Efficient spline orthogonal basis for representation of density functions

Area of Science:

• Functional Data Analysis
• Bayesian Statistics
• Spline Theory

Background:

• Probability density functions (PDFs) are functional data with scale invariance and unit integral constraints.
• Bayes spaces methodology and centered log-ratio (CLR) transformation are used to process PDFs in Lebesgue spaces.
• Standard B-spline bases are unsuitable for CLR-transformed data due to the zero integral constraint.

Purpose of the Study:

• To develop an orthogonal spline basis suitable for CLR-transformed probability density functions.
• To address the lack of orthogonality in the recently developed Z B-splines.
• To enhance computational efficiency and data interpretability in functional data analysis.

Main Methods:

• Construction of a novel orthogonal spline basis, termed Z B-splinets, derived from Z B-splines.
• Incorporation of the zero integral property specific to CLR-transformed density functions.
• Demonstration of the Z B-splinet approach on two empirical datasets.

Main Results:

• An efficient method for constructing orthogonal Z B-splines (Z B-splinets) is presented.
• Z B-splinets offer computational efficiency compared to non-orthogonal bases.
• The localized basis supports of Z B-splinets improve data interpretability, particularly in functional principal component analysis.

Conclusions:

• Z B-splinets provide a computationally efficient and interpretable tool for analyzing functional data represented as probability densities.
• The orthogonality and locality properties make Z B-splinets advantageous for various statistical applications.
• The proposed method is validated through practical application on empirical datasets.