| Signal regression and image regression, in which the outcomes are scalars and the predictors are one-dimensional signals or multidimensional images, are of interest in many scientific fields. The principal statistical challenge is how to reduce the dimension of the predictors in what would otherwise be a severely ill-posed problem. A pair of novel methods, functional principal component regression (FPCR) and functional partial least squares (FPLS), combine two existing approaches to the dimension reduction problem: selection of most relevant components, as is done in ordinary principal component regression (PCR) and partial least squares (PLS), and restriction of the coefficient function to the span of a spline basis. Chapter 1 outlines several existing signal regression methods and presents two ways to define FPCR/FPLS, based, respectively, on regularized components and regularized regression. Simulations and real data analyses with chemometric data, in Chapter 2 demonstrate the strong performance of the regularized-regression form of FPCR/FPLS, compared with the regularized-components form as well as with existing methods. Chapter 3 discusses how to incorporate covariates in FPCR/FPLS, and presents some results on the generalized cross-validation and restricted maximum likelihood approaches to smoothing parameter selection for a, broader class of serniparametric regression models. In Chapter 4, FPCR and FPLS are extended from signals to images and from linear to generalized linear models. Chapter 5 applies FPCR and FPLS, in their linear and logistic regression versions, to neuroimaging data sets in which the predictors are maps of serotonin receptor and transporter binding in the brain. |
