Reliable estimation and approximation of probability density functions is fundamental for their further processing. However, their specific properties, i.e. scale invariance and relative scale, prevent the use of standard methods of spline approximation and have to be considered when building a suitable spline basis. Bayes Hilbert space methodology allows to account for these properties of densities and enables their conversion to a standard Lebesgue space of square integrable functions using the centered log-ratio transformation. As the transformed densities fulfill a zero integral constraint, the constraint should likewise be respected by any spline basis used. Bayes Hilbert space methodology also allows to decompose bivariate densities into their interactive and independent parts with univariate marginals. As this yields a useful framework for studying the dependence structure between random variables, a spline basis ideally should admit a corresponding decomposition. This paper proposes a new spline basis for (transformed) bivariate densities respecting the desired zero integral property. We show that there is a one-to-one correspondence of this basis to a corresponding basis in the Bayes Hilbert space of bivariate densities using tools of this methodology. Furthermore, the spline representation and the resulting decomposition into interactive and independent parts are derived. Finally, this novel spline representation is evaluated in a simulation study and applied to empirical geochemical data.

翻译：概率密度函数的可靠估计与逼近是其进一步处理的基础。然而，密度函数的特定性质（即尺度不变性和相对尺度）阻碍了标准样条逼近方法的使用，在构建合适的样条基时必须予以考虑。贝叶斯希尔伯特空间方法能够解释密度的这些性质，并可通过中心化对数比变换将其转换为标准勒贝格平方可积函数空间。由于变换后的密度满足零积分约束，因此所使用的任何样条基也应尊重这一约束。贝叶斯希尔伯特空间方法还能将二元密度分解为其交互部分和具有单变量边际的独立部分。由于这为研究随机变量之间的依赖结构提供了有用框架，理想情况下样条基应支持相应的分解。本文提出了一种新的样条基，用于（变换后的）二元密度，并满足所需的零积分性质。我们证明，利用该方法论的工具，该基与二元密度贝叶斯希尔伯特空间中的相应基存在一一对应关系。此外，推导了样条表示及其向交互部分和独立部分的分解。最后，通过仿真研究评估了这种新型样条表示，并将其应用于实测地球化学数据。