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Item Random discrete groups of Möbius transformations : probabilities and limit set dimensions : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand(Massey University, 2017) O'Brien, Graeme KThis thesis addresses two areas related to the quantification of discrete groups. We study "random" groups of Möbius transformations and in particular random two-generator groups; that is, groups where the generators are selected randomly. Our intention is to estimate the likelihood that such groups are discrete and to calculate the expectation of their associated geometric and topological parameters. Computational results of the author [55] that indicate a low probability of a random group being discrete are extended and we also assess the expected Hausdorff dimension of the limit set of a discrete group. In both areas of research analytic determinations are correlated with computational results. Our results depend on the precise notion of randomness and we introduce geometrically natural probability measures on the groups of all Möbius transformations of the circle and the Riemann sphere.Item The edge slide graph of the n-dimensional cube : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand(Massey University, 2017) Al Fran, Howida AdelThe goal of this thesis is to understand the spanning trees of the n-dimensional cube Qn by understanding their edge slide graph. An edge slide is a move that “slides” an edge of a spanning tree of Qn across a two-dimensional face, and the edge slide graph is the graph on the spanning trees of Qn with an edge between two trees if they are connected by an edge slide. Edge slides are a restricted form of an edge move, in which the edges involved in the move are constrained by the structure of Qn, and the edge slide graph is a subgraph of the tree graph of Qn given by edge moves. The signature of a spanning tree of Qn is the n-tuple (a1; : : : ; an), where ai is the number of edges in the ith direction. The signature of a tree is invariant under edge slides and is therefore constant on connected components. We say that a signature is connected if the trees with that signature lie in a single connected component, and disconnected otherwise. The goal of this research is to determine which signatures are connected. Signatures can be naturally classified as reducible or irreducible, with the reducible signatures being further divided into strictly reducible and quasi-irreducible signatures. We determine necessary and sufficient conditions for (a1; : : : ; an) to be a signature of Qn, and show that strictly reducible signatures are disconnected. We conjecture that strict reducibility is the only obstruction to connectivity, and present substantial partial progress towards an inductive proof of this conjecture. In particular, we reduce the inductive step to the problem of proving under the inductive hypothesis that every irreducible signature has a “splitting signature” for which the upright trees with that signature and splitting signature all lie in the same component. We establish this step for certain classes of signatures, but at present are unable to complete it for all. Hall’s Theorem plays an important role throughout the work, both in characterising the signatures, and in proving the existence of certain trees used in the arguments.Item Microbial co-existence and stable equilibria in a mechanistic model of enteric methane production : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū Campus, New Zealand(Massey University, 2016) Wang, YuanchengGlobally, 14.5% of all anthropogenic greenhouse gases come from ruminants. One of these is methane, which is produced in the rumen of ruminant animals. Feed is degraded by microbes to produce volatile fatty acids (which are absorbed by the animal) and hydrogen (which is metabolized by methanogens to form methane). The dynamics of hydrogen production and metabolism are subject to thermodynamic control imposed by the hydrogen concentration. Existing models to estimate methane production are based on calculation of hydrogen balances without considering the presence of methanogens and do not include thermodynamic control. In this project, a model is developed based on glucose-hydrogenmethanogen dynamics to estimate methane production and illustrates a co-existence of microbes that employs different fermentation pathways competing for the same food source in the rumen. Glucose was chosen as an example of a fermentable feed component. A thermodynamic term was integrated into a Monod-type model to represent the thermodynamic control of hydrogen concentration on the rates of hydrogen generation and hydrogen metabolism. Results of this model suggest that the microbial community composition and the combination of the different pathways are determined by the rumen environment, biological parameters of the microbes and the feedback imposed by substrate and product concentrations. The mathematical enunciation of this model is therefore consistent with biological expectations. This model could be expanded to include plant polymer degradation rate, feeding level and feeding frequency to explore their effects on methane production. This model could also be integrated into models of whole rumen function to address more complex questions. It would also support experimentation with animals for understanding factors that control methane formation and to explore methane mitigation strategies.Item Image registration under conformal diffeomorphisms : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand(Massey University, 2017) Tufail, Muhammad YousufImage registration is the process of finding an alignment between two or more images so that their appearance matches. It has been widely studied and applied to several fields, including medical imaging and biology (where it is related to morphometrics). In biology, one motivation for image registration comes from the work of Sir D'Arcy Thompson. In his book On Growth and Form he presented several examples where a grid superimposed onto a two-dimensional image of one species was smoothly deformed to suggest a transformation to an image of another species. His examples include relationships between species of fish and comparison of human skulls with higher apes. One of Thompson's points was that these deformations should be as `simple' as possible. In several of his examples, he uses what he calls an isogonal transformation, which would now be called conformal, i.e., angle-preserving. His claims of conformally-related change between species were investigated further by Petukhov, who used Thompson's grid method as well as computing the cross-ratio (which is an invariant of the Möbius group, a finite-dimensional subgroup of the group of conformal diffeomorphisms) to check whether sets of points in the images could be related by a Möbius transformation. His results suggest that there are examples of growth and evolution where a Möbius transformation cannot be ruled out. In this thesis, we investigate whether or not this is true by using image registration, rather than a point-based invariant: we develop algorithms to construct conformal transformations between images, and use them to register images by minimising the sum-of-squares distance between the pixel intensities. In this way we can see how close to conformal the image relationships are. We develop and present two algorithms for constructing the conformal transformation, one based on constrained optimisation of a set of control points, and one based on gradient flow. For the first method we consider a set of different penalty terms that aim to enforce conformality, based either on discretisations of the Cauchy-Riemann equations, or geometric principles, while in the second the conformal transformation is represented as a discrete Taylor series. The algorithms are tested on a variety of datasets, including synthetic data (i.e., the target is generated from the source using a known conformal transformation; the easiest possible case), and real images, including some that are not actually conformally related. The two methods are compared on a set of images that include Thompson's fish example, and a small dataset demonstrating the growth of a human skull. The conformal growth model does appear to be validated for the skulls, but interestingly, not for Thompson's fish.Item Shewhart methodology for modelling financial series : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Palmerston North, New Zealand(Massey University, 2017) Premarathna, Liyana Pathiranaralalage Nadeeka DilrukshiQuality management techniques are widely used in industrial applications for monitoring observable process variation. Among them, the scientific notion of Shewhart principles is vital for understating variations in any type of process or service. This study extensively investigates and demonstrates Shewhart methodology for financial data. Extremely heavy tails noted in the empirical distribution of stock returns led to the development of new parametric probability distributions for pricing assets and forecasting market risk. Standard asset pricing models have also extended to account the first four (excess) moments in return distributions. These approaches remain complex, but yet they are inadequate for capturing extreme volatility caused by infrequent market events. It is well known that the security markets are always subjected to a certain amount of variability caused by noise-traders and other frictional price changes. Unforeseen events which are happening in the world may lead to huge market losses. This research shows that Shewhart methodology for partitioning data into common and special cause variations adds value to modelling stock returns. Applicability of the proposed method is discussed using several scenarios occurring in an industrial process and a financial market. A set of new propositions based on Shewhart methodology is formed for finer description of the statistical properties in stock returns. Research issues which are related to the first four moments, co-moments and autocorrelation in stock returns are identified. New statistical tools such as difference control charts, odd-even analysis and estimates for co-moments are proposed to investigate the new propositions and research issues. Finally, several risk measures are proposed, and considered with respect to investor’s preferences. The research issues are investigated using partitioned data from S&P 500 stocks and the findings show that inmost of the scenarios, contradictory conclusions were made as a result of special cause variations. A modelling approach based on common and special cause variations is therefore expected to lead appropriate asset pricing and portfolio management. New statistical tools proposed in this study can be used to other time series data; a new R-package called QCCTS (Quality Control Charts for Time Series) is developed for this purpose.Item Acceptance sampling for food quality assurance : this dissertation is submitted for the degree of Doctor of Philosophy in Statistics, Institute of Fundamental Sciences, Massey University(Massey University, 2017) Santos-Fernández, EdgarAcceptance sampling plays a crucial role in food quality assurance. However, safety inspection represents a substantial economic burden due to the testing costs and the number of quality characteristics involved. This thesis presents six pieces of work on the design of attribute and variables sampling inspection plans for food safety and quality. Several sampling plans are introduced with the aims of providing a better protection for the consumers and reducing the sample sizes. The effect of factors such as the spatial distribution of microorganisms and the analytical unit amount is discussed. The quality in accepted batches has also been studied, which is relevant for assessing the impact of the product in the public health system. Optimum design of sampling plans for bulk materials is considered and different scenarios in terms of mixing efficiency are evaluated. Single and two-stage sampling plans based on compressed limits are introduced. Other issues such as the effect of imperfect testing and the robustness of the plan have been also discussed. The use of the techniques is illustrated with practical examples. We considered numerous probability models for fitting aerobic plate counts and presence-absence data from milk powder samples. The suggested techniques have been found to provide a substantial sampling economy, reducing the sample size by a factor between 20 and 80% (when compared to plans recommended by the international Commission on Microbiological Specification for Food (ICMSF) and the CODEX Alimentarius). Free software and apps have been published, allowing practitioners to design more stringent sampling plans. Keywords: Bulk material, Composite samples, Compressed limit, Consumer Protection, Double sampling plan, Food safety, Measurement errors, Microbiological testing, Sampling inspection plan.Item Image registration using finite dimensional lie groups : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand(Massey University, 2016) Zarredooghabadi, RaziyehD'Arcy Thompson was a biologist and mathematician who, in his 1917 book `On Growth and Form', posited a `Theory of Transformations', which is based on the observation that a smooth, global transformation of space may be applied to the shape of an organism so that its transformed shape corresponds closely to that of a related organism. Image registration is the computational task of finding such transformations between pairs of images. In modern applications in areas such as medical imaging, the transformations are often chosen from the infinite-dimensional diffieomorphism group. However, this differs from Thompson's approach where the groups are chosen to be as simple as possible, and are generally finite-dimensional. The main exception to this is the similarity group of translation, rotation, and scaling, which is used to pre-align images. In this thesis the set of planar Lie groups are investigated and applied to image registration of the types of images that Thompson considered. As these groups are smaller, successful registration in these groups provides more specific information about the relationship between the images than diffeomorphic registration does, as well as providing faster implementations. We build a lattice of the Lie groups showing which are subgroups of each other, and the groups are used to perform image registration by minimizing the L2-norm of the difference between the group-transformed source image and the target image. A robust, practical, and efficient algorithm for registration in Lie groups is developed and tested on a variety of image types. Each successful registration returns a point in a Lie group. Given several related images (such as the hooves of several animals) it is possible to find smooth curves that pass through the Lie group elements used to relate the various images. These curves can then be employed to interpolate points between the set of images or to extrapolate to new images that have not been seen before. We discuss the mathematics behind this and demonstrate it on the images that Thompson used, as well as other datasets of interest. Finally, we consider using a sequence of the planar Lie groups to perform registration, with the output from one group being used as the input to the next. We call this multiregistration, and have identified two types: where the smallest group is a subgroup of the next smallest, and so on up a chain, and where the groups are not directly related, i.e., separated on the lattice. We demonstrate experimentally that multiregistration can provide more information about the relationship between images than simple registration. In addition, we show that transformations that cannot be obtained by a single registration in any of the groups considered can be successfully reached.Item Tree-based models for poverty estimation : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Manawatu(Massey University, 2016) Bilton, Penelope AThe World Food Programme utilises the technique of poverty mapping for efficient allocation of aid resources, with the objective of achieving the first two United Nations Sustainable Development Goals, elimination of poverty and hunger. A statistical model is used to estimate levels of deprivation across small geographical domains, which are then displayed on a poverty map. Current methodology employs linear mixed modelling of household income, the predictions from which are then converted to various area-level measures of well-being. An alternative technique using tree-based methods is developed in this study. Since poverty mapping is a small area estimation technique, the proposed methodology needs to include auxiliary information to improve estimate precision at low levels, and to take account of complex survey design of the data. Classifcation and regression tree models have, to date, mostly been applied to data assumed to be collected through simple random sampling, with a focus on providing predictions, rather than estimating uncertainty. The standard type of prediction obtained from tree-based models, a "hard" tree estimate, is the class of interest for classification models, or the average response for regression models. A \soft" estimate equates to the posterior probability of being poor in a classification tree model, and in the regression tree model it is represented by the expectation of a function related to the poverty measure of interest. Poverty mapping requires standard errors of prediction as well as point estimates of poverty, but the complex structure of survey data means that estimation of variability must be carried out by resampling. Inherent instability in tree-based models proved a challenge to developing a suitable variance estimation technique, but bootstrap resampling in conjunction with soft tree estimation proved a viable methodology. Simulations showed that the bootstrap based soft tree technique was a valid method for data with simple random sampling structure. This was also the case for clustered data, where the method was extended to utilise the cluster bootstrap and to incorporate cluster effects into predictions. The methodology was further adapted to account for stratification in the data, and applied to generate predictions for a district in Nepal. Tree-based estimates of standard error of prediction for the small areas investigated were compared with published results using the current methodology for poverty estimation. The technique of bootstrap sampling with soft tree estimation has application beyond poverty mapping, and for other types of complex survey data.Item Modular forms and two new integer sequences at level 7 : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand(Massey University, 2016) O'Brien, Lynette AnneInteger sequences resulting from recurrence relations with polynomial coefficients are rare. Two new integer sequences have been discovered and are the main result in this thesis. They consist of a three-term quadratic recurrence (n+1)²c₇(n+1) = (26n² + 13n + 2)c₇(n) + 3(2n - 2)c₇(n-1) with initial conditions c₇(-1) = 0 and c₇(0) = 1, and a five-term quartic recurrence (n + 1)⁴u₇(n + 1) = -Pu₇(n) - Qu₇(n - 1) - Ru₇(n-2) - Su₇(n - 3) where P = 26n⁴ + 52n³+ 58n² + 32n + 7, Q = 267n⁴ + 268n² + 18, R = 1274n⁴ - 2548n³ + 2842n² - 1568n + 343, S = 2401(n - 1)⁴ with initial conditions u₇(0) = 1 and u₇(-1) = u₇(-2) = u₇(-3) = 0. The experimental procedure used in their discovery utilizes the mathematical software Maple. Proofs are given that rely on the theory of modular forms for level 7, Ramanujan's Eisenstein series, theta functions and Euler products. Differential equations associated with theta functions are solved to reveal these recurrence relations. Interesting properties are investigated including an analogue of Clausen's identity, asymptotic behaviour of the sequences and finally two conjectures for congruence properties are given.Item Bayesian Modelling of Direct and Indirect Effects of Marine Reserves on Fishes : A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand.(Massey University, 2016) Smith, Adam Nicholas HowardThis thesis reviews and develops modern advanced statistical methodology for sampling and modelling count data from marine ecological studies, with specific applications to quantifying potential direct and indirect effects of marine reserves on fishes in north eastern New Zealand. Counts of snapper (Pagrus auratus: Sparidae) from baited underwater video surveys from an unbalanced, multi-year, hierarchical sampling programme were analysed using a Bayesian Generalised Linear Mixed Model (GLMM) approach, which allowed the integer counts to be explicitly modelled while incorporating multiple fixed and random effects. Overdispersion was modelled using a zero-inflated negative-binomial error distribution. A parsimonious method for zero inflation was developed, where the mean of the count distribution is explicitly linked to the probability of an excess zero. Comparisons of variance components identified marine reserve status as the greatest source of variation in counts of snapper above the legal size limit. Relative densities inside reserves were, on average, 13-times greater than outside reserves. Small benthic reef fishes inside and outside the same three reserves were surveyed to evaluate evidence for potential indirect effects of marine reserves via restored populations of fishery-targeted predators such as snapper. Sites for sampling were obtained randomly from populations of interest using spatial data and geo-referencing tools in R—a rarely used approach that is recommended here more generally to improve field-based ecological surveys. Resultant multispecies count data were analysed with multivariate GLMMs implemented in the R package MCMCglmm, based on a multivariate Poisson lognormal error distribution. Posterior distributions for hypothesised effects of interest were calculated directly for each species. While reserves did not appear to affect densities of small fishes, reserve-habitat interactions indicated that some endemic species of triplefin (Tripterygiidae) had different associations with small-scale habitat gradients inside vs outside reserves. These patterns were consistent with a behavioural risk effect, where small fishes may be more strongly attracted to refuge habitats to avoid predators inside vs outside reserves. The approaches developed and implemented in this thesis respond to some of the major current statistical and logistic challenges inherent in the analysis of counts of organisms. This work provides useful exemplar pathways for rigorous study design, modelling and inference in ecological systems.