Functional linear regression is a useful extension of simple linear
regression and has been investigated by many researchers. However, functional
variable selection problems when multiple functional observations exist, which
is the counterpart in the functional context of multiple linear regression, is
seldom studied. Here we propose a method using group smoothly clipped absolute
deviation penalty (gSCAD) which can perform regression estimation and variable
selection simultaneously.

A single-index model (SIM) provides for parsimonious multi-dimensional
nonlinear regression by combining parametric (linear) projection with
univariate nonparametric (non-linear) regression models. We show that a
particular Gaussian process (GP) formulation is simple to work with and ideal
as an emulator for some types of computer experiment as it can outperform the
canonical separable GP regression model commonly used in this setting.

Recently nonparametric functional model with functional responses has been
proposed within the functional reproducing kernel Hilbert spaces (fRKHS)
framework. Motivated by its superior performance and also its limitations, we
propose a Gaussian process model whose posterior mode coincide with the fRKHS
estimator. The Bayesian approach has several advantages compared to its
predecessor. Firstly, the multiple unknown parameters can be inferred together
with the regression function in a unified framework.

We study the problem of estimating time-varying coefficients in ordinary
differential equations. Current theory only applies to the case when the
associated state variables are observed without measurement errors as presented
in \cite{chenwu08b,chenwu08}. The difficulty arises from the quadratic
functional of observations that one needs to deal with instead of the linear
functional that appears when state variables contain no measurement errors. We
derive the asymptotic bias and variance for the previously proposed two-step
estimators using quadratic regression functional theory.