Splinets -- Orthogonal Splines for Functional Data Analysis

This study introduces an efficient workflow for functional data analysis in classification problems, utilizing advanced orthogonal spline bases. The methodology is based on the flexible Splinets package, featuring a novel spline representation designed for enhanced data efficiency. The focus here is to show that the novel features make the package a powerful and efficient tool for advanced functional data analysis. Two main aspects of spline implemented in the package are behind this effectiveness: 1) Utilization of Orthonormal Spline Bases -- the workflow incorporates orthonormal spline bases, known as splinets, ensuring a robust foundation for data representation; 2) Consideration of Spline Support Sets -- the implemented spline object representation accounts for spline support sets, which refines the accuracy of sparse data representation. Particularly noteworthy are the improvements achieved in scenarios where data sparsity and dimension reduction are critical factors. The computational engine of the package is the dyadic orthonormalization of B-splines that leads the so-called splinets -- the efficient orthonormal basis of splines spanned over arbitrarily distributed knots. Importantly, the locality of $B$-splines concerning support sets is preserved in the corresponding splinet. This allows for the mathematical elegance of the data representation in an orthogonal basis. However, if one wishes to traditionally use the $B$-splines it is equally easy and efficient because all the computational burden is then carried in the background by the splinets. Using the locality of the orthogonal splinet, along with implemented algorithms, the functional data classification workflow is presented in a case study in which the classic Fashion MINST dataset is used. Significant efficiency gains obtained by utilization of the package are highlighted including functional data representation through stable and efficient computations of the functional principal components. Several examples based on classical functional data sets, such as the wine data set, showing the convenience and elegance of working with Splinets are included as well.