地球与空间观测遇见复杂代数：从复数到八元数的多元自回归时间序列分析
Earth and space observations meet complex algebras: from complex to octonions for multivariate autoregressive time series analysis
- Faculty of Engineering and Science, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与科学学院）
- Data Observatory Foundation, ANID Technology Center No. DO210001（数据观测基金会，ANID技术中心No. DO210001）
- Millennium Institute of Astrophysics, ICM-ANID ICN12_009（天体物理学千年研究所，ICM-ANID ICN12_009）
- Department of Mathematics and Computer Science, Faculty of ScienceUniversidad de Santiago（圣地亚哥大学理学院数学与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对非规则采样多元时间序列，提出基于超复数（如八元数）自回归的离散时间框架，结合状态空间与卡尔曼滤波估计，有效捕捉动态和跨变量交互，并在遥感和天文数据中验证。
AI中文摘要：
许多数据集以在非规则时间间隔观测到的多元时间序列形式出现，尤其是在地球观测和天文测量中。经典时间序列模型假设时间是离散的且观测在等间隔时刻发生，这限制了它们在观测间隔变化时捕捉动态的能力。现有的非规则采样方法依赖于连续时间公式，这些公式假设观测间隔足够小。我们通过引入一个离散时间框架来解决这一局限性，该框架能够适应非规则的观测间隔，同时保留多元依赖结构。我们提出一个基于超复数自回归过程的非规则观测多元时间序列分析框架。该框架将多元观测嵌入到超复数代数结构中，允许在单个数学实体中表示多个变量及其相互作用，同时纳入非规则观测间隔。由此产生的过程具有结构化的矩阵表示，使得模型能够表示为状态空间系统。这一表述提供了一种估计方法，其中模型参数可通过卡尔曼滤波技术推断。状态空间表示允许在非规则采样间隔下进行递归估计和预测。该方法的性能通过数据和应用于遥感及天文数据集得到展示，这些数据集中的观测发生在非均匀时间间隔。结果表明，该方法能够捕捉传统时间序列技术难以建模的动态和跨变量相互作用。所提出的框架为分析非规则采样的多元时间序列提供了一种方法，并为跨学科建模观测数据开辟了可能性。
英文摘要：
Many datasets arise as multivariate time series observed at irregular time intervals, particularly in Earth observation and astronomical measurements. Classical time-series models assume that time is discrete and observations occur at equally spaced intervals, limiting their ability to capture dynamics when observation gaps vary. Existing approaches to irregular sampling rely on continuous-time formulations, which assume that observation intervals are sufficiently small. We address this limitation by introducing a discrete-time framework that accommodates irregular observation gaps while preserving multivariate dependence structures. We introduce a framework for the analysis of irregularly observed multivariate time series based on hypercomplex autoregressive processes. The framework embeds multivariate observations into a hypercomplex algebraic structure, allowing multiple variables and their interactions to be represented within a single mathematical entity while incorporating irregular observation gaps. The resulting process admits a structured matrix representation, which enables the model to be expressed as a state-space system. This formulation provides an estimation methodology in which model parameters can be inferred using Kalman filtering techniques. The state-space representation allows recursive estimation and prediction in the presence of irregular sampling intervals. The performance of the methodology is illustrated through data and applications to remote sensing and astronomical datasets, where observations occur at nonuniform time intervals. The results demonstrate that the approach captures dynamics and cross-variable interactions difficult to model using traditional time-series techniques. The proposed framework provides a methodology for analyzing irregularly sampled multivariate time series and opens possibilities for modeling observational data across disciplines.
