Functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued. In recent years, samples of time‐varying object data such as time‐varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. We propose metric covariance, a novel association measure for paired object data lying in a metric space (Ω,d) that we use to define a metric autocovariance function for a sample of random Ω‐valued curves, where Ω generally will not have a vector space or manifold structure. The proposed metric autocovariance function is non‐negative definite when the squared semimetric d² is of negative type. Then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for Ω‐valued functional data, including time‐varying probability distributions, covariance matrices and time dynamic networks. Analogues of functional principal components for time‐varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real‐valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space Ω. We establish asymptotic consistency of the sample‐based estimators for the corresponding population targets under mild metric entropy conditions on Ω and continuity of the Ω‐valued random curves. These concepts are illustrated with samples of time‐varying probability distributions for human mortality, time‐varying covariance matrices derived from trading patterns and time‐varying networks that arise from New York taxi trips.

中文翻译：

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随时间变化的随机对象的功能模型

功能数据分析提供了一个流行的功能模型工具箱，用于分析具有实际价值的随机函数的样本。近年来，越来越多地收集了时变对象数据（例如不在矢量空间中的时变网络）的样本。可以将这些数据视为缺少局部或全局线性结构的常规度量空间的元素，因此无法应用已成功用于功能数据分析（例如功能主成分分析）的通用方法。我们提出了度量协方差，这是一种用于度量空间（Ω，d），我们用于为随机Ω值曲线的样本定义度量自协方差函数，其中Ω通常不会具有向量空间或流形结构。当平方的半度量d ²为负类型时，建议的度量自协方差函数是非负定的。然后，以自协方差函数为核的线性算子的本征函数可以用作对象函数主成分分析的构建块用于Ω值的功能数据，包括时变概率分布，协方差矩阵和时间动态网络。时变物体的功能主成分的类似物是通过应用Fréchet均值和在特征函数方向上应用随机物体轨迹的距离函数的投影获得的，从而得出实值Fréchet分数。使用广义Fréchet积分的概念，我们构造了对象函数主成分它位于度量空间Ω中。我们在温和的度量熵条件下，基于Ω估计和Ω值随机曲线的连续性，为相应的总体目标建立了基于样本的估计量的渐近一致性。这些概念以人类死亡的时变概率分布样本，从交易模式得出的时变协方差矩阵和纽约出租车旅行产生的时变网络的样本进行了说明。