Splines, knots, and penalties

Abstract 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. WIREs Comp Stat 2010 2 637–653 DOI: 10.1002/wics.125 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation