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The P-splines of Eilers and Marx (1996) combine a B-spline basis with adiscrete quadratic penalty on the basis coefficients, to produce a reduced rankspline like smoother. P-splines have three properties that make them verypopular 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 andpenalty, rather than the order of penalty controlling the order of the basis asin spline smoothing; iii) it is very easy to set up the B-spline basisfunctions and penalties. The discrete penalties are somewhat less interpretablein terms of function shape than the traditional derivative based splinepenalties, but tend towards penalties proportional to traditional splinepenalties in the limit of large basis size. However part of the point ofP-splines is not to use a large basis size. In addition the spline basisfunctions arise from solving functional optimization problems involvingderivative based penalties, so moving to discrete penalties for smoothing maynot always be desirable. The purpose of this note is to point out that thethree properties of basis-penalty sparsity, mix-and-match penalization and easeof setup are readily obtainable with B-splines subject to derivative basedpenalization. The penalty setup typically requires a few lines of code, ratherthan the two lines typically required for P-splines, but this one offdisadvantage seems to be the only one associated with using derivative basedpenalties. As an example application, it is shown how basis-penalty sparsityenables efficient computation with tensor product smoothers of scattered data.

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