Generalised smoothing in functional data analysis

The incorporation of model-based penalties in a penalised regression frame- work (generalised smoothing) has been the subject of many publications, most notably: Cao and Ramsay (2007); Heckman and Ramsay (2000); Ramsay and Silverman (2005); Ramsay et al. (2007). Generalised smooth- ing facilitates the estimation of the parameters of an ordinary di erential equation (ODE) from noisy data without the speci cation of an explicit expression of the functional entity described by the ODE. This is a notable consequence of the smoothing procedure however it is not its primary aim. Generalised smoothing aims to obtain an estimated functional entity that adheres to the data and incorporates domain speci c information de ned by the ODE. The existing methodology for the estimation of the param- eters in generalised smoothing is hindered by the absence of an explicit expression in terms of the parameters of the ODE for the penalty within penalised tting criterion. The aim of this research is to obtain this ex- plicit expression for penalties de ned by B{spline basis functions in order to facilitate the development of the estimation procedure. The recursive algorithm developed by de Boor (2001) is the predominant methodology for the evaluation of B-spline basis functions over a given in- terval. While this algorithm is a fast and numerically stable method for nding a point on a B-spline curve given the domain, it does not explicitly provide knowledge of the internal structure of the B-spline functions. This work introduces an alternative representation of B{spline basis functions in terms of the underlying polynomials that comprise the B{spline. This alterative representation of B{spline basis functions produces generalised penalties which can be written explicitly in terms of the parameters of the ODE. A joint estimation strategy in which the penalised least squares cri- terion is minimised with respect to the parameters of the B-spline and the parameters of the ODE is developed. Finally this joint estimation strat- egy is shown to produce estimates of both parameters that have a higher accuracy and are more computationally e cient than estimates developed by existing methods.