The Nonparametric Estimation of Branching Curves
将样条平滑方法扩展到估计在特定时间点分支的曲线，通过惩罚粗糙度来拟合分支曲线系统，并用向日葵实验数据展示该方法在园艺和树木栽培中的应用。
Abstract The spline smoothing approach to nonparametric regression is extended to cover the estimation of curves that branch at certain specified points in time. This extension is motivated and illustrated by data from an experiment on sunflowers. Apart from providing insight into this particular data-analytic problem, the work described demonstrates the power of the roughness penalty approach in coping with curve estimation of a nonstandard kind. Particularly in the context of certain experiments in horticulture and arboriculture, the experimental material may be treated identically to the control up to a certain known time part of the way through the experiment. To display and investigate the effect of treatments applied during growth it is of interest to estimate a branching curve. In the simplest case we want to study two growth curves that are common up to a certain known time and then diverge (or branch) after the application of a treatment to part of the experimental material. The spline-smoothing approach to nonparametric regression as discussed, for example, by Silverman (1985) can be extended to deal with branching curves by defining a roughness penalty that measures the local variation in all branches of the curve system. The usual residual sum of squares is penalized by the addition of a multiple of this roughness penalty, and the resulting penalized sum of squares can be minimized. The solution branching curve is made up of a number of cubic spline branches that fit together according to rules that are derived in the present article and that allow the curve system to be found by solving a set of sparse linear equations. A nonparametric approach to this curve estimation problem is virtually essential, because it is hard to contemplate extensions of the usual parametric growth curve models that allow for branching curves in any way that is not rather artificial. The estimation technique is illustrated by data from an experiment on sunflowers in which each of four treatments diverges from the control at a different specified time. The branching curve estimates give an informative presentation of the data and of the structure of the experiment and indicate the relative merits of the treatments.