Reciprocal components, reciprocal curves, and partial least squares

Multivariate structure-seeking techniques are being applied and studied extensively in many areas of science. In psychometrics and chemometrics, the algorithmic data analysis method known as Partial Least Squares forms the base analysis procedure for many modelling situations. In this dissertation the underlying paradigm of Partial Least Squares is shown to be a fundamental statistical concept similar to principal components, canonical correlations and linear regression. This new theoretical method is termed Reciprocal Components in the linear case and in the more general nonlinear case--Reciprocal Curves.; These new Reciprocal Analysis methods are derived from modelling structurally related partitions of a random vector by summarizing the respective random vector subspaces with isometric conditional expectation surfaces. Theories are given concerning the interpretation and use of these new methods. In addition, iterative Algorithms are proposed and demonstrated that allow for computation of these surfaces in the theoretical and discrete, sample data setting. In the discrete data setting the algorithm converges to general geometric structures as suggested by the geometric orientation of the data in variables space.