In functional data analysis, some regions of the domain of the functions can be of more interest than others owing to the quality of measurement, relative scale of the domain, or simply some external reason (e.g. interest of stakeholders). Weighting the domain is of interest particularly with probability density functions (PDFs), as derived from distributional data, which often aggregate measurements of different quality or are affected by scale effects. A weighting scheme can be embedded into the underlying sample space of a PDF when it is considered as continuous compositions applying the theory of Bayes spaces. The origin of a Bayes space is determined by a given reference measure, and this can be easily changed through the well‐known chain rule. This work provides a formal framework for defining weights through a reference measure, and it is used to develop a weighting scheme on the bounded domain of distributional data. The impact on statistical analysis is illustrated through an application to functional principal component analysis of income distribution data. Moreover, a novel centred log‐ratio transformation is proposed to map a weighted Bayes space into an unweighted $L^2$ space, enabling to use most tools developed in functional data analysis (e.g. clustering and regression analysis) while accounting for the weighting scheme. The potential of our proposal is shown on a real case study using Italian income data.

中文翻译：

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加权功能数据分析中的概率密度域

在功能数据分析中，由于度量的质量，领域的相对规模或仅仅是某些外部原因（例如，利益相关者的利益），功能领域的某些区域可能比其他领域更受关注。特别是从分布数据得出的概率密度函数（PDF）中，对域进行加权是很有意义的，概率密度函数通常会汇总不同质量的测量结果，或者受比例效应的影响。当采用贝叶斯空间理论将加权方案视为连续组成时，可以将加权方案嵌入到PDF的基础样本空间中。贝叶斯空间的原点由给定的参考度量确定，并且可以通过众所周知的链规则轻松更改。这项工作提供了一个正式的框架，可以通过参考度量来定义权重，并用于在分布数据的有界域上开发加权方案。通过应用到收入分配数据的功能主成分分析中，可以说明对统计分析的影响。此外，提出了一种新颖的中心对数比变换，以将加权的贝叶斯空间映射为未加权的${\mathrm{大号}}^2$空间，可以使用大多数在功能数据分析（例如，聚类和回归分析）中开发的工具，同时考虑加权方案。使用意大利收入数据的真实案例研究显示了我们提议的潜力。