AI 中文总结
fdWasserstein包基于最优传输几何，为R用户提供协方差算子的距离、均值、检验、主成分分析及软聚类等功能，可用于比较函数数据群体的协方差差异。
AI 中文摘要
数据越来越多地以曲线集合的形式出现——如语音记录、生长轨迹、一天的传感器读数——每个观测值都是一个完整的函数而非单个数值。针对此类数据的常见问题是平均曲线在不同组间的差异，但平均仅反映了部分情况：两组曲线可能均值几乎相同，但其围绕均值的波动方式却存在显著差异，而这种变异性——即曲线内的协变模式——往往承载着科学信号。在该层面比较群体意味着比较它们的协方差算子，而协方差相关统计因协方差的非线性而受损。fdWasserstein包为R用户提供了进行此类比较的函数，其核心是最优传输的几何，在此框架下协方差算子可被合理地平均、对比和插值。该包提供协方差算子间的Procrustes-Wasserstein距离、它们的Fréchet均值（重心）、用于检验多个协方差相等的ANOVA型置换检验、协方差变异的主成分分析，以及基于协方差结构的曲线的熵正则化软聚类。我们概述了其基本思想，讨论了实现方式，并在该包附带的音素数据上演示了完整工作流程。
英文摘要
Data increasingly arrive as collections of curves - a voice recording, a growth trajectory, a day of sensor readings - where each observation is a whole function rather than a single number. The usual question asked of such data is how the average curve differs from one group to the next. But the average is only half the picture: two populations of curves can share almost the same mean and still differ profoundly in how they fluctuate around it, and it is often this variability - the pattern of covariation within a curve - that carries the scientific signal. Comparing populations at this level means comparing their covariance operators, and statistics on covariances is impaired by their non-linearity. The fdWasserstein package equips R users with functions to make such comparisons. It is centered on the geometry of optimal transport, under which covariance operators can be meaningfully averaged, contrasted, and interpolated. It provides the Procrustes-Wasserstein distance between covariance operators, their Frechet mean (barycenter), an ANOVA-type permutation test for the equality of several covariances, principal component analysis of covariance variation, and an entropy-regularized soft clustering of curves by their covariance structure. We outline the underlying ideas, discuss the implementation and demonstrate the complete workflow on the phoneme data shipped with the package.
