Possible topics for Mathematics and Statistics honours thesis in 2009

General algebra
Associate Professor Brian Davey, B.Davey@latrobe.edu.au

The 'classical' kinds of algebras include groups, rings and vector spaces. I am mainly interested in more 'modern' kinds of algebras, like lattices and Boolean algebras, which arise in logic and theoretical computer science. Possible topics in general algebra include:

1. CSP's. General algebra has recently been used as a tool for improving our understanding of the complexity of Constraint Satisfaction Problems, which frequently arise as practical problems in computer science (for example, in scheduling shift workers). This project would review the background and investigate some recent papers on the topic.

2. Primal and quasi-primal algebras. Primal and quasi-primal algebras are important generalisations of the two-element Boolean algebra, which is familiar from discrete mathematics, and arise in many parts of general algebra. The aim of this project will be to find simple and effective tests for quasi-primality based on some old and some recent tests for primality.

3. Graph algebras. Every graph (in the sense of discrete mathematics) can be converted in a natural way into an algebra with a single binary operation. This project will review the many applications that graph algebras have found and will include some recent papers illustrating their use on the fascinating interface between complexity theory and algebra.

4. NU algebras. Algebras with a near-unanimity term are an important generalisation of lattices. This project will review the various properties that these algebras possess and study some very recent papers, including the proof that it is decidable whether or not a finite algebra actually has a near-unanimity term.

Algebras of programs
Dr Marcel Jackson, m.g.jackson@latrobe.edu.au

There are a number of logical and algebraic approaches to analysing the correctness of computer programs.  This project would involve an exposition of some of these approaches and their interrelationships, and lead to an investigation into the relationship between these and algebras of functions and relations.  This project is best suited to students with some extra background in logic, or with a strong background in algebra with a computer science interest.  No knowledge of programming is necessary, since the focus is on algorithms, rather than actual programming languages.

Algebras of functions and relations
Dr Marcel Jackson, m.g.jackson@latrobe.edu.au

This topic concerns the problem of classifying various algebras of functions and binary relations.  For example: the Cayley representation for groups shows that groups are precisely the algebras arising as systems of permutations of a set under the operations of composition and inverse.  What about algebras of binary relations under the operation of composition and intersection?  There are a plethora of such questions, many motivated from the kind of issue considered in the previous topic “Algebras of programs”.

Computational complexity in algebra
Dr Marcel Jackson, m.g.jackson@latrobe.edu.au

How hard is it to decide if two arbitrary finite graphs are isomorphic? What about two finite groups?  This project would involve a study of computational questions such as these: the focus would be on general algebraic structures, and connections between computational problems and logical properties of algebras.  This project could be directed to computational problems on finite structures (a popular current trend) or for infinite structures (more classical, but still active and important).

Simon's Theorem (automata and semigroups)
Dr Marcel Jackson, m.g.jackson@latrobe.edu.au

Finite state automata (as encountered in, say MAT1DM) and algebraic structures known as "semigroups" have a close relationship: finite semigroups all arise as the algebra of input transitions of a FSA under composition, and every finite semigroup gives rise to a finite automata for which it is the algebra of transitions.  Simon's Theorem relates languages recognised by machines that read a fixed multiple number of bits at a time to an elementary structural property of finite semigroups. It has a number of different proofs.  This topic is suitable for a student with an algebra background. Dr Marcel Jackson will also consider other proposals in semigroup theory or universal algebra.

To discover and explain features of integrable systems
Dr Peter Van Der Kamp, P.VanDerKamp@latrobe.edu.au

Amongst all differential/difference equations, or mappings/correspondences, the integrable ones are the nice ones. They always carry special structures. Examples are a Lax-pair, infinitely many symmetries, sufficiently many integrals of motion, and polynomial growth (as opposed to exponential growth) of degrees of iterates and multivaluedness. Often one is able to calculate the first few terms of an infinite series from which one can guess the general form. At this point you want to write down a proof! What does the identity, which makes it happen, look like?

The Inverse Problem in the Calculus of Variations
Associate Professor Geoff Prince, g.prince@latrobe.edu.au

When are the solutions of a system of second order ordinary differential equations (ode’s)



the solution of an Euler-Lagrange equation



This is a famous question in the calculus of variations, important in differential geometry, mechanics and relativity. This project involves an exploration of this inverse problem for some important classes of differential equations using differential geometry and computer algebra.

Numerical integration by arcs
Associate Professor Geoff Prince, g.prince@latrobe.edu.au

Numerical integration of differential equations uses approximation by straight line segments to construct solutions. This project explores what seems to be a completely new idea: approximation by circular arcs. It involves development and implementation of analogues of Euler's method and Runge-Kutta methods and comparison with the traditional algorithms.

Associate Professor Geoff Prince will also consider proposals in differential geometry, differential equations, relativity and mechanics.

Symbolic Dynamics and Formal Languages
Dr John Banks, j.banks@latrobe.edu.au

The branch of dynamical systems theory known as symbolic dynamics interacts in many ways with the theory of formal languages, a theory of fundamental importance in many areas of computer science. This project would explore some of the many connections between the two theories. Some examples of specific areas that might be investigated include the connection between sofic shifts and regular languages, the application of the theory of context free languages to symbolic dynamics, the application of symbolic dynamics to coding problems such as the coding of data for hard disk storage.

The Effect of the Choice of Value-at-Risk Methodology on Subsequent “Backtesting”
Associate Professor Paul Kabaila, p.kabaila@latrobe.edu.au

Value-at-Risk is a measure of financial risk that is very widely used in the banking sector. The Bank for International Settlements provides the regulatory framework for banks by setting minimal capital requirements for banks in terms of a 99% Value-at-Risk. There are a number of different methods of finding a 99% Value-at-Risk. Banks are free to choose the methodology that they will use for finding it. However, the quality of a Value-at-Risk found by a bank is required to be evaluated by “backtesting” over the most recent twelve months of data. The bank faces financial penalties if backtesting indicates that the method that was used by the bank is faulty. The purpose of this Honours project is to compare the backtesting properties of two different methods of finding Value-at-Risk.

Confidence Intervals for the Difference Between the Treatment Effects Obtained After a Preliminary Hypothesis Test in a Two-Treatment, Two-Period Crossover Trial
Associate Professor Paul Kabaila, p.kabaila@latrobe.edu.au

The two-treatment, two-period crossover trial design is widely used in clinical trials. The purpose of this design is to compare two treatments labelled A and B. Patients are randomly allocated to either (a) a group that receives treatment A first and then (after the effect of treatment A has presumably worn off) treatment B or (b) a group that receives treatment B first and then (after the effect of treatment B has presumably worn off) treatment A. A well-known weakness of this design is that if a treatment has not worn off before the second treatment is applied (this is called a carryover effect) then the statistical inference for the difference between the treatment effects can be severely biased. One method of trying to overcome this weakness is to perform a preliminary hypothesis test for the presence of carryover effect before finding the confidence interval for the difference between the treatment effects. The purpose of this Honours project is to examine (based on Freeman, P.R., 1989. The performance of the two-stage analysis of the two-treatment, two-period crossover trial. Statistics in Medicine, vol 8, pages 1421−1432) the coverage probability properties of these confidence intervals.

Associate Professor Paul Kabaila will also consider topics in time series predictions, confidence intervals utilizing uncertain prior information exact inference from count data.

WKS sampling reconstructions
Dr Andriy Olenko, a.olenko@latrobe.edu.au

WKS sampling is a process of representing continuous signals by sequences of numbers (called samples). A classical approach to this problem is provided by Shannon’s sampling theorem which states that if a signal is band-limited, then it is uniquely determined by its sampled values. This result is one of the basic tools in signal processing. In direct numerical implementations we consider the truncated variant of Shannon’s sampling theorem. Restoring continuous signals with given accuracy from discrete samples or assessing the information lost in the sampling process are fundamental problems.   This project will analyse truncation errors and the information lost for deterministic and stochastic signals.  Applications in signal processing and information compression will be considered.

Clustering Large Financial Data
Dr Andriy Olenko, a.olenko@latrobe.edu.au

There is a growing need for a more automated system of partitioning data sets into groups, or clusters. Clustering techniques can be used to discover natural groups in data sets and to identify abstract structures that might reside there, without having any background knowledge of the characteristics of the data. Clustering has been used in a variety of areas, including computer vision, data mining, bioinformatics (gene expression analysis), and information retrieval, to name just a few.  This project focuses on a few of the most important clustering algorithms.  Applications in stocks clustering with actual data will be considered.

Tauberian theorems in probability theory
Dr Andriy Olenko, a.olenko@latrobe.edu.au

In probability theory and statistics Tauberian theorems are applied to analysing the asymptotic behavior of stochastic processes, record processes, random permutations, infinitely divisible random variables, etc. We use Abelian and Tauberian theorems to describe a relationship between the asymptotic behaviour at the origin of the spectrum of random processes and that at infinity of integrals of the random process.  This project will analyse Tauberian theorems for various random variables and processes.

Dr Andriy Olenko will also consider proposals in such areas as spatial statistics, limit theorems, actuarial mathematics and Shannon sampling theory.  

A study of principal Hessian directions
Dr Luke Prendergast, luke.prendergast@latrobe.edu.au

Modern day researchers are often faced with the onerous task of analysing data sets that consist of measurements for a large number of variables.  Such data sets are commonly referred to has ‘high-dimensional data’ and many classical methods are either not applicable in this setting or struggle to return meaningful results.  In regression analysis high-dimensional data consists of a large number of predictor variables.  It is often the case that simple models such as the Multiple Linear Regression (MLR) model are not appropriate yet the large number of predictor variables makes it difficult to visually determine more complex structure.  Principal Hessian directions (pHd) is a dimension reduction method that seeks to reduce the dimension of the predictor space allowing for the visual detection of elusive regression structure.  The purpose of this project is to show, via Stein’s Lemma, that pHd is a useful dimension reduction tool under certain conditions for the predictor.  Interestingly, pHd also often performs well when theoretically it should fail and this project will therefore include a study into this phenomenon. 

Improving estimated sufficient summary plots through the trimming of influential observations
Dr Luke Prendergast, luke.prendergast@latrobe.edu.au

Ordinary Least Squares (OLS) can be useful for a wide variety of assumed underlying regression models.  Whilst OLS influence diagnostics have been developed for the Multiple Linear Regression (MLR) model which are capable of detecting influential observations in practice, the usefulness of these diagnostics is limited when considering regression models that allow for more complex dependency between the response and predictor variables.  This project will consider a robustness analysis of OLS under a general single-index model that will lead to the construction of influence diagnostics that are useful for a wide range of models.   This section of the project will require that the student has previously been exposed to power series expansions.  Various trimming algorithms will also be considered that provide vastly superior results for both simulated and real-life data.

Dr Luke Prendergast will also consider topics in robustness properties of statistical estimators and dimension reduction methods.

Combining Correlated Statistical Evidence
Associate Professor Robert Staudte, r.staudte@latrobe.edu.au

In the reference 'Meta Analysis' listed below, a new approach to calibrating and combining statistical evidence is proposed and applied to independent experiments. However, sometimes the evidence measures are not independent, such as when experiments are correlated or when one has two evidence measures from one experiment for two different alternatives hypotheses, and wants to compare them. Such is the case in meta-analysis, e.g. when choosing between the fixed, unequal effects model and a random effects model.

Reference:

Meta Analysis: a Guide to Calibrating and Combining Statistical Evidence, Wiley (2008), by E. Kulinskaya, S. Morgenthaler and R.G. Staudte.

Variance stabilizing the risk difference estimate for two Poisson risks
Associate Professor Robert Staudte, r.staudte@latrobe.edu.au

In the above reference are provided some different methods for finding the evidence for different Poisson parameters in independent experiments. None are wholly satisfactory, so the purpose of this project is to not only study what is known, but find a better approach to this problem.

Value at Risk and expected shortfall for financial time series models
Dr Ajay Chandra, a.chandra@latrobe.edu.au

One of the aims of financial risk management is the accurate calculation of the magnitude and probabilities of large potential losses due to extreme events such as stock market crashes, currency crises, trading scandals, or large bond defaults. To get an insight into these problems, we typically evaluate risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) based on financial time series models such as autoregressive conditional heteroskedastic (ARCH) or generalised ARCH (GARCH). This project would definitely give some exposure in finance setting.

Dr Ajay Chandra will also consider proposals for topics in time series, multivariate analysis and nonparametric analysis.