Robust functional principal component analysis for non-Euclidean random objects

Summary Functional data analysis offers a diverse toolkit of statistical methods tailored to analysing samples of real-valued random functions. Recently, samples of time-varying random objects, such as time-varying networks, have been increasingly encountered in data analysis. These data structures represent elements within general metric spaces that lack local or global linear structures, rendering traditional functional data analysis methods inapplicable. Moreover, the existing methodology for time-varying random objects does not work well in the presence of outlying objects. In this article, we propose a robust method for analysing time-varying random objects. Our method employs pointwise Fréchet medians and constructs pointwise distance trajectories between the individual sample functions and the sample Fréchet median curve. This representation effectively transforms time-varying objects into functional data. A novel robust approach to functional principal component analysis, based on a Winsorized $ U $-statistic estimator of the covariance structure, is introduced. The proposed robust analysis of these distance trajectories is able to identify key features of object trajectories over time and is useful for downstream analysis. One contribution of this work is to provide a theoretical basis for establishing the asymptotic equicontinuity of time-varying $ M $-estimators located in a general metric space. To illustrate the efficacy of our approach, numerical studies focusing on dynamic networks and time-varying spherical data are conducted. The results indicate that the proposed method exhibits good all-round performance and surpasses existing approaches in terms of robustness, showcasing its superior performance in handling time-varying object data.