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Efficient spline orthogonal basis for representation of density functions

Jana Burkotová, Ivana Pavlů, Hiba Nassar, Jitka Machalová, Karel Hron

TL;DR

The paper addresses efficient representation of probability density functions (PDFs) by using Bayes spaces and the centred log-ratio transformation to $L^2_0(I)$. It introduces zero-integral ZB-splines and develops ZB-splinets via a dyadic orthogonalization to obtain an orthonormal, locally supported basis. The authors prove reduced total support and fewer inner-product computations for ZB-splinets compared with Gram-Schmidt variants, and demonstrate functional principal component analysis on a demographic dataset. The work advances practical density-function FDA by enabling fast, interpretable analyses and suggests a future R package for broader use.

Abstract

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.

Efficient spline orthogonal basis for representation of density functions

TL;DR

The paper addresses efficient representation of probability density functions (PDFs) by using Bayes spaces and the centred log-ratio transformation to . It introduces zero-integral ZB-splines and develops ZB-splinets via a dyadic orthogonalization to obtain an orthonormal, locally supported basis. The authors prove reduced total support and fewer inner-product computations for ZB-splinets compared with Gram-Schmidt variants, and demonstrate functional principal component analysis on a demographic dataset. The work advances practical density-function FDA by enabling fast, interpretable analyses and suggests a future R package for broader use.

Abstract

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 -spline basis is no longer suitable. Recently, a new spline basis incorporating this zero integral property, called -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 -splines basis, called -splinets. The advantages of the -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.
Paper Structure (16 sections, 2 theorems, 42 equations, 13 figures, 3 tables)

This paper contains 16 sections, 2 theorems, 42 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. ZB-spline basis
  3. Bayes spaces
  4. B-splines
  5. ZB-splines
  6. Orthogonalization
  7. Smoothing splines
  8. Dyadic orthogonalization
  9. Splinets
  10. ZB-splinets
  11. Efficiency of the $Z\!B$-splinets
  12. Application
  13. Functional principal component analysis
  14. Demographic example
  15. Results and discussion
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\Delta \Lambda=\{\lambda_i\}_{i=-k}^{g+k+1}$ be a dyadic set of knots with equispaced $g$ inner knots exknots where $g= (2^N-1)(k+1)-k$ for $N> 0$. Then the relative total support of orthogonalized $Z\!B$-splines of degree $k\in \mathbb{N}_0$

Figures (13)

  • Figure 1: $B$-splines of degree 2 ( left), the corresponding $Z\!B$-splines of degree 1 ( right).
  • Figure 2: $B$-splines of degree 3 ( left), the corresponding $Z\!B$-splines ( right).
  • Figure 3: $B$-splines of degree 1 left, the corresponding $B$-splinets right
  • Figure 4: $Z\!B$-splines of degree 1 ( left), the corresponding $Z\!B$-splinets ( right).
  • Figure 5: $Z\!B$-splines of degree 2 ( left), the corresponding $Z\!B$-splinets ( right).
  • ...and 8 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof