Samples of dynamic or time-varying networks and other random object data such as time-varying probability distributions are increasingly encountered in modern data analysis. Common methods for time-varying data such as functional data analysis are infeasible when observations are time courses of networks or other complex non-Euclidean random objects that are elements of general metric spaces. In such spaces, only pairwise distances between the data objects are available and a strong limitation is that one cannot carry out arithmetic operations due to the lack of an algebraic structure. We combat this complexity by a generalized notion of mean trajectory taking values in the object space. For this, we adopt pointwise Fr\'echet means and then construct pointwise distance trajectories between the individual time courses and the estimated Fr\'echet mean trajectory, thus representing the time-varying objects and networks by functional data. Functional principal component analysis of these distance trajectories can reveal interesting features of dynamic networks and object time courses and is useful for downstream analysis. Our approach also makes it possible to study the empirical dynamics of time-varying objects, including dynamic regression to the mean or explosive behavior over time. We demonstrate desirable asymptotic properties of sample based estimators for suitable population targets under mild assumptions. The utility of the proposed methodology is illustrated with dynamic networks, time-varying distribution data and longitudinal growth data.

翻译：动态或时变网络样本以及其他随机对象数据（如时变概率分布）在现代数据分析中日益常见。当观测数据为网络或其他复杂非欧几里得随机对象的时间序列时，这些对象作为一般度量空间的元素，使得函数数据分析等时变数据常用方法不再适用。在此类空间中，仅可获得数据对象间的成对距离，且由于缺乏代数结构而无法执行算术运算，这是一个重大限制。我们通过引入在对象空间中取值的广义均值轨迹概念来应对这一复杂性。具体而言，我们采用逐点弗雷歇均值，进而构建个体时间序列与估计的弗雷歇均值轨迹之间的逐点距离轨迹，从而将时变对象和网络表示为函数型数据。对这些距离轨迹进行函数主成分分析，能够揭示动态网络与对象时间序列的显著特征，并有助于后续分析。该方法还可用于研究时变对象的经验动态特性，包括随时间变化的均值回归或爆发性行为。我们在温和假设下证明了基于样本的估计量对于相应总体目标具有理想的渐近性质。通过动态网络、时变分布数据及纵向生长数据等案例，验证了所提方法的实用性。