdl.handle.net/1974/6
Queen's UniversityInformationSat, 29 Aug 2015 03:13:47 GMT2015-08-29T03:13:47ZThe Channel Imagehttp://qspace.library.queensu.ca:80/retrieve/122/Jeffery_hall.jpg
http://hdl.handle.net/1974/6
Prediction and Filtering of Stationary Processes: Yaglom’s Method and Minimax Filtering
http://hdl.handle.net/1974/13493
Title: Prediction and Filtering of Stationary Processes: Yaglom’s Method and Minimax Filtering
Authors: Mascher, Philipp
Abstract: The aim of this work is to give a basic introduction to the theory of stationary stochastic processes, particularly to the somewhat specialized problem of prediction and filtering of such processes. Kolmogorov was the first to make
a contribution to its solution using involved mathematical theory. In the years following the publication of Wiener’s famous book, the theory gained considerable popularity from the applied sciences, particularly radio engineering. In this work, we shall present Yaglom’s method to solving the problems considered in Wiener’s book. This alternative approach is entirely based on rather basic facts from Hilbert space theory and the theory of complex variables. As it turns out, the theory of filtering of stationary processes heavily relies on spectral properties of the processes. In particular, Yaglom’s approach assumes complete knowledge of the spectral densities. In this work, however, we shall not be concerned with the problem of estimating such quantities based on a finite sample. Instead, in order to account for uncertainty as frequently encountered in practice, we shall discuss the problem of minimax filtering which has emerged from the practical need of allowing for incomplete knowledge about spectral properties.Tue, 11 Aug 2015 04:00:00 GMThttp://hdl.handle.net/1974/134932015-08-11T04:00:00ZOptimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
http://hdl.handle.net/1974/13457
Title: 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 15:57:18.667Tue, 28 Jul 2015 04:00:00 GMThttp://hdl.handle.net/1974/134572015-07-28T04:00:00ZLinearization and Stability of Nonholonomic Mechanical Systems
http://hdl.handle.net/1974/13149
Title: Linearization and Stability of Nonholonomic Mechanical Systems
Authors: Yang, Steven
Abstract: The stability of an equilibrium point of a nonlinear system is typically analyzed in two ways: (1) stability of its linearization, and (2) Lyapunov stability. An unconstrained simple mechanical system is a type of nonlinear system with a special structure, and so the methods for stability analysis can be specialized for this particular class of nonlinear systems. For a simple mechanical system subject to velocity constraints, the situation becomes more complicated. If the constraints are holonomic, then the problem can simply be reduced to that of an unconstrained simple mechanical system by restricting analysis to a certain submanifold of the configuration space. If the constraints are nonholonomic, this approach cannot be taken. In this report we study the differences and additional complexities that arise in these nonholonomic mechanical systems, and derive results with regards to linearization and stability of its equilibria.Tue, 23 Jun 2015 04:00:00 GMThttp://hdl.handle.net/1974/131492015-06-23T04:00:00ZOptimal Quantization and Approximation in Source Coding and Stochastic Control
http://hdl.handle.net/1974/13147
Title: 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 12:20:57.086Tue, 23 Jun 2015 04:00:00 GMThttp://hdl.handle.net/1974/131472015-06-23T04:00:00Z