All neuroimaging modalities have their own strengths and limitations. A current trend is toward interdisciplinary approaches that use multiple imaging methods to overcome limitations of each method in isolation. At the same time neuroimaging data is increasingly being combined with other non-imaging modalities, such as behavioral and genetic data. The data structure of many of these modalities can be expressed as time-varying multidimensional arrays (tensors), collected at different time-points on multiple subjects. Here, we consider a new approach for the study of neural correlates in the presence of tensor-valued brain images and tensor-valued predictors, where both data types are collected over the same set of time points. We propose a time-varying tensor regression model with an inherent structural composition of responses and covariates. Regression coefficients are expressed using the B-spline technique, and the basis function coefficients are estimated using CP-decomposition by minimizing a penalized loss function. We develop a varying-coefficient model for the tensor-valued regression model, where both predictors and responses are modeled as tensors. This development is a non-trivial extension of function-on-function concurrent linear models for complex and large structural data where the inherent structures are preserved. In addition to the methodological and theoretical development, the efficacy of the proposed method based on both simulated and real data analysis (e.g., the combination of eye-tracking data and functional magnetic resonance imaging (fMRI) data) is also discussed.

翻译：所有神经影像模态均有其自身优势与局限性。当前趋势是采用跨学科方法，利用多种影像技术克服单一方法的局限。与此同时，神经影像数据正越来越多地与其他非影像模态（如行为与遗传数据）相结合。其中许多模态的数据结构可表示为随时间变化的多维数组（张量），这些数据在多个受试者的不同时间点采集。本文针对张量值脑影像与张量值预测变量同时存在（且两类数据在同一时间点集合上采集）的情况下研究神经关联的新方法。我们提出一种时变张量回归模型，该模型具有响应变量与协变量的内在结构组成。回归系数采用B样条技术表达，基函数系数通过CP分解并最小化惩罚损失函数进行估计。我们为张量值回归模型开发了变系数模型，其中预测变量与响应变量均建模为张量。这一方法是对复杂大结构数据中函数并发线性模型的重要扩展，且保留了内在结构。除方法论与理论发展外，本文还讨论了基于模拟与真实数据分析（如眼动追踪数据与功能磁共振成像数据的结合）所提出的方法的有效性。