Mathematics and Statisticshttp://hdl.handle.net/10523/962018-03-22T00:23:00Z2018-03-22T00:23:00ZSemiparametric dispersal kernels in stochastic spatiotemporal epidemic modelsLuo, Peihttp://hdl.handle.net/10523/79392018-03-21T13:02:05Z2018-03-20T23:01:38ZSemiparametric dispersal kernels in stochastic spatiotemporal epidemic models
2018
Luo, Pei
The dispersal kernel plays a fundamental role in stochastic spatiotemporal epidemic models. By quantifying the rate at which an infectious source infects a susceptible individual in terms of their separation distance, the dispersal kernel is able to account for the observed spatial characteristics of an epidemic. The aim of this thesis is to construct a dispersal kernel which belongs to a semiparametric family. We introduce a new concept called the natural bridge basis in order to build the semiparametrized dispersal kernel. We use data from a citrus canker epidemic in Florida to illustrate and examine our approach. We find features of the semiparametrized dispersal kernel which were not previously evident in parametrized dispersal kernels.
2018-03-20T23:01:38ZModelling strategies to improve genetic evaluation for the New Zealand sheep industryHolmes, John Barretthttp://hdl.handle.net/10523/78852018-03-06T13:02:08Z2018-03-05T20:24:56ZModelling strategies to improve genetic evaluation for the New Zealand sheep industry
2018
Holmes, John Barrett
The question of how best to optimise the accuracy of genetic evaluation for livestock populations has been given new life by the advent of genomics. Therefore we will investigate methods of evaluating and/or improving the accuracy of genetic evaluation in ways applicable to genotyped populations, while trying to maximise computational efficiency. We will explore modelling strategies with utility outside animal breeding, including examples of these potential non-animal breeding applications.
2018-03-05T20:24:56ZMass loss due to gravitational waves with a cosmological constantSaw, Vee-Liemhttp://hdl.handle.net/10523/77052018-02-28T01:08:30Z2017-11-07T20:08:00ZMass loss due to gravitational waves with a cosmological constant
2017
Saw, Vee-Liem
The theoretical basis for the energy carried away by gravitational waves that an isolated gravitating system emits was first formulated by Hermann Bondi during the 1960s. Recent findings from looking at distant supernovae reveal that the rate of expansion of our Universe is accelerating, which may be well-explained by including a positive cosmological constant into the Einstein field equations for general relativity. By solving the Newman-Penrose equations (which are equivalent to the Einstein field equations), we generalise this notion of Bondi mass-energy and thereby provide a firm theoretical description of how an isolated gravitating system loses energy as it radiates gravitational waves in a Universe that expands at an accelerated rate.
2017-11-07T20:08:00ZThe structure of GCR and CCR groupoid C*-algebrasvan Wyk, Daniel Willemhttp://hdl.handle.net/10523/75832017-10-10T23:52:08Z2017-10-10T23:50:12ZThe structure of GCR and CCR groupoid C*-algebras
2017
van Wyk, Daniel Willem
We remove the assumptions of amenability in two theorems of Clark about C*-algebras of locally compact groupoids. The first result is that if the groupoid C*-algebra is GCR, or equivalently then the groupoid's orbits are locally closed. We prove the contrapositive. We begin by constructing a direct integral representation of the groupoid C*-algebra with respect to a measure on the groupoid's unit space. If the orbits are not locally closed, then there is a non-trivial ergodic measure on the unit space. We adapt a known result for transformation groups to groupoids, which shows that the direct integral representation cannot be type I if the measure on the unit space is non-trivially ergodic.
The second result is that if the groupoid C*-algebra is CCR, then the groupoid's orbits are closed. Here we show that if a representation of a stability subgroup is induced to a representation of the groupoid C*-algebra, then the induced representation is equivalent to a representation as multiplication operators acting on a vector-valued L2-space. If we assume the groupoid C*-algebra is CCR, but an orbit is not closed, then the equivalence of two representations as multiplication operators leads to a contradiction.
2017-10-10T23:50:12ZInverse problems in evolutionary biologyHiscott, Gordonhttp://hdl.handle.net/10523/74302018-02-28T01:24:14Z2017-07-07T04:02:45ZInverse problems in evolutionary biology
2017
Hiscott, Gordon
In this thesis, we explore three techniques which could be used to increase the efficiency of analyses in evolutionary genetics while still producing reasonably accurate results. The first of these methods improves the efficiency of analyses based on Markov chain Monte Carlo (MCMC) through the application of delayed acceptance sampling, an MCMC method with an additional proposal step in which an acceptance probability is computed from computationally less expensive approximate likelihoods. Rejection at the additional decision step should allow software like SNAPP (``SNP and AFLP Phylogenies") to avoid unnecessary computation of full likelihoods and, therefore, run more efficiently. The second method we discuss combines dynamic programming with classical numerical integration methods to compute likelihoods with respect to continuous trait models on trees. This method assumes explicitly known transition densities, but is efficient and has a relatively fast convergence rate. We apply the method to a threshold model which combines continuous traits with discrete observations. The third method we look at is another dynamic programming integration algorithm, except that this algorithm takes advantage of a basis function approximation of likelihood functions. This method allows for numerical solutions to PDEs to be applied directly and the use of Chebyshev polynomials as the basis functions make the method easy to implement. We apply this method to the computation of the likelihood given a genetic data set generated by diffusion processes.
2017-07-07T04:02:45ZModeling Continuous Time Series With Many ZerosWang, Yunanhttp://hdl.handle.net/10523/73972017-06-25T23:02:07Z2017-06-25T23:00:58ZModeling Continuous Time Series With Many Zeros
2017
Wang, Yunan
Earthquake activity is generally modeled using point processes as earthquake events usually occur at random times and locations. Recent studies have found it mathematically challenging and computationally complex to incorporate a point process model into a hidden Markov model to describe long-term seismicity. Given that earthquake data can be discretized in time to consider daily or hourly energy release, time series models could be a useful method for earthquake data analysis. Time series models can account for the autocorrelation of earthquakes. However, one issue that arises with the earthquake occurrence data is that there is a substantial proportion of time when no earthquake is recorded. This thesis proposes a class of two-part autoregressive (2PAR) models for continuous time series data with excess zeros. We employ a Bernoulli variable to model the excess zeros in the data, and use autoregressive processes to describe the serial correlation. Using this class of 2PAR models, we can model correlations that exist in either zeros or nonzeros in the data. We have proposed a class of residual analysis to check the goodness-of-fit of the proposed models. We also introduced a forecasting procedure using simulation to check the performance of the models.
We carried out a simulation study which shows that the estimators are unbiased and consistent, and the residual analysis and forecasting procedure for the proposed models are promising. We applied the proposed models to the energy indices obtained from the total stress release per hour from the 2010 Darfield earthquake sequence. The results reveal that the 2PAR models with serial correlation in both the presence probability and the earthquake energy indices captured the main features of the data. A retrospective forecasting experiment suggested that the proposed models provide higher information gain against a reference model.
2017-06-25T23:00:58ZAnalogues of Leavitt path algebras for higher-rank graphsPangalela, Yosafat Eka Prasetyahttp://hdl.handle.net/10523/72792018-02-28T01:38:12Z2017-04-06T20:59:28ZAnalogues of Leavitt path algebras for higher-rank graphs
2017
Pangalela, Yosafat Eka Prasetya
Directed graphs and their higher-rank analogues provide an intuitive framework to study a class of C*-algebras which we call graph algebras. The theory of graph algebras has been developed by a number of researchers and also influenced other branches of mathematics: Leavitt path algebras and Cohn path algebras, to name just two.
Leavitt path algebras for directed graphs were developed independently by two groups of mathematicians using different approaches. One group, which consists of Ara, Goodearl and Pardo, was motivated to give an algebraic framework of graph algebras. Meanwhile, the motivation of the other group, which consists of Abrams and Aranda Pino, is to generalise Leavitt's algebras, in which the name Leavitt comes from. Later, Abrams and now with Mesyan introduced the notion of Cohn path algebras for directed graphs. Interestingly, both Leavitt path algebras and Cohn path algebras for directed graphs can be viewed as algebraic analogues of C*-algebras of directed graphs.
In 2013, Aranda Pino, J. Clark, an Huef and Raeburn introduced a higher-rank version of Leavitt path algebras which we call Kumjian-Pask algebras. At their first appearance, Kumjian-Pask algebras were only defined for row-finite higher-rank graphs with no sources. Clark, Flynn and an Huef later extended the coverage by also considering locally convex row-finite higher-rank graphs. On the other hand, Cohn path algebras for higher rank graphs still remained a mystery.
This thesis has two main goals. The first aim is to introduce Kumjian-Pask algebras for a class of higher-rank graphs called finitely-aligned higher-rank graphs. This type of higher-rank graph covers both row-finite higher-rank graphs with no sources and locally convex row-finite higher-rank graphs. Therefore, we give a generalisation of the existing Kumjian-Pask algebras. We also establish the graded uniqueness theorem and the Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras of finitely-aligned higher-rank graphs.
The second aim is to introduce a higher-rank analogue of Cohn path algebras. We then study the relationship between Kumjian-Pask algebras and Cohn path algebras and use this to investigate properties of Cohn path algebras. Finally, we establish a uniqueness theorem for Cohn path algebras.
2017-04-06T20:59:28ZC*-algebras generated by semigroups of partial isometriesTolich, Ilijahttp://hdl.handle.net/10523/72772018-02-28T01:38:08Z2017-04-06T04:29:56ZC*-algebras generated by semigroups of partial isometries
2017
Tolich, Ilija
This thesis examines the C*-algebras associated to semigroups of partial isometries. There are many interesting examples of C*-algebras generated by families of partial isometries, for example the C*-algebras associated to directed graphs and the C*-algebras associated to inverse semigroups.
In 1992 Nica introduced a class of partially ordered groups called quasi-lattice ordered groups, and studied the C*-algebras generated by semigroups of isometries satisfying a covariance condition. We have adapted Nica's construction for semigroups of partial isometries associated to what we call doubly quasi-lattice ordered groups. For each doubly quasi-lattice ordered group we construct two algebras: a concretely defined reduced algebra, and a universal algebra generated by a covariant family of partial isometries. We examine when representations of the universal algebra are faithful, and this gives rise to a notion of amenability for doubly quasi-lattice ordered groups.
We prove several recognition theorems for amenability. In particular, we prove that the universal and reduced algebras are isomorphic if and only if the doubly quasi-lattice ordered group is amenable. Further, we prove that if there is an order preserving homomorphism from a doubly quasi-lattice ordered group to an amenable group, then the quasi-lattice ordered group is amenable and the associated universal algebra is nuclear.
2017-04-06T04:29:56Z