Advances in Object Oriented Data Analysis

Abstract

The analysis of object-oriented data has gained prominence in modern statistics. Examples include dynamic transportation networks, geochemical data and count data. Such object-oriented data, often referred to as random objects, often lie within general metric spaces which may lack local or global linear structures, rendering traditional statistical data analysis methods inapplicable. Moreover, probabilistic modeling for these objects often involves intractable likelihoods, a challenge even in Euclidean contexts. This thesis develops novel statistical methodologies addressing these challenges, with contributions in in both nonparametric and parametric methods, covering dimension reduction, change point detection, parameter estimation, and statistical inference.

Chapter 2 introduces a novel robust functional principal component analysis framework based on a Winsorized U-statistic covariance estimator, tailored for time-varying random objects. This approach effectively identifies key features of dynamic objects, enhancing downstream analyses. Numerical studies on dynamic networks demonstrate the method's superior robustness and comprehensive performance compared to existing techniques. Chapter 3 presents a novel procedure to precisely identify and localize abrupt changes in the distribution of non-Euclidean random objects with periodic behavior. The proposed procedure is flexible and broadly applicable, accommodating a variety of suitable change point detectors for random objects. Through extensive simulations on time-varying graph Laplacians and a specific change point detector, we highlight its advantages over the most competitive method in the literature, where periodicity blurs the changes that the procedure aims to discover. A real-world application on the New York City Citi Bike sharing system showcases the interpretability and effectiveness of the method in identifying meaningful change points linked to historical events. Chapter 4 addresses intractable likelihood estimation by extending the widely used score-matching framework. A unified asymptotic theory for generalized score matching is developed under the independence assumption, accommodating both continuous and discrete data. Real-world and simulated analyses demonstrate the method's strong theoretical foundation and practical utility. Show more