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P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data

机译：具有导数罚分的P样条和不均匀分布数据的张量积平滑

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摘要

The P-splines of Eilers and Marx (Stat Sci 11:89-121, 1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as reduced rank smoothers: (i) the basis and the penalty are sparse, enabling efficient computation, especially for Bayesian stochastic simulation; (ii) it is possible to flexibly 'mix-and-match' the order of B-spline basis and penalty, rather than the order of penalty controlling the order of the basis as in spline smoothing; (iii) it is very easy to set up the B-spline basis functions and penalties. The discrete penalties are somewhat less interpretable in terms of function shape than the traditional derivative based spline penalties, but tend towards penalties proportional to traditional spline penalties in the limit of large basis size. However part of the point of P-splines is not to use a large basis size. In addition the spline basis functions arise from solving functional optimization problems involving derivative based penalties, so moving to discrete penalties for smoothing may not always be desirable. The purpose of this note is to point out that the three properties of basis-penalty sparsity, mix-and-match penalization and ease of setup are readily obtainable with B-splines subject to derivative based penalization. The penalty setup typically requires a few lines of code, rather than the two lines typically required for P-splines, but this one off disadvantage seems to be the only one associated with using derivative based penalties. As an example application, it is shown how basis-penalty sparsity enables efficient computation with tensor product smoothers of scattered data.
机译：Eilers和Marx的P样条曲线（Stat Sci 11：89-121，1996）结合了B样条曲线基和离散的基于基系数的二次罚函数，从而产生了更平滑的降阶样条曲线。 P样条具有三个特性，使其在降低秩的平滑器中非常受欢迎：（i）基础和罚分稀疏，可以进行有效的计算，尤其是对于贝叶斯随机模拟而言； （ii）可以灵活地“混合匹配” B样条基和罚分的顺序，而不是像样条平滑中那样，罚分的顺序控制基数的顺序； （iii）建立B样条基函数和惩罚非常容易。在功能形状方面，离散惩罚比传统的基于衍生样条的惩罚难于解释，但是在大的基础尺寸的限制下，倾向于倾向于与传统的样条惩罚成比例的惩罚。但是，P样条曲线的重点不是使用大的基数。另外，样条基函数是通过解决涉及基于导数的罚分的函数优化问题而产生的，因此，移至离散罚分进行平滑可能并不总是理想的。本说明的目的是指出，对B样条进行基于派生的惩罚，可以轻松获得基本惩罚稀疏性，混合匹配匹配惩罚和易于设置这三个属性。惩罚设置通常需要几行代码，而不是P样条通常需要的两行，但是这种一次性的缺点似乎是与使用基于派生的惩罚相关的唯一缺点。作为一个示例应用程序，它显示了基本罚分稀疏度如何通过分散数据的张量积平滑器实现高效计算。

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