Two essays on complex tensor data

Abstract

Modern applications across various fields often involve the collection and analysis of data that is structured in complex tensor formats. Skillfully harnessing the multidimensional information and complex structure inherent in tensors can greatly enhance our ability to extract valuable insights from the data. In this research, we study two statistical problems that are specific to complex tensor data.

The first essay considers a functional linear regression with two dimensional functional predictor as input and estimates the continuous coefficient function. To avoid the inefficiency of the classical method involving estimation of a two-dimensional coefficient function, we propose a functional bilinear regression model, and introduce an innovative three-term penalty to impose roughness penalty in the estimation. The proposed estimator exhibits minimax optimal property for prediction under the framework of reproducing kernel Hilbert space. An iterative generalized cross-validation approach is developed to choose tuning parameters, which significantly improves the computational efficiency over the traditional cross-validation approach. The statistical and computational advantages of the proposed method over existing methods are further demonstrated via simulated experiments, the Canadian weather data, and a biochemical long-range infrared light detection and ranging data.

The second essay considers an Independent Component Analysis (ICA) problem for multiple groups of matrix-valued data. We first propose a novel one-subject ICA model, which use both non-Gaussianity and time correlation to identify factors. This unique approach enables the model to "adapt" various type of factor structure. In contrast, existing methods rely on either non-Gaussianity or time correlation alone, limiting their applicability to specific factor structures. Furthermore, we extend this framework to multi-group structured data and develop multi-group ICA method to to identify factor structures of stock markets across multiple countries. The proposed model accommodates differences in factor structures between developed countries and emerging markets while maintaining homogeneity within each group. The statistical merits of the proposed method over existing methods are demonstrated by numerical experiments. Furthermore, we employ the superior forecast of size and momentum factor returns and design a market timing strategy to achieve significant improvement on the performance of various long-short zero-cost portfolios.