  In this paper, we introduce the persistence transformation, a novel
methodology in Topological Data Analysis (TDA) for applications in time series
data which can be obtained in various areas such as science, politics, economy,
healthcare, engineering, and beyond. This approach captures the enduring
presence or `persistence' of signal peaks in time series data arising from
Morse functions while preserving their positional information. Through rigorous
analysis, we demonstrate that the proposed persistence transformation exhibits
stability and outperforms the persistent diagram of Morse functions (with
respect to filtration, e.g., the upper levelset filtration). Moreover, we
present a modified version of the persistence transformation, termed the
reduced persistence transformation, which retains stability while enjoying
dimensionality reduction in the data. Consequently, the reduced persistence
transformation yields faster computational results for subsequent tasks, such
as classification, albeit at the cost of reduced overall accuracy compared to
the persistence transformation. However, the reduced persistence transformation
finds relevance in specific domains, e.g., MALDI-Imaging, where positional
information is of greater significance than the overall signal height. Finally,
we provide a conceptual outline for extending the persistence diagram to
accommodate higher-dimensional input while assessing its stability under these
modifications.
