Zhang Rongqian, Zhang Yupeng, Liu Yuyao, Guo Yunjie, Shen Yueyang, Deng Daxuan, Qiu Yongkai Joshua, Dinov Ivo D
Department of Statistics, University of Michigan, Ann Arbor, MI 48109, USA.
Statistics Online Computational Resource, Department of Health Behavior and Biological Sciences, University of Michigan, Ann Arbor, MI 48109, USA.
Neural Comput Appl. 2022 Apr;34(8):6377-6396. doi: 10.1007/s00521-021-06789-8. Epub 2022 Jan 16.
Many modern techniques for analyzing time-varying longitudinal data rely on parametric models to interrogate the time-courses of univariate or multivariate processes. Typical analytic objectives include utilizing retrospective observations to model current trends, predict prospective trajectories, derive categorical traits, or characterize various relations. Among the many mathematical, statistical, and computational strategies for analyzing longitudinal data, tensor-based linear modeling offers a unique algebraic approach that encodes different characterizations of the observed measurements in terms of state indices. This paper introduces a new method of representing, modeling, and analyzing repeated-measurement longitudinal data using a generalization of event order from the positive reals to the complex plane. Using complex time (kime), we transform classical time-varying signals as 2D manifolds called kimesurfaces. This kime characterization extends the classical protocols for analyzing time-series data and offers unique opportunities to design novel inference, prediction, classification, and regression techniques based on the corresponding kimesurface manifolds. We define complex time and illustrate alternative time-series to kimesurface transformations. Using the Laplace transform and its inverse, we demonstrate the bijective mapping between time-series and kimesurfaces. A proposed general tensor regression based linear model is validated using functional Magnetic Resonance Imaging (fMRI) data. This kimesurface representation method can be used with a wide range of machine learning algorithms, artificial intelligence tools, analytical approaches, and inferential techniques to interrogate multivariate, complex-domain, and complex-range longitudinal processes.
许多用于分析随时间变化的纵向数据的现代技术都依赖参数模型来探究单变量或多变量过程的时间进程。典型的分析目标包括利用回顾性观察来对当前趋势进行建模、预测未来轨迹、推导分类特征或刻画各种关系。在众多用于分析纵向数据的数学、统计和计算策略中，基于张量的线性建模提供了一种独特的代数方法，该方法根据状态指标对观测测量的不同特征进行编码。本文介绍了一种使用从正实数到复平面的事件顺序推广来表示、建模和分析重复测量纵向数据的新方法。使用复时间（kime），我们将经典的随时间变化的信号转换为称为kime曲面的二维流形。这种kime表征扩展了用于分析时间序列数据的经典协议，并为基于相应的kime曲面流形设计新颖的推理、预测、分类和回归技术提供了独特的机会。我们定义了复时间，并说明了从时间序列到kime曲面变换的替代方法。使用拉普拉斯变换及其逆变换，我们展示了时间序列和kime曲面之间的双射映射。使用功能磁共振成像（fMRI）数据对提出的基于一般张量回归的线性模型进行了验证。这种kime曲面表示方法可与广泛的机器学习算法、人工智能工具、分析方法和推理技术一起使用，以探究多变量、复域和复范围的纵向过程。
