Data collected in the biomedical and social sciences by means of questionnaires is in most instances qualitative in nature. Such data, typically set out in the form of (multi-dimensional) contingency tables, is usually subjected to hypothesis testing in order to assess the interrelationships between the questions. Prior to undertaking confirmatory procedures, we argue that exploratory techniques should be used to gain a "feel" for the data. Correspondence Analysis (an exploratory data analysis procedure) and Log-linear Model building (a confirmatory data analysis procedure) are discussed before an investigation is undertaken to ascertain whether they can be used in conjunction. We found that correspondence analysis : (i) detects questions that are "strictly" independent/unrelated, (ii) detects pairwise relationships between questions (2-factor interactions) and thus can be used to suggest a splitting of large data sets into two or more subsets of questions that are independent, each of which can be analysed separately, and (iii) cannot be used to select log-linear models in general because it does not detect higher order interactions.

Includes bibliography. / This study initially set out to consider the possibility of constructing an adaptive robust estimation procedure for the standard linear regression model when the disturbance vector deviated from normality, however, after the initial success in that field it seemed only appropriate that the approach be extended to robust location parameter estimation. This is a particular case of the regression model and an area in which a number of different estimators have been proposed and a great deal of comparative research work done. Due to the wider scope of such research the greater part of the thesis is devoted to this field of research which led to many interesting and useful results and conclusions.

Includes bibliographical references. / A dissimilarity coefficient for estimating the dissimilarity between two bird atlas distributions is developed. This coefficient is based on the Euclidean distance concept. The atlas distributions are compared over all quarter degree grid cells. Existing coefficients are not suitable for the comparison of distributions with different total areas and species with different mean reporting rates. In each grid cell the reliability of the reporting rates depends on the number of checklists collected for the grid cell. Weights are used to solve this problem. To solve the problem of different levels of abundance and conspicuousness of species, the reporting rates are sorted into percentiles, using five or 10 categories for the strictly positive reporting rates. Each grid cell is weighted by a function of the number of checklists collected for the grid cell. The coefficient is scaled by the maximum possible sum of the differences which would occur if there is no overlap between the two distributions, so that the dissimilarity coefficient lies between zero (a perfect match) and one (no overlap). A variety of these coefficients are investigated and compared. The continuity of observed reporting rates in a spatial cellular map is an indication of spatial autocorrelation present, especially between observations which are in close vicinity. We are particularly interested in measuring and comparing the continuity of the reporting rates in the bird distributions from The Atlas of Southern African Birds. The variogram, developed in geostatistics, estimates this spatial autocorrelation. The classical variogram estimator, however, is dependent on the scale of measurement and assumes that the data are intrinsically stationary. The bird atlas distribution maps contain trend and the variance of each observation (reporting rate) is a function of the number of checklists collected for the grid cell and the underlying probability of encountering the species in the grid cell. The approach of removing this binomial measurement error from the variogram developed by McNeill (1991) is investigated but not found satisfactory. A weighted variogram, where each squared difference is weighted by a function of the smaller number of checklists, is developed. To make the variogram values comparable between species a function of the mean reporting rates is used to scale the variogram. We were particularly interested in the first variogram value of each species distribution, 2y(1). The bird distribution maps in The Atlas of Southern African Birds show the raw observed reporting rates. Each of these reporting rates is a random variable dependent on sampling error due to binomial variation based on the number of checklists collected for the grid cell and on the underlying probability of encountering the species. The distribution maps show this measurement error. It is believed that a smoothed version of the bird distribution maps will to some extent improve the statement these observed distributions are aiming to make. Single-step regression methods are investigated for a fast approach to this problem. These cause problems because of frequent 'zero' observed reporting rates and because they smooth the maps too heavily. Generalized Linear Models are investigated and this iterative procedure is applied to model the reporting rates with a binomial distribution on square blocks of nine grid cells where a value for the central cell is 'predicted' in each regression. This approach is especially suited to accommodate the binomial distribution characteristics and is found to smooth the bird atlas distributions well. Because only a local window is taken for each regression, the spatial autocorrelation is adequately included in the spatial explanatory variables.

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The thesis is presented in two parts, (a) "Nonparametric Analysis of Variance", and (b) "An Asymptotic Expansion of the Null Distributions of Kruskal and Wallis's and Friedman's Statistics". In the first part we present a number of new nonparametric tests designed for a variety of experimental situations. These tests are all based on a so-called "matching" principle. The range of situations covered by the tests are: (i) Two-way analysis of variance with a general alternative hypothesis (without interaction). (ii) Two-way analysis of variance with an ordered alternative hypothesis (without interaction). (iii) Interaction in two-way analysis of variance, both the univariate and. multivariate cases. (iv) Latin square designs. (v) Second-order interaction in three-way analysis of variance. (vi) Third-order interaction in four-way analysis of variance. The validity of the tests is supported by a series of simulation studies which were performed with a number of different distributions. In the second part of the thesis we develop an asymptotic expansion for the construction of improved approximations to the null distributions of Kruskal and Wallis's (1952) and Friedman's (1937) statistics. The approximation is founded on the method of steepest descents, a procedure that is better known in Numerical Analysis than in Statistics. In order to implement this approximation it was necessary to derive the third and fourth moments of the Kruskal-Wallis statistic and the fourth moment of Friedman's statistic. Tables of approximate critical values based on this approximation are presented for both statistics.

Mathematical modelling of gene regulatory networks is a relatively new area which is playing an important role in theoretical and experimental investigations that seek to open the door to understanding the real mechanisms that take place in living systems. The current thesis concentrates on studying the animal stress-response gene regulatory network by seeking to predict the consequence of environmental hazards caused by chemical mixtures (typical of industrial pollution). Organisms exposed to pollutants display multiple defensive stress responses, which together constitute an interlinked gene network (the Stress-Response Network; SRN). Multiple SRN reporter-gene outputs have been monitored during single and combined chemical exposures in transgenic strains of two invertebrates, Caenorhabditis elegans and Drosophila melanogaster. Reporter expression data from both species have been integrated into mathematical models describing the dynamic behaviour of the SRN and incorporating its known regulatory gene circuits. We describe some mathematical models of several types of different stress response networks, incorporating various methods of activation and inhibition, including formation of complexes and gene regulation (through several known transcription factors). Although the full details of the protein interactions forming these types of circuits are not yet well-known, we seek to include the relevant proteins acting in different cellular compartments. We propose and analyse a number of different models that describe four different stress response gene networks and through a combination of analytical (including stability, bifurcation and asymptotic) and numerical methods, we study these models to gain insight on the effect of several stresses on gene networks. A detailed time-dependent asymptotic analysis is performed for relevant models in order to clarify the roles of the distinct biochemical reactions that make up several important proteins production processes. In two models we were able to verify the theoretical predictions with the corresponding laboratory experimental observations that carried out by my coworkers in Britain and India.

The estimation of the location in which an impact event took place from its resultant impact spatter bloodstain pattern can be a significant investigative issue in the reconstruction of a crime scene. The bloodstain pattern analysis methods through which an estimate is constructed utilise the established bloodstain pattern analysis principles of spatter bloodstain directionality, impact angle calculation, and straight-line trajectory approximation. Uncertainty, however, can be shown to be present in the theoretical definition and practical approximation of an impact site; the theoretical justification for impact angle calculation; spatter bloodstain sample selection; the dimensional measurement of spatter bloodstain morphologies; the inability to fully incorporate droplet flight dynamics; and the limited numerical methods used to describe mathematical estimates. An experimental computer-based research design was developed to investigate this uncertainty. A series of experimental impact spatter patterns were created, and an exhaustive spatter bloodstain recording methodology developed and implemented. A computer application was developed providing a range of analytical approaches to the investigation of estimate uncertainty, including a three-dimensional computer graphic virtual investigative environment. The analytical computer application was used to generate a series of estimates using a broad spatter bloodstain sampling strategy, with six potentially probative estimates analysed in detail. Two additional pilot projects investigating the utility of a sampled photographic recording methodology and an automated image analysis approach to spatter bloodstain measurement were also conducted. The results of these analyses indicate that, with further development, the application of similar analytical approaches to the construction and investigation of an estimate could prove effective in minimising the effect that estimate uncertainty might have on informing the conclusions of this forensic reconstructive process, and thereby reaffirm the scientific expert evidential status of estimate techniques within legal contexts.

We here analyse a number of class-D amplifier topologies. Class-D amplifiers operate by converting an audio input signal into a high-frequency square wave output, whose lower-frequency components can accurately reproduce the input. Their high power efficiency and potential for low distortion makes them suitable for use in a wide variety of electronic devices. By calculating the outputs from a classical class-D design implementing different sampling schemes we demonstrate that a more recent method, called the Fourier transform/Poisson resummation method, has many advantages over the double Fourier series method, which is the traditional technique employed for this analysis. We thereby show that when natural sampling is used the input signal is reproduced exactly in the low-frequency part of the output, with no distortion. Although this is a known result, our calculations present the method and notation that we later develop. The classical class-D design is prone to noise, and therefore negative feedback is often included in the circuit. Subsequently we incorporate the Fourier transform/Poisson resummation method into a formalised and succinct analysis of a first-order negative feedback amplifier. Using perturbation expansions we derive the audio-frequency part of the output, demonstrating that negative feedback introduces undesirable distortion. Here we reveal the next order terms in the output compared with previous work, giving further insight into the nonlinear distortion. We then further extend the analysis to examine two more complex negative feedback topologies, namely a second-order and a derivative negative feedback design. Modelling each of these amplifiers presents an increased challenge due to the differences in their respective circuit designs, and in addition, for the derivative negative feedback amplifier we must consider scaling regimes based on the relative magnitudes of the frequencies involved. For both designs we establish novel expressions for the output, including the most significant distortion terms.