A generalised smoother for linear ordinary differential equations

This paper introduces an o-the shelf smoother for linear ordinary differential equations.
This smoother is obtained by enhancing the functional data analysis estimation procedure for approximating discrete data by a function when the data is assumed to adhere to a specified linear ordinary differential equation (ODE) motivated from domain specific information. The estimate of the functional object is obtained minimising a penalised residual sum of squares where the roughness penalty is the L2 norm of the differential operator which contains the linear ODE. The almost universal approach to evaluating this roughness penalty involves using numerical integration techniques. Carey et al. (2012) developed a recursive algorithm that identifies the coefficients of the piecewise polynomials that comprise the B-spline basis function. In this paper, we propose utilising this alternative expression for the B-spline basis functions in order to produce an exact expression for the roughness penalty. Explicit expressions are derived for each of the estimated parameters that minimise the penalised residual sum of squares and for the
uncertainty quantification for these estimates.

Carey, M., G. G. E. Gath, and K. Hayes (2012). A universal matrix algorithm for the computation of the coecients of the polynomials comprising b–spline basis functions. Submitted. (ref. Dott. Laura Sangalli)