标题： 时间序列中的时间箭头 显示英文标题

标题： The Arrow of Time in Multivariate Time Series

摘要： 我们证明，如果所有创新项都是正态分布的，满足（线性）多变量自回归滑动平均（VARMA）模型的时间序列，在反向时间方向上也满足相同的模型假设。 如果创新项是非高斯的，这种可逆性就会失效。 这意味着在假设VARMA过程具有非高斯噪声的情况下，时间的方向变得可检测。 因此，我们的工作为一种已被用于推断视频片段方向的算法提供了理论依据。 我们提出了一种稍作修改的实用算法，用于估计给定样本的时间方向，并证明了其一致性。 我们进一步研究了算法性能如何依赖于样本大小、时间序列的维度数量以及过程的阶数。 对经济领域真实数据的应用表明，考虑多变量过程而非单变量过程在估计时间方向方面可能更有益。 我们的结果扩展了早期关于单变量时间序列的工作。 它与因果推断的概念有关，其中最近的方法利用误差项的非高斯性来进行因果结构学习。 显示英文摘要

摘要： We prove that a time series satisfying a (linear) multivariate autoregressive moving average (VARMA) model satisfies the same model assumption in the reversed time direction, too, if all innovations are normally distributed. This reversibility breaks down if the innovations are non-Gaussian. This means that under the assumption of a VARMA process with non-Gaussian noise, the arrow of time becomes detectable. Our work thereby provides a theoretic justification of an algorithm that has been used for inferring the direction of video snippets. We present a slightly modified practical algorithm that estimates the time direction for a given sample and prove its consistency. We further investigate how the performance of the algorithm depends on sample size, number of dimensions of the time series and the order of the process. An application to real world data from economics shows that considering multivariate processes instead of univariate processes can be beneficial for estimating the time direction. Our result extends earlier work on univariate time series. It relates to the concept of causal inference, where recent methods exploit non-Gaussianity of the error terms for causal structure learning.