On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Silica bonded stationary phases: Innovative functionalization methods and novel chromatographic separations based on molecular recognition

The research work included in this thesis examines the synthesis, characterization and chromatographic evaluation of novel bonded silica stationary phases. Innovative methods of preparation of silica hydride intermediates and octadecylsilica using a “green chemistry” approach eliminate the use of toxic organic solvents and exploit the solvating power and enhanced diffusivity of supercritical carbon dioxide to produce phases with a surface coverage of bonded ligands which is comparable to, or exceeds, that achieved using traditional organic solvent-based methods. A new stationary phase is also discussed which displays chromatographic selectivity based on molecular recognition. Chapter 1 introduces the chemistry of silica stationary phases, the retention mechanisms and theories on which reversed-phase liquid chromatography and hydrophilic interaction chromatograpy are based, the art and science of achieving a well packed liquid chromatography column, the properties of supercritical carbon dioxide and molecular recognition chemistry. Chapter 2 compares the properties of silica hydride materials prepared using supercritical carbon dioxide as the reaction medium with those synthesized in an organic solvent. A higher coverage of hydride groups on the silica surface is seen when a monofunctional silane is reacted in supercritical carbon dioxide while trifunctional silanes result in a phase which exhibits different properties depending on the reaction medium used. The differing chromatographic behaviour of these silica hydride materials prepared using supercritical carbon dioxide and using organic solvent are explored in chapter 3. Chapter 4 focusses on the preparation of octadecylsilica using mono-, di- and trifunctional alkoxysilanes in supercritical carbon dioxide and in anhydrous toluene. The surface coverage of octadecyl groups, as calculated using thermogravimetric analysis and elemental analysis, is highest when a trifunctional alkoxysilane is reacted with silica in supercritical carbon dioxide. A novel silica stationary phase is discussed in chapter 5 which displays selectivity for analytes based on their hydrogen bonding capabilities. The phase is also highly selective for barbituric acid and may have a future application in the solid phase extraction of barbiturates from biological samples.

The contribution of warlordism to the disintegration of the western Roman army (c. AD 395-480)

My thesis investigates the dynamics behind the changing nature of the leadership of the western Roman army in the fifth century through the concept of ‘warlordism’. I carried this out by analyzing those cases of insubordination and military unrest in the officer class of the western Roman army, which can be shown to be linked to the slow decline of central authority and the imperial office in the period 395-480. My thesis demonstrates that theories of ‘Warlordism’, as developed in social sciences, can be useful for both the late Imperial west as for other eras of ancient history, such as the late Roman republic. Warlordism was a way of continuing politics, if necessary by military means, when commanders found themselves outside the legitimate framework. Unlike the case of usurpation of the imperial office, when there was little hope of achieving permanent recognition and acceptance, it offered insubordinate officers a chance of returning to the ruling imperial regime depending on circumstances and the success of their resistance. I propose that warlordism functioned as an alternative to usurpation, a tool for military dissidence, fuelled by an economy of violence. Contrary to modern warlordism, the warlordism of the fifth century AD represented a transient phase which no imperial commander was willing to prolong indefinitely. At some stage, given the means, warlords in the western Roman army wanted to become part of the imperial echelon again. Yet these alternative methods of violent opposition, and the acquisition of force through private means, ensured the breakdown of the state’s monopoly on violence and the disintegration of centralized armies. What started as an accidental revolution became a new form of military rule.