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.

翻译：本文提出了一种基于贝叶斯空间与希尔伯特空间结构的框架，利用分段周期样条逼近和分析圆周密度数据。通过中心化对数比率变换，密度函数被表示为标准实值函数$L^2$空间的子空间元素，从而在保留分布相对性质及周期结构的同时，可运用函数型数据分析工具。我们构造了具有零积分约束的周期样条系数表示方法，并推导了光滑样条与惩罚样条的矩阵形式，实现了高效估计与计算。该方法被应用于长期风向数据，不仅提供了平滑且可解释的密度估计，还支持进一步的统计分析（包括函数型回归）。结果表明了所提方法的实际适用性及其向复杂密度值数据扩展的潜力。