Compositional Periodic Spline Approximation for Circular Density Data in Bayes Spaces

TL;DR

This paper introduces a new method for approximating circular density data using compositional periodic splines within Bayes spaces, enabling effective analysis of directional data.

Contribution

It develops a novel spline-based framework in Bayes spaces with matrix formulations for efficient estimation and applies it to wind direction data analysis.

Findings

Provides smooth, interpretable density estimates for wind directions

Enables functional regression analysis of circular data

Demonstrates practical relevance and potential extensions

Abstract

This paper proposes a novel framework for the approximation and analysis of circular density data using compositional periodic splines within Bayes spaces with the Hilbert space structure. By applying the centered log-ratio transformation, densities are represented in a subspace of the standard $L^2$ space of real-valued functions, which enables the use of functional data analysis tools while preserving the relative nature of distributions and their periodic structure. A coefficient-based construction of periodic splines with a zero-integral constraint is developed, together with matrix formulations for both smoothing splines and penalized splines, allowing efficient estimation and implementation. The methodology is applied to long-term wind direction data, where it provides smooth and interpretable density estimates and supports further statistical analysis, including functional regression. The results demonstrate the practical relevance of the proposed approach and its potential for extensions to more complex density-valued data.