The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated. / Whole document restricted, but available by request, use the feedback form to request access.

This thesis consists of two parts. The first part proposes a new model – the Markov-modulated Hawkes process with stepwise decay (MMHPSD) to investigate the seismicity rate. The MMHPSD is a self-exciting process which switches among different states, in each of which the process has distinguishable background seismicity and decay rates. Parameter estimation is developed via the expectation maximization algorithm. The model is applied to data from the Landers earthquake sequence, demonstrating that it is useful for modelling changes in the temporal patterns of seismicity. The states in the model can capture the behavior of main shocks, large aftershocks, secondary aftershocks and a period of quiescence with different background rates and decay rates. The state transitions can then explain the seismicity rate changes and help indicate if there is any seismicity shadow or relative quiescence. The second part of this thesis develops statistical methods to examine earthquake sequences possessing ancillary data, in this case groundwater level data or GPS measurements of deformation. For the former, signals from groundwater level data at Tangshan Well, China, are extracted for the period from 2002 to 2005 using a moving window method. A number of different statistical techniques are used to detect and quantify coseismic responses to P, S, Love and Rayleigh wave arrivals. The P phase arrivals appear to trigger identifiable oscillations in groundwater level, whereas the Rayleigh waves amplify the water level movement. Identifiable coseismic responses are found for approximately 40 percent of magnitude 6+ earthquakes worldwide. A threshold in the relationship between earthquake magnitude and well–epicenter distance is also found, satisfied by 97% of the identified coseismic responses, above which coseismic changes in groundwater level at Tangshan Well are most likely. A non-linear filter measuring short-term deformation rate changes is introduced to extract signals from GPS data. For two case studies of a) deep earthquakes in central North Island, New Zealand, and b) shallow earthquakes in Southern California, a hidden Markov model (HMM) is fitted to the output from the filter. Mutual information analysis indicates that the state having the largest variation of deformation rate contains precursory information that indicates an elevated probability for earthquake occurrence.

The New Zealand Ministry of Fisheries relies on fishery assessments to determine suitable catch quotas for exploited fisheries. Currently, 628 fish stocks are managed in New Zealand using the Quote Management System, which includes the 8 com- mercial flatfish species caught within the Exclusive Economic Zone. These eight species of flatfish, which includes four species of flounder, two species of sole, brill and turbot, are currently managed using a combined catch quota. Since these eight species are managed using a common catch quota, there is concern that some of the individual species may be under or over-fished. This thesis describes work involving the flatfish species caught in the FLA3 man- agement area, around the south island of New Zealand. The FLA3 management area contains three key species: New Zealand sole, lemon sole, and sand flounder. Due to the nature and limitations of the data available, simple biomass dynamic models were applied to these species. The maximum likelihood and Bayesian goodness of fit techniques were used to estimate the model parameters. Three models were used: the Fox model, the Schaefer model and the Pella-Tomlinson model with m = 3. As a mathematical/statistical exercise, these models were used to conduct a risk analysis to analyse the advantages and disadvantages of six management options for setting a TACC. However, because of issues over the way that the parameter K has been modelled (due to necessity caused by the lack of data), this should not be seen as an appropriate method for estimating the fish stock. Conclusions were drawn from the results regarding suitable future action for the assessment and management of flatfish stock in FLA3.

A useful measure of quality of a global optimisation algorithm such as simulated annealing is the length of time it must be run to reach a global optimum within a certain accuracy. Such a performance measure assists in choosing and tuning algorithms. This thesis proposes an approach to obtaining such a measure through successive approximation of a generic stochastic global optimisation algorithm with a sequence of stochastic processes culminating in backtracking adaptive search. The overall approach is to approximate the progress of an optimisation algorithm with that of a model process, backtracking adaptive search. The known convergence rate of the model then provides an estimator of the unknown convergence rate of the original algorithm. Parameters specifying this model are chosen based on observation of the optimisation algorithm. The optimisation algorithm may first be approximated with a time-inhomogeneous Markovian process defined on the problem range. The distribution of the number of iterations to convergence for this averaged range process is shown to be identical with that of the original process. This process is itself approximated by a time-homogeneous Markov process in the range, the asymptotic averaged range process. This approximation is defined for all Markovian optimisation algorithms and a weak condition under which its convergence time closely matches that of the original algorithm is developed. The asymptotic averaged range process is of the same form as backtracking adaptive search, the final stage of approximation. Backtracking adaptive search is an optimisation algorithm which generalises pure adaptive search and hesitant adaptive search. In this thesis the distribution of the number of iterations for which the algorithm runs in order to reach a sufficiently extreme objective function level is derived. Several examples of backtracking adaptive search on finite problems are also presented, including special cases that have received attention in the literature. Computational results of the entire approximation framework are reported for several examples. The method can be applied to any optimisation algorithm to obtain an estimate of the time required to obtain solutions of a certain quality. Directions for further work in order to improve the accuracy of such estimates are also indicated.

Researchers are frequently faced with the problem of analyzing incomplete and often unbalanced genotype-by-environment (GxE) matrices which arise as a trials programme progresses over seasons. The principal data for this investigation, arising from a ten year programme of onion trials, has less than 2,300 of the 49,200 combinations from the 400 genotypes and 123 environments. This 'sparsity' renders standard GxE methodology inapplicable. Analysis of this data to identify onion varieties that suit the shorter, hotter days of tropical and subtropical locations therefore presented a unique challenge. Removal of some data to form a complete GxE matrix wastes information and is consequently undesirable. An incomplete GxE matrix can be analyzed using the additive main effects and multiplicative interaction (AMMI) model in conjunction with the EM algorithm but proved unsatisfactory in this instance. Cluster analysis has been commonly used in GxE analyses, but current methods are inadequate when the data matrix is incomplete. If clustering is to be applied to incomplete data sets, one of two routes needs to be taken: either the clustering procedure must be modified to handle the missing data, or the missing entries must be imputed so that standard cluster analysis can be performed. A new clustering method capable of handling incomplete data has been developed. 'Two-stage clustering', as it has been named, relies on a partitioning of squared Euclidean distance into two independent components, the GxE interaction and the genotype main effect. These components are used in the first and second stages of clustering respectively. Two-stage clustering forms the basis for imputing missing values in a GxE matrix, so that a more complete data array is available for other GxE analyses. 'Two-stage imputation' estimates unobserved GxE yields using inter-genotype similarities to adjust observed yield data in the environment in which the yield is missing. This new imputation method is transferrable to any two-way data situation where all observations are measured on the same scale and the two factors are expected to have significant interaction. This simple, but effective, imputation method is shown to improve on an existing method that confounds the GxE interaction and the genotype main effect. Future development of two-stage imputation will use a parameterization of two-stage clustering in a multiple imputation process. Varieties recommended for use in a certain environment would normally be chosen using results from similar environments. Differing cluster analysis approaches were applied, but led to inconsistent environment clusterings. A graphical summary tool, created to ease the difficulty in identifying the differences between pairs of clusterings, proved especially useful when the number of clusters and clustered observations were high. 'Cluster influence diagrams' were also used to investigate the effects the new imputation method had on the qualitative structure of the data. A consequence of the principal data's sparsity was that imputed values were found to be dependent on the existence of observable inter-genotype relationships, rather than the strength of these observable relationships. As a result of this investigation, practical recommendations are provided for limiting the detrimental effects of sparsity. Applying these recommendations will enhance the future ability of two-stage imputation to identify those onion varieties that suit tropical and subtropical locations.

This thesis investigates the analysis of the extended model of animal growth proposed by Oliviera et al (personal communication, July 2009). This mechanistic model of animal growth based on a detailed representation of energy dynamics focussing on the interaction between four compartment of body composition; nutrient level, fat content, visceral protein and non-visceral protein. The model is mathematically analysed and the behaviour of the model for different feeding level is examined. The animal growth model exhibits thresholds typical of nonlinear systems and multiple stable steady states which have distinct basins of stability which depend on the value of the large number of physiologically-determined parameters. These have not been previously explored theoretically and these are done in this thesis. The model demonstrates richer behaviour where path-following techniques are used to explore the distribution in parameter space of the varying phenomenology.

In any population of cells, individual cells grow for some period of time and then divide into two or more parts, called daughters. To describe this process mathematically, we need to specify functions describing the growth rate, size at division, and proportions into which each cell divides. In this thesis, it is assumed that the growth rate of a cell can be determined precisely from its size, but that both its size at division and the proportions into which it divides may be described stochastically, by probability density functions whose parameters are dependent on cell size and age (or birth-size). Special cases are also considered where all cells with the same birth-size divide at the same size, or where all cells divide exactly in half. We consider a population of cells growing and dividing steadily, such that the total cell population is increasing, but the proportion of cells in any size class remains constant. In Chapter 1, equations are derived which need to be solved in order to deduce the shape of the steady size distribution (or steady size/age or size/birth-size distributions) from any given growth rate and probability distributions describing the division rate and division proportions. In the general case, a Fredholm-type integral equation is obtained, but if the probability of cell division depends on cell size only (i.e. not age or birth-size), and all cells divide into equal-sized daughters, then we obtain a functional differential equation. In two special cases, the resulting equations simplify considerably, and it is these cases which are explored further in this thesis. The first is where the probability of a cell dividing in any instant of time is a constant, independent of cell age or size. In Chapter 2, the functional differential equation resulting when cells divide into equal-sized daughters is solved for the special case where the growth rate is constant, and in an appendix the case where the growth rate is described by a power law is dealt with. The second case which simplifies is where the time-independent part of the growth rate of a cell is proportional to cell size. This case is particularly important, as it is a good first-order approximation to the real cell growth rate in some structured tissues, and in some bacteria. The special case in which this leads to a functional differential equation is discussed in Chapter 3, and the integral equation arising in the general case is dealt with in Chapter 4. Finally, the conditions under which the integral operator in Chapter 4 will be both square-integrable and non-factorable are discussed in Chapter 5. It is shown that if these conditions are satisfied then a unique, stable, steady size distribution will exist.

This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.

The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated. / Whole document restricted, but available by request, use the feedback form to request access.

This thesis reports on a study of underground fluid flow and boiling processes which take place in hydrothermal eruptions. A conceptual model is presented for the eruptive process and a laboratory scale physical model confirming the effectiveness of this process is described. A mathematical formulation of the underground flow problem is given for two fluid flow regimes: two-phase homogeneous mixture (HM) flow and separable two-phase (SP) flow. Solutions to the system of equations obtained are solved under the simplifying assumptions of two-dimensional steady isothermal flow and transient non-isothermal horizontal flow. The main contribution of the study on steady isothermal flows is a description of how the ground flow may recover following a hydrothermal eruption. A numerical technique developed for plotting the streamlines in this case (and verified against analytic results) may also have applications in solving the steady non-isothermal flow problem. The main contribution of the study on the transient horizontal flow problem is a comparison of the differing predictions of HM and SP flow. The rate at which a boiling front progresses through a porous medium and the degree of boiling which occurs is described for each fluid flow regime. A set of horizontal physical experiments and numerical simulations have also been carried out for comparison with the mathematical model. Qualitative results for these three models agree. Suggestions given for improvements to the design of the physical experiment provide a basis for future study into the type of flow which occurs in hydrothermal eruptions