Efficient spline orthogonal basis for representation of density functions

• Main Authors: Burkotová, Jana, Pavlů, Ivana, Nassar, Hiba, Machalová, Jitka, Hron, Karel
• Format: Journal Article
• Language: English
• Published: 03-05-2024
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis; Statistics - Methodology
• DOI: 10.48550/arxiv.2405.02231

Probability density functions form a specific class of functional data
objects with intrinsic properties of scale invariance and relative scale
characterized by the unit integral constraint. The Bayes spaces methodology
respects their specific nature, and the centred log-ratio transformation
enables processing such functional data in the standard Lebesgue space of
square-integrable functions. As the data representing densities are frequently
observed in their discrete form, the focus has been on their spline
representation. Therefore, the crucial step in the approximation is to
construct a proper spline basis reflecting their specific properties. Since the
centred log-ratio transformation forms a subspace of functions with a zero
integral constraint, the standard $B$-spline basis is no longer suitable.
Recently, a new spline basis incorporating this zero integral property, called
$Z\!B$-splines, was developed. However, this basis does not possess the
orthogonal property which is beneficial from computational and application
point of view. As a result of this paper, we describe an efficient method for
constructing an orthogonal $Z\!B$-splines basis, called $Z\!B$-splinets. The
advantages of the $Z\!B$-splinet approach are foremost a computational
efficiency and locality of basis supports that is desirable for data
interpretability, e.g. in the context of functional principal component
analysis. The proposed approach is demonstrated on an empirical demographic
dataset.