  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.
