Topological data analysis (TDA) approaches are becoming increasingly popular for studying the dependence patterns in multivariate time series data. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes, which can be altered by various neurological impairments such as epileptic seizures. Existing TDA approaches rely on the notion of distance between data points that is symmetric by definition for building graph filtrations. For brain dependence networks, this is a major limitation that constrains practitioners to using only symmetric dependence measures, such as correlations or coherence. However, it is known that the brain dependence network may be very complex and can contain a directed flow of information from one brain region to another. Such dependence networks are usually captured by more advanced measures of dependence such as partial directed coherence, which is a Granger causality based dependence measure. These dependence measures will result in a non-symmetric distance function, especially during epileptic seizures. In this paper we propose to solve this limitation by decomposing the weighted connectivity network into its symmetric and anti-symmetric components using matrix decomposition and comparing the anti-symmetric component prior to and post seizure. Our analysis of epileptic seizure EEG data shows promising results.

翻译：拓扑数据分析（Topological Data Analysis, TDA）方法在分析多变量时间序列数据的依赖模式方面日益受到关注。特别是，大脑网络中的多种依赖模式可能与特定任务及认知过程相关，这些模式可能因癫痫发作等神经功能障碍而发生改变。现有TDA方法依赖于数据点间的距离概念（该距离在定义上具有对称性）来构建图过滤。对于大脑依赖网络而言，这构成了一个主要限制，迫使研究者只能使用对称的依赖度量（如相关性或相干性）。然而，已知大脑依赖网络可能极为复杂，且包含从一个脑区到另一个脑区的有向信息流。此类依赖网络通常通过更先进的依赖度量（如部分有向相干性，一种基于格兰杰因果关系的依赖度量）来捕获。这些依赖度量将产生非对称的距离函数，在癫痫发作期间尤为显著。本文提出通过矩阵分解将加权连接网络分解为对称分量和反对称分量，并比较发作前后反对称分量的方法来解决这一局限性。我们对癫痫发作脑电图数据的分析展示了令人鼓舞的结果。