Recent studies have actively employed persistent homology (PH), a topological data analysis technique, to analyze the topological information in time series data. Many successful studies have utilized graph representations of time series data for PH calculation. Given the diverse nature of time series data, it is crucial to have mechanisms that can adjust the PH calculations by incorporating domain-specific knowledge. In this context, we introduce a methodology that allows the adjustment of PH calculations by reflecting relevant domain knowledge in specific fields. We introduce the concept of featured time series, which is the pair of a time series augmented with specific features such as domain knowledge, and an influence vector that assigns a value to each feature to fine-tune the results of the PH. We then prove the stability theorem of the proposed method, which states that adjusting the influence vectors grants stability to the PH calculations. The proposed approach enables the tailored analysis of a time series based on the graph representation methodology, which makes it applicable to real-world domains. We consider two examples to verify the proposed method's advantages: anomaly detection of stock data and topological analysis of music data.

翻译：近年来的研究积极采用持续同调（PH）这一拓扑数据分析技术来分析时间序列数据中的拓扑信息。许多成功的研究利用时间序列数据的图表示进行PH计算。鉴于时间序列数据的多样性，有必要建立能够通过融入领域特定知识来调整PH计算的机制。在此背景下，我们提出了一种能够通过反映特定领域的相关领域知识来调整PH计算的方法。我们引入了特征时间序列的概念，即增强特定特征（如领域知识）的时间序列与一个影响向量的配对，该影响向量为每个特征分配一个值以微调PH结果。随后我们证明了所提方法的稳定性定理，该定理表明调整影响向量可使PH计算保持稳定性。所提方法基于图表示方法学实现对时间序列的定制化分析，使其能够应用于现实领域。我们通过两个实例验证所提方法的优势：股票数据的异常检测与音乐数据的拓扑分析。