Phil's research is in the field of mathematical modelling, and is chiefly concerned with the theory of stochastic processes and applications in ecology, epidemiology, parasitology, telecommunications and chemical kinetics.

A current project: General Stochastic Models for Branching

This is joint work with Hanjun Zhang, Anyue Chen (University of Liverpool and Junping Li (Central South University, Changsha). A common feature of branching models is that particles or individuals behave independently producing descendants according to the same rule. However, since particles may interact, through collision or some other mechanism, this branching property may be lost. For this reason, more general branching models have been proposed. We are studying a particularly interesting class, which we call the weighted (or non-linear) Markov branching processes. We are examining questions concerning the existence and uniqueness of such processes, and criteria for extinction.

Iadine has research intersests in mathematical modelling and decision making in ecology and conservation biology.

A current project: Strategies for managing invasive species in space: deciding whether to eradicate, contain or control

Invasive species are a major threat to ecosystems worldwide. Once they have established, even a determined commitment to control, contain or eradicate, it is often difficult to decide on the most efficient and effective management strategy due to the complex interaction of factors such as the extent of the invasion, the ecology of the species, the dynamics of the system, and how the species responds to different management actions. The decision-making process is exacerbated further by our inability to observe these systems perfectly.

Using partially observable Markov decision process (POMDP) and graph theory, this project investigates the role space plays in optimally managing invasive species and will produce the first spatially explicit decision support tool for managing cryptic invasive species. We aim to provide a general tool and rules of thumbs that can be applied to a range of species in diverse situations.

Hanjun is known internationally for contributions to Markov process theory.

Martin is interested in applications of probability to the biological sciences. The bulk of his research to date has addressed problems in population genetics.

Thesis title: Discrete-time Stochastic Metapopulation Models

A metapopulation is a population that occupies several geographically separated habitat patches. Although the individual patches may become empty through local extinction, they may be recolonized through migration from other patches. There is considerable empirical evidence which suggests that a balance between migration and extinction is reached that enables metapopulations to persist for long periods, and there has been considerable interest in developing methods that account for the persistence of these populations and which provide an effective means of studying their long-term behaviour before extinction occurs.

For many populations extinction and colonization happens in distinct phases, often at different stages in the organism's life cycle, and the natural stochastic model is a (time-inhomogeneous) Markov chain in discrete time. One of the problems with existing discrete-time metapopulation models is their inability to properly model the colonization process. We remedy this by incorporating a simple device to account for the colonization potential of occupied patches. We will develop deterministic and distributional approximation techniques to analyse these models that build on methods we have developed for "mainland-island" models.

Thesis title: Hitting Times for Markov Population Processes Subject to Catastrophes

(Degree conferred December 2005)

Ben has worked on a range of problems in stochastic modelling of complex biological systems. He has determined the extinction probabilities and expected extinction times for the Markovian catastrophe process in continuous time, with a general transition rate function, and has given necessary and sufficient conditions for explosivity. Ben has developed truncation procedures for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. He has developed theory for first-exit time problems in the context of general piecewise-deterministic processes, providing a general, robust numerical procedure for estimating first-exit times and implemented this using techniques from interval analysis.

Ben is a research scientist in the Cancer Epidemiology Unit at the University of Oxford having previously held a postdoctoral position in the School of Biological Sciences at the University of Bristol.

Thesis title: Long Time Scale Simulations of Biological Molecular Systems

(PhD conferred December 2007)

Ben works primarily on long time-scale molecular dynamics. Traditionally, molecular processes are seen from a classical physics perspective and use various forward integration algorithms to provide thermodynamic information from trajectories. These techniques are primarily limited by computational resource constraints. A series of new algorithms has been proposed which achieves low resolution trajectories of any time scale. One of the difficulties of these approaches is estimation of the overall time in which a molecular process takes place. Ben is using mean first passage times to provide an initial trajectory through the molecules' conformation space. This approach reduces errors introduced by poor time-scale estimation. The practitioner is also provided with a starting point for a trajectory search using more traditional deterministic algorithms.

Ben is currently a medical student at Flinders University School of Medicine.

Thesis title: Topics in Quasi Stationarity of Markov Chains

(Degree awarded July 2007)

Quasi-stationary distributions are tools that allow one model the long-term behaviour of processes that "die out". Convergence of standard truncation methods for evaluating quasi-stationary distributions is not always guaranteed, and it is desirable to have algorithms that avoid truncation. An algorithm which avoids truncation in computing stationary distributions is the GTH Algorithm. It completely avoids subtraction, and it was shown that the algorithm computes stationary distributions with low relative error, and even extremely small stationary probabilities with high accuracy. A simple principle has been proposed: that many algorithms in non-negative arithmetic produce results with low relative error. Nick has been exploring how this principle applies to the evaluation of quasi-stationary distributions. He has examined an algorithm for computing the dominant eigenvectors which uses non-negative arithmetic and which gives demonstrably low relative error.

Nicholas is a risk consultant with Energy Edge (Brisbane) Ltd.

Thesis title: Trigonometric Scores Rank Procedures with Applications to Long-tailed Distributions

(Degree conferred May 2006)

Long-tailed distributions have become extremely popular for modelling stochastic noise in many applications including image analysis, finance and environmental data analysis. However, often the tail behaviour of such distributions is not precisely known and nonparametric statistical procedures are evoked to perform inference about the location and scale characteristics. Olena's work proposes several new rank procedures that are efficient for a wide range of unimodal, symmetric, long-tailed distributions.

Olena is a Lecturer in Biometrics in the School of Land, Crop and Food Sciences, The University of Queensland.

Thesis title: Stochastic Models of Election Timing

(Degree conferred July 2006)

Dharma has made several major contributions to the study of election forecasting. He has derived a model for the early election call problem that accounts for the possibility of a government using control tools, termed "boosts", to induce shocks in the opinion polls by making timely policy announcements or economic actions. These actions improve the government's popularity and have some impact on the early-election exercise boundary. He is presently working on some theoretical extensions the basic framework. He is studying a bounded mean-reverting process, used in the pricing of energy options and in election forecasting. He aims to provide conditions for existence and uniqueness of a bounded mean-reverting stochastic differential equation whose drift coefficient does not satisfy either of the usual Lipschitz or linear growth conditions.

Dharma is a Lecturer in the Department of Mathematics, Parahyangan Catholic University, Indonesia.

Thesis title: Statistical Inference for Stochastic Processes

Stochastic processes have been used to model a wide range of phenomena such as population dynamics, chemical reactions, epidemics and telecommunications traffic. However, the statistical methods for these processes have not received a great deal of attention. There are two key aspects of statistical inference that will be investigated: parameter estimation for stochastic processes, and optimal design of experiments that can be formulated as stochastic processes. Whilst the former has received attention by a number of authors, the latter is a largely unexplored, with great potential to improve the utility of stochastic processes as statistical models in an experimental context.

He proposes the use of Gaussian diffusion approximations as a powerful tool for obtaining analytical approximations to Fisher's information matrix, which is required for the optimal design of experiments, but which can be extremely difficult to obtain using a direct analytical approach. The Gaussian diffusion approximation will easily couple with numerical optimisation procedures to obtain optimal designs. Recent work has shown that the use of these methods allows for advances in parameter estimation for stochastic processes by allowing the likelihood function to be approximated.

Thesis title: Density Dependent Markov Population Processes: Models and Methodology

(Degree conferred February 2007)

Joshua has developed continuous-time Markov chain models for metapopulations that inhabit a dynamic landscape. He has established deterministic and diffusion approximations for these processes, and derived normal approximations to their (quasi-)stationary distributions. He has also investigated the costs and decisions of controlling populations that have a negative impact on their habitat. For two commonly used control regimes, suppression and reduction, he has given population managers direction on how best to choose the control parameters. Joshua has compared the predicted extinction time estimates derived from continuous-time Markov chain models with the estimates from their appropriate Ornstein-Uhlenbeck approximating diffusion and a simple Brownian motion approximation. In joint work with Thomas Taimre he developed a method for estimating the parameters of a wide class of continuous-time Markov chains called density-dependent Markov chains. The only other known approach to estimating parameters for such processes is computationally infeasible when the state space, or uncertainty in the parameter values, is too large. This new procedure makes use of the above-mentioned diffusion approximations, and in the situations where the approach is most commonly applied, the estimates improve as the state space increases in size. Several applications of this procedure are currently under investigation.

Joshua currently holds a Research Fellowship at King's College Cambridge UK, having previously held a post-doctoral research assistantship at the University of Warwick.

Thesis title: On the Analysis of Absorbing Markov Processes

(PhD conferred May 2008)

David's PhD thesis is concerned with the analysis of absorbing discrete-state Markov processes. He has looked at the problem of establishing the existence of quasi-stationary distributions (QSDs). He has proved results on exponential convergence rates and has established the existence of a QSD for a particular chemical reaction model.

David has studied a well-known and important exponential convergence rate: Kingman's decay parameter. He has adapted results of Mu-Fa Chen to give explicit bounds for the decay parameter of a birth-death process in terms of the transition rates, an immediate corollary of which is a necessary and sufficient condition for the decay parameter to be positive, and has analysed these bounds analytically and numerically.

David has also investigated an application of the analysis of absorbing Markov chains in ecology. He considered a threatened species occupying a habitat which consists of a number of discrete patches. He extended existing models to allow for the possibility that one can protect certain patches from disturbances. He discussed deterministic approximations, which become both computationally necessary, and mathematically more accurate, as the system size becomes large. He has used both the full stochastic model and the deterministic approximation to investigate the effect on the viability of the population of two management options: creating more patches, or protecting existing patches from disturbance events. The optimal management plan has been determined under a given a total budget and per-patch costs for these two possible actions.

David currently holds a research fellowship in the School of Mathematical Sciences, University of Nottingham UK.

Thesis title: Volume Weighted Average Price Options

(Degree conferred July 2007)

Antony developed methods for the valuation of a Volume Weighted Average Price option (VWAP). This is an option which has a strike which is a VWAP. He obtained a number of results about these options including an approximation to the price by moment matching and also a series solution. Antony also investigated a numerical solution to the partial differential equation that describes the price of the option by finite differences. This procedure presents a number of challenges; simple finite difference methods are impractical due to the curse of dimensionality, so alternating direction implicit and splitting methods were investigated.

Antony is a risk consultant with Energy Edge (Brisbane) Ltd, having previously held a position on the financial risk management team of Pricewaterhouse Coopers (Auckland, New Zealand).

Thesis title: Cross-Entropy Methods for Multi-Agent Systems

Thomas is focusing on the theory and applications of the Cross-Entropy method. The aim is to develop a picture of the behaviour of Cross-Entropy algorithms (and extensions thereof) in relation to a range of optimisation and estimation problems. In conjunction with this investigation, a toolbox of Cross-Entropy algorithms for Matlab is also being developed. The toolbox will contain a collection of algorithms capable of competently tackling many classes of optimisation problems, with low time-cost to the user. A continuing aspect of this research is examining the links between the Cross-Entropy method and related techniques, one of which is the so-called "Probability Collectives" approach, a substantially similar technique to the Cross-Entropy method, albeit arrived at from different theoretical considerations. This aspect of the research has been partly motivated by potential extensions to the original Cross-Entropy approach. The work undertaken to date shows promise for optimisation applications in the multi-agent setting. With reference to the Cross-Entropy method itself, two substantial applications to optimisation problems have been: to the Euclidean Clustering Problem; and (with Joshua Ross) in an application to the estimation of the parameters of a class of continuous-time Markov chains. Thomas is continuing to consider applications of the Cross-Entropy method and algorithms in a variety of optimisation and estimation settings; developing the Cross-Entropy toolbox further; examining links between the Cross-Entropy method and other techniques, and developing extensions to the current algorithms.