A comment on the orthogonalization of B-spline basis functions and their derivatives

Through the use of a matrix representation for B-splines presented by Qin (Vis. Comput. 16:177-186, 2000) we are able to reexamine calculus operations on B-spline basis functions. In this matrix framework the problem associated with generating orthogonal splines is reexamined, and we show that this approach can simplify the operations involved to linear matrix operations. We apply these results to a recent paper (Zhou et al. in Biometrika 95:601-619, 2008) on hierarchical functional data analysis using a principal components approach, where a numerical integration scheme was used to orthogonalize a set of B-spline basis functions. These orthogonalized basis functions, along with their estimated derivatives, are then used to construct estimates of mean functions and functional principal components. By applying the methods presented here such algorithms can benefit from increased speed and precision. An R package is available to do the computations.