Combining dimension reduction methods

Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Engineering and Mathematical Sciences, Faculty of Science, Technology & Engineering, La Trobe University, Bundoora, Victoria.

Given the constant growth in data storage capacities available due to advances in technology, the analysis of high-dimensional data is becoming increasingly important in modern day statistics. One of the challenges faced with high-dimensional data is graphical visualization which allows the relationship(s) between response and predictor variables to be detected. Methods such as Sliced Inverse Regression (SIR), Sliced Average Variance Estimation (SAVE) and Principal Hessian Directions (PHD) provide a way to overcome this problem by reducing the dimension of the data without loss of information. However, each of these methods has respective advantages and disadvantages. The purpose of this thesis is to provide advances to the combination of dimension reduction methods which can provide significant improvement to estimation in many contexts. The contribution of this thesis is three-fold. First, a new approach to combining dimension reduction methods is proposed which involves the iterative application of these methods. This approach is applicable to both continuous and discrete response data. Second, a novel way of choosing dimension reduction parameters based on influence measures is introduced, as the number of parameters can increase when these methods are combined. Third, given that maximum ‘separation’ of classes is of major benefit in discriminant analysis, the choice of parameters is considered based on new visual and quantitative measures of separation.