project-main.tex

% Thesis class does bad things to figure float, mostly because of the double
% spacing. So save off the kernel version of float spacing for reinstatement.
\makeatletter
\let\kernel@xfloat\@xfloat
\makeatother

% Use the class supplied by the University of Calgary
\documentclass[12pt]{ucalgthes1}

% External references
\usepackage[letterpaper,top=1in, bottom= 1in, left= 1in, right= 1in]{geometry}
\usepackage{longtable}
\usepackage{hyperref}
\usepackage[all]{hypcap}
\usepackage{mathptmx}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[retainorgcmds]{IEEEtrantools}
\usepackage{mathrsfs}
\usepackage[canadian]{babel}
\usepackage{bbm}
\usepackage{listings}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{algorithm}
\usepackage{algpseudocode}
%\usepackage[section]{placeins}

% Syntax highlighting for julia
\include{project-julia}

% To get Tikz and PGF to work we reinstate the original float
\makeatletter
\def\@xfloat#1[#2]{
        \kernel@xfloat#1[#2]%
        \def\baselinestretch{1}%
        \@normalsize \normalsize
}
\makeatother

% We know return to our originally scheduled programming
\usepackage{pgf}
\usepackage{tikz}

% For the Markov chain diagrams
\usetikzlibrary{arrows,automata}

% Static definitions
\title{The Application of Lie Theory to Markov Processes\\ \bigskip
Computation of the Maximum Likelihood Estimator of the Generator of Continuous Time 
Homogeneous Markov Processes on Finite-State Spaces from Stopped Random Variables}
\author{Aaron Geoffrey Sheldon}
\thesisyear{2016}
\thesis{project}
\newcommand{\thesistitle}{The Application of Lie Theory to Markov Processes}
\monthname{August}
\dept{GRADUATE PROGRAM IN ``Statistics''}
\degree{``Masters of Science''}

% Mathematical elements
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{observation}{Observation}
\newtheorem{proposition}{Proposition}
\theoremstyle{definition}\newtheorem{definition}{Definition}
\renewcommand{\IEEEproofindentspace}{0em}
\renewcommand{\IEEEQED}{\IEEEQEDopen}
\interdisplaylinepenalty=0

% Start of main
\begin{document}
	\pagenumbering{gobble}%wierd hack to prevent hyperref warnings due to numbering of the title page
	\makethesistitle
	\pagenumbering{roman}
	\setcounter{page}{2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\chapter*{UNIVERSITY OF CALGARY \\ FACULTY OF GRADUATE STUDIES}
%\thispagestyle{empty}
%The undersigned certify that they have read, and recommend
%to the Faculty of Graduate Studies for acceptance, a \Thesis\ entitled
%``\thesistitle'' submitted by \Author\
%in partial fulfillment of the requirements for the degree of
%\Degree.\\
%
%                 Substitute  List of Examiners
%
%\begin{signing}{Department of Academic Computing}
%\signline
%Chairman, Dr.~John D.~Doe \\
%Department of Academic Computing \\
%Services  \\
%\signline
%Chairman, Dr.~John D.~Doe \\
%Department of Academic Computing \\
%Services  \\
%\signline
%Chairman, Dr.~John D.~Doe \\
%Department of Academic Computing \\
%Services  \\
%\signline
%Chairman, Dr.~John D.~Doe \\
%Department of Academic Computing \\
%Services  \\
%\newsigncolumn         use this command to start a new column if necessary
%\newsigncolumn
%\signline
%Chairman, Dr.~John D.~Doe \\
%Department of Academic Computing \\
%Services  \\
%\signline
%Dr.~Jane Smith \\
%Department of Academic Computing  \\
%\signline
%Dr.~A.~B.~Brown \\
%Department of Academic Computing  \\
%\end{signing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	% Abstract
	\newpage
	\phantomsection
	\altchapter{\bf{Abstract}}
	Individually Lie theory and the probability theory of stochastic processes have been
	highly productive fields of investigation for more than a century; yet they remain ripe
	for cross pollination. In particular, the application of algebraic and analytic results
	from Lie theory can yield novel computational methods for the estimation of generators of
	continuous time homogeneous Markov processes on finite-state spaces. In this project we
	derive the minimal Lie algebra that contains the generators of continuous time homogeneous
	Markov processes on finite-state spaces. Taking advantage of the guarantees of algebraic 
	and analytic closure we construct Pad\'{e} approximations for the Taylor series expressions
	of the first and second order Fr\'{e}chet derivatives of the exponential map. This further
	allows for the proposal of a Newton-Raphson algorithm for maximum likelihood estimation of
	the generator of a continuous time homogeneous Markov process on a finite-state space from
	stopped random variables.

	% Acknowledgements
	\newpage
	\phantomsection
	\altchapter{\bf{Acknowledgements}}
	First and foremost is my love and gratitude to my loving family; who withstood the long
	stretches of my physical presence, but mental absence while I developed, and explored the
	mathematical geography I describe in this report.
	
	Second, I am deeply appreciative of my supervisor who has had the extraordinary patience 
	to survive the many years of rumination it took for me to develop the meager ideas 
	expressed in this report.

	% Contents listings
	\begin{singlespace}
		\newpage
		\phantomsection
		\tableofcontents
		\pagestyle{plain}
		\newpage
		\phantomsection
		\listoftables
		\pagestyle{plain}
		\newpage
		\phantomsection
		\listoffigures
		\pagestyle{plain}
		\newpage
		\phantomsection
		\listofalgorithms
		\pagestyle{plain}
		\clearpage
		\clearpage
	\end{singlespace}
	\newpage
	\phantomsection
	\chapter*{\bf{List of Symbols, Abbreviations and Nomenclature}\hfill} 
	\addcontentsline{toc}{chapter}{List of Symbols}
	\listofsymbols
	\pagestyle{plain}
	\clearpage
	\pagenumbering{arabic}

	% Content of project
	\include{project-chapter-1}
	\include{project-chapter-2}
	\include{project-chapter-3}
	\include{project-chapter-4}
	\include{project-chapter-5}
	\bibliography{project-bibliography} 
	\bibliographystyle{plain}
	\appendix
	\include{project-appendix-1}
\end{document}