测量时间之矢：多元时间序列中方向结构的识别、估计与推断
Measuring the Arrow of Time: Identification, Estimation, and Inference for Directional Structure in Multivariate Time Series
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中文总结 AI 辅助
本文提出基于滞后协方差的多元时间序列方向结构测量方法，经实验验证可区分相关网络等方法的失效，为相关应用提供基础参考。
中文摘要 AI 辅助
科学领域的诸多问题具有相同形式：多个耦合序列被共同观测，分析者不仅想知道它们是否协同变化，还想知道哪个序列先变化以及变化强度如何。本文基于一个核心思路构建了一套完整方法：耦合系统的方向正是记录倒放时行为发生变化的那部分。仅基于同期协方差的工具（相关矩阵、距离度量、生成树、无向中心性、主成分）不包含任何方向信息：可逆系统和循环系统在同一时间点记录的每次采样都可具有完全相同的协方差。形式上，方向是由滞后协方差携带的循环矩阵；对于线性系统，其消失恰好对应统计时间可逆性，特征映射将该表征推广到非线性系统；在高斯基准下，其幅度是所识别循环的熵产生泛函，即正向记录与反向记录间每单位时间散度的二次分量。围绕该估计量，我们构建了一种移除一阶偏差的交叉拟合估计量、删除块刀切标准误，以及在规定块原假设下精确的随机化检验，还包含一族校正方法和非线性扩展。抽样理论明确了时间之矢何时可测，设计层将传输与时钟排序分离。一个包含四个已知答案的系统实验室将该方法与相关网络、格兰杰因果关系、传递熵及连通性指数进行对比，报告了各方法作为方向度量的失效情况，包括我们自己的方法。两种语言的完整算法和示例使本文成为一系列应用的基础参考。
英文摘要
Many questions across the sciences take the same form: several coupled series are observed together, and the analyst wants to know not merely that they move together but which one moves first, and how strongly. This paper sets out a complete method built on one organising idea: the direction of a coupled system is exactly the part of its behaviour that changes when the record is played backwards. Tools built on contemporaneous covariance alone (correlation matrices, distance measures, spanning trees, undirected centralities, principal components) carry no information about direction: a reversible system and a circulating one can share identical covariance at every sampling of the same point-in-time record. Formally, direction is a circulation matrix carried by the lagged covariance. Its vanishing is exactly statistical time reversibility for linear systems, feature maps carry the characterisation to nonlinear ones, and under the Gaussian benchmark its magnitude is an entropy-production functional of the identified circulation, the quadratic component of the divergence per unit time between the forward and reversed records. Around this estimand we build a cross-fitted estimator removing first-order bias, delete-block jackknife standard errors, and a randomisation test exact under its stated block null, with a familywise correction and a nonlinear extension. A sampling theory says when the arrow is measurable at all, and a design layer separates transmission from the ordering of clocks. A laboratory of four systems with known answers compares the method with correlation networks, Granger causality, transfer entropy, and connectedness indices, reporting the failures of each when read as a measure of direction, including our own. Complete algorithms and worked examples in two languages make the paper the base reference for a series of applications.
