分布数据的多因素分析
Multiple factor analysis of distributional data
摘要：
在符号数据分析（SDA）的框架中，分布变量是多值变量的一种特殊情况：每个单元由一组分布（例如直方图，密度函数或分位数函数）表示，每个变量一个。因子分析方法是进行降维和可视化的主要探索工具。在目前的工作中，我们使用多因素分析（MFA）的方法来分析数据描述的分布变量。每个分布变量都导出一组与每个分布的分位数相关的新数值变量。我们称这些新的变量为\textit{quantile variables}和一组分位数变量相关的分布是一个块中的MFA方法。因此，MFA是在分位数变量的并列表上执行的。\\我们表明，在分析中分解的标准是一个近似的变异性的基础上一个合适的度量分布之间：平方$L_2$瓦瑟斯坦距离。模拟和实际分布数据的应用证实了该方法的有效性。阶乘平面上的结果的解释是通过与分布的几个特征（位置、尺度和形状）相关的新的解释工具来执行的。
英文摘要：
In the framework of Symbolic Data Analysis (SDA), distribution-variables are a particular case of multi-valued variables: each unit is represented by a set of distributions (e.g. histograms, density functions or quantile functions), one for each variable. Factor analysis (FA) methods are primary exploratory tools for dimension reduction and visualization. In the present work, we use Multiple Factor Analysis (MFA) approach for the analysis of data described by distributional variables. Each distributional variable induces a set new numeric variable related to the quantiles of each distribution. We call these new variables as \textit{quantile variables} and the set of quantile variables related to a distributional one is a block in the MFA approach. Thus, MFA is performed on juxtaposed tables of quantile variables. \\ We show that the criterion decomposed in the analysis is an approximation of the variability based on a suitable metrics between distributions: the squared $L_2$ Wasserstein distance. \\ Applications on simulated and real distributional data corroborate the method. The interpretation of the results on the factorial planes is performed by new interpretative tools that are related to the several characteristics of the distributions (location, scale and shape).
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