Splines, knots, and penalties

Penalized splines have gained much popularity as a flexible tool for smoothing and semi-parametric models. Two approaches have been advocated: (1) use a B-spline basis, equally spaced knots, and difference penalties [Eilers PHC, Marx BD. Flexible smoothing using B-splines and penalized likelihood (with Comments and Rejoinder). Stat Sci 1996, 11:89-121.] and (2) use truncated power functions, knots based on quantiles of the independent variable and a ridge penalty [Ruppert D, Wand MP, Carroll RJ. Semiparametric Regression. New York: Cambridge University Press; 2003]. We compare the two approaches on many aspects: numerical stability, quality of the fit, interpolation/extrapolation, derivative estimation, visual presentation and extension to multidimensional smoothing. We discuss mixed model and Bayesian parallels to penalized regression. We conclude that B-splines with difference penalties are clearly to be preferred.

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