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.

Properties of molecules in tunnel junctions

Molecular tunnel junctions involve studying the behaviour of a single molecule sandwiched between metal leads. When a molecule makes contact with electrodes, it becomes open to the environment which can heavily influence its properties, such as electronegativity and electron transport. While the most common computational approaches remain to be single particle approximations, in this thesis it is shown that a more explicit treatment of electron interactions can be required. By studying an open atomic chain junction, it is found that including electron correlations corrects the strong lead-molecule interaction seen by the ΔSCF approximation, and has an impact on junction I − V properties. The need for an accurate description of electronegativity is highlighted by studying a correlated model of hexatriene-di-thiol with a systematically varied correlation parameter and comparing the results to various electronic structure treatments. The results indicating an overestimation of the band gap and underestimation of charge transfer in the Hartree-Fock regime is equivalent to not treating electron-electron correlations. While in the opposite limit, over-compensating for electron-electron interaction leads to underestimated band gap and too high an electron current as seen in DFT/LDA treatment. It is emphasised in this thesis that correcting electronegativity is equivalent to maximising the overlap of the approximate density matrix to the exact reduced density matrix found at the exact many-body solution. In this work, the complex absorbing potential (CAP) formalism which allows for the inclusion metal electrodes into explicit wavefunction many-body formalisms is further developed. The CAP methodology is applied to study the electron state lifetimes and shifts as the junction is made open.