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Efficient spline orthogonal basis for representation of density functions.

Jana Burkotová1, Ivana Pavlů1, Hiba Nassar2

  • 1Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, Olomouc, Czech Republic.

Journal of Applied Statistics
|March 16, 2026
PubMed
Summary

Researchers developed Z B-splinets, an orthogonal spline basis for probability density functions. This method offers computational efficiency and localized data representation, improving functional data analysis.

Keywords:
62G0762R1065D0765D10Bayes spaceSpline approximationefficiencyfunctional data analysisorthogonalizationprobability density function

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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.