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Torus graphs for multivariate phase coupling analysis
Annals of Applied Statistics ( IF 1.8 ) Pub Date : 2020-06-29 , DOI: 10.1214/19-aoas1300
Natalie Klein 1 , Josue Orellana 1 , Scott L Brincat 2 , Earl K Miller 2 , Robert E Kass 1
Affiliation  

Angular measurements are often modeled as circular random variables, where there are natural circular analogues of moments, including correlation. Because a product of circles is a torus, a $d$-dimensional vector of circular random variables lies on a $d$-dimensional torus. For such vectors we present here a class of graphical models, which we call torus graphs, based on the full exponential family with pairwise interactions. The topological distinction between a torus and Euclidean space has several important consequences. Our development was motivated by the problem of identifying phase coupling among oscillatory signals recorded from multiple electrodes in the brain: oscillatory phases across electrodes might tend to advance or recede together, indicating coordination across brain areas. The data analyzed here consisted of 24 phase angles measured repeatedly across 840 experimental trials (replications) during a memory task, where the electrodes were in 4 distinct brain regions, all known to be active while memories are being stored or retrieved. In realistic numerical simulations, we found that a standard pairwise assessment, known as phase locking value, is unable to describe multivariate phase interactions, but that torus graphs can accurately identify conditional associations. Torus graphs generalize several more restrictive approaches that have appeared in various scientific literatures, and produced intuitive results in the data we analyzed. Torus graphs thus unify multivariate analysis of circular data and present fertile territory for future research.

中文翻译:

多变量相位耦合分析的圆环图

角度测量通常被建模为循环随机变量,其中有力矩的自然循环模拟,包括相关性。因为圆的乘积是一个圆环,一个 $d$ 维的圆形随机变量向量位于一个 $d$ 维的圆环上。对于这样的向量,我们在这里展示了一类图形模型,我们称之为圆环图,基于具有成对交互的完整指数族。环面和欧几里德空间之间的拓扑区别有几个重要的后果。我们的开发是由识别从大脑中多个电极记录的振荡信号之间的相位耦合问题推动的:电极间的振荡相位可能倾向于一起前进或后退,表明大脑区域之间的协调。此处分析的数据包括 24 个相位角,在记忆任务期间通过 840 次实验试验(重复)重复测量,其中电极位于 4 个不同的大脑区域,所有这些区域在存储或检索记忆时都处于活动状态。在现实的数值模拟中,我们发现标准的成对评估,称为锁相值,无法描述多元相相互作用,但圆环图可以准确识别条件关联。圆环图概括了各种科学文献中出现的几种更具限制性的方法,并在我们分析的数据中产生了直观的结果。因此,圆环图统一了循环数据的多变量分析,并为未来的研究提供了肥沃的领域。
更新日期:2020-06-29
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