Splinets -- splines through the Taylor expansion, their support sets and orthogonal bases

Krzysztof PodgÓrski

A new representation of splines that targets efficiency in the analysis of functional data is implemented. The efficiency is achieved through two novel features: using the recently introduced orthonormal spline bases, the so-called {\it splinets} and accounting for the spline support sets in the proposed spline object representation. The recently-introduced orthogonal splinets are evaluated by {\it dyadic orthogonalization} of the $B$-splines. The package is built around the {\it Splinets}-object that represents a collection of splines. It treats splines as mathematical functions and contains information about the support sets and the values of the derivatives at the knots that uniquely define these functions. Algebra and calculus of splines utilize the Taylor expansions at the knots within the support sets. Several orthonormalization procedures of the $B$-splines are implemented including the recommended dyadic method leading to the splinets. The method bases on a dyadic algorithm that can be also viewed as the efficient method of diagonalizing a band matrix. The locality of the $B$-splines in terms of the support sets is, to a great extend, preserved in the corresponding splinet. This together with implemented algorithms utilizing locality of the supports provides a valuable computational tool for functional data analysis. The benefits are particularly evident when the sparsity in the data plays an important role. Various diagnostic tools are provided allowing to maintain the stability of the computations. Finally, the projection operation to the space of splines is implemented that facilitates functional data analysis. An example of a simple functional analysis of the data using the tools in the package is presented. The functionality of the package extends beyond the splines to piecewise polynomial functions, although the splines are its focus.

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