A spline function is a curve constructed from polynomial segments that are subject to conditions or continuity at their joints. In this paper, we shall present the algorithm of the cubic smoothing spline and we shall justify its use in estimating trends in time series. Considerable effort has been devoted over several decades to developing the mathematics of spline functions. Much of the intere...

In this paper some of the relationships between optimal control and statistics are examined. We produce generalized, smoothing splines by solving an optimal control problem for linear control systems, minimizing the -norm of the control signal, while driving the scalar output of the control system close to given, prespecified interpolation points. We then prove a convergence result for the smoo...

Periodic smoothing splines appear for example as generators of closed, planar curves, and in this paper they are constructed using a controlled two point boundary value problem in order to generate the desired spline function. The procedure is based on minimum norm problems in Hilbert spaces and a suitable Hilbert space is defined together with a corresponding linear affine variety that capture...

The optimal solution to the problem of driving the output of a single-input, single-output linear control system close to given waypoints is analyzed. Furthermore, this smooth interpolation is done while the state space is constrained by an infinite dimensional, non-negativity constraint on the derivative of the output spline function. For the case when the acceleration is controlled directly t...

Bayesian hierarchical models have been used for smoothing splines, thin-plate splines, and L-splines. In analyzing high dimensional data sets, additive models and backfitting methods are often used. A full Bayesian analysis for such models may include a large number of random effects, many of which are not intuitive, so researchers typically use noninformative improper or nearly improper priors...