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.

Ross has research experience in a number of areas of probability and statistics, from Lévy processes and stochastic processes displaying long memory, to Bayesian nonparametrics, and computation for Bayesian statistics and time series analysis.

A current project: Statistical inference for partially observed metapopulations

A common type of metapopulation model is a presence-absence model in discrete time with a Markov dependence structure. This type of model only tracks whether or not each patch within the metapopulation is occupied and the future state of the metapopulation depends only on the present state. When all patches within the metapopulation are observed, the likelihood function can be written explicitly which makes estimation relatively straightforward. However, it is not always possible to perform a complete census of a metapopulation. This may be due to costs associated with a census or due to accessibility or perhaps the locations of some patches are unknown. In these situations we may attempt to make inference about the metapopulation based only on the observations from a small number of patches.

In theory one can still perform likelihood based inference given only partially observation of the metapopulation. To form the likelihood function based on partial observation, one takes the likelihood function based on complete observation and integrates out those unobserved patches. Typically, the necessary integration is performed using simulation based methods. However, when the number of patches is large, the situation where partial observation would be most common, accurate integration can be very challenging.

An alternative to using an exact likelihood is to use an approximate likelihood based letting the number of patches increase to infinity. We provide certain conditions under which the model for the observed patches converges to a time-inhomogeneous Markov chain. This provides an explicit approximate likelihood function which can then be used in either maximum likelihood estimation (MLE) or Bayesian estimation. Under rather mild conditions, we show that the exact MLE converges to the proposed approximate MLE as the number of patches increases. Similarly for the Bayesian estimate, we prove convergence of the exact posterior distribution to the proposed approximate posterior distribution.

Unfortunately, when the exact likelihood function is replaced by the asymptotic approximation, a loss of efficiency is incurred. To investigate the degree of the efficiency loss, we carry out a simulation study using two metapopulation models. The exact MLE and Bayesian estimates will be evaluated using a sequential Monte Carlo algorithm.

Martin is interested in applications of probability to the biological sciences. The bulk of his research to date has addressed problems in population genetics. Martin is presently Research Officer in the Bioinformatics Division at the Walter and Eliza Hall Institute (Melbourne).

Hanjun is known internationally for contributions to Markov process theory. Hanjun presently holds a position in the School of Mathematics and Computational Science at Xiangtan University.

Thesis title (provisional): 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

(PhD conferred December 2005)

Ben worked on a range of problems in stochastic modelling of complex biological systems. He determined the extinction probabilities and expected extinction times for the Markovian catastrophe process in continuous time, with a general transition rate function, and gave necessary and sufficient conditions for explosivity. Ben 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 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 (provisional): Animal Movement Between Populations Deduced from Family Trees

The aim is to develop a new method for estimating animal movements using information contained in family trees. Movement estimates are essential to population models that assist natural resource managers to plan species recovery and to predict the effect of future challenges, such as human-mediated activities and climate change. I will evaluate ways of constructing family trees from genetic data and develop a statistic that describes animal movement between populations that is based on the families whose members were sampled in more than one population; empirical data has been sourced from a long-term mark-recapture study of dugongs in Moreton Bay, and new samples from two adjacent populations.

Thesis title: Topics in Quasi Stationarity of Markov Chains

(MPhil conferred 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 explored how this principle applies to the evaluation of quasi-stationary distributions. He 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: 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 were 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 used 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 (provisional): Fault Detection in Complex and Distributed Systems

The primary goal is to develop a theoretical framework based on Markov processes in order to detect, identify and isolate faults in complex and distributed systems. The aim is to improve overall safety and reduce any negative impact on the environment due to a fault. There are three main tasks. The first is development of local stochastic models, which need to be capable of interpreting the local environment's state. At the core is estimation of transition probabilities. The second is extracting the features of local models in the case of non-faulty and faulty operating conditions. In order to assist local models to achieve satisfactory results, design and implementation of a multi-agent system is proposed. The next task is planning an optimal action to protect the environment. Finally, the framework has to allow for the possibility of incorporating local expert knowledge about the system.

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

(PhD 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

(PhD conferred July 2006)

Dharma made several major contributions to the study of election forecasting. He 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 (provisional): Optimal Design for Statistical Inference in 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.

Daniel 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

(PhD conferred February 2007)

Joshua developed continuous-time Markov chain models for metapopulations that inhabit a dynamic landscape. He established deterministic and diffusion approximations for these processes, and derived normal approximations to their (quasi-)stationary distributions. He 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 gave population managers direction on how best to choose the control parameters. Joshua 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 looked at the problem of establishing the existence of quasi-stationary distributions (QSDs). He proved results on exponential convergence rates and established the existence of a QSD for a particular chemical reaction model.

David studied a well-known and important exponential convergence rate: Kingman's decay parameter. He 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 analysed these bounds analytically and numerically.

David 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 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 was 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 (provisional): Models for Spatially Structured Metapopulations

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.

I will begin by looking at basic patch-occupancy models that merely record which patches are occupied. The main aim is to exploit recent developments in stochastic network theory by adapting models that were developed originally for the study of telecommunications systems. By recording the numbers of individuals in the various patches we can incorporate local patch dynamics, spatial structure and migration patterns. I will adopt the powerful diffusion approximation technique that has been used so effectively in the analysis of patch-occupancy models.

Thesis title: Volume Weighted Average Price Options

(PhD 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: Advances in Cross-Entropy Methods

(PhD conferred May 2009)

The Cross-Entropy Method is a technique used for solving estimation, simulation and optimisation problems. Thomas Taimre's work has helped consolidate our understanding of the Cross-Entropy Method and several of its generalisations, as well as its connection with importance sampling and other Monte Carlo methods. Thomas has elucidated several new major applications of cross-entropy methodology to optimisation problems. He has developed new methods within the generalised cross-entropy framework which enable one to construct state- and time-dependent importance sampling algorithms, and he has developed a new algorithm for counting solutions to difficult binary-encoded problems.

Thomas is presently a postdoctoral fellow within the Department of Mathematics at the University of Queensland.

Thesis title (provisional): Minimum Risk Optimal Portfolio Allocation: a Game Theoretic Approach

My research problem concerns allocating capital among a set of risky assets so as to obtain an optimal portfolio allocation. I will use a game theoretic approach based on the notion of Conditional Value at Risk. This is a novel approach to optimal portfolio management. Since Conditional Value at Risk is a coherent risk measure it can potentially reduce the likelihood that a portfolio will suffer substantial losses.

I will adopt the coalitional games concept interpreting the different portfolios as different players. The Aumann Shapley Principle of game theory will be used to compute allocations. If we consider risk measure as a linear optimization problem, then the Shapely value can be computed more easily. Since Conditional Value at Risk is a coherent risk measure, providing optimization shortcuts through linear programming techniques, it can be used here as a risk measure. Both game theory and Conditional Value at Risk have been used independently in portfolio management, and I expect that in combination they will prove to be very effective.