Generalized penalty for circular coordinate representation

<p style='text-indent:20px;'>Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization dimension reduction. We follow framework circular coordinate representation, which allows perform reduction high-dimensional datasets on torus using persistent cohomology. In this paper, we propose method adapt take into account roughness coordinates in change-point applications. To do that, use generalized penalty function instead an <inline-formula><tex-math id="M1">\begin{document}$ L_{2} $\end{document}</tex-math></inline-formula> traditional algorithm. provide simulation experiments real analyses support our claim with will detect change under different sampling schemes while preserving structures.</p>