  One of the most important problems arising in time series analysis is that of
bifurcation, or change point detection. That is, given a collection of time
series over a varying parameter, when has the structure of the underlying
dynamical system changed? For this task, we turn to the field of topological
data analysis (TDA), which encodes information about the shape and structure of
data. The idea of utilizing tools from TDA for signal processing tasks, known
as topological signal processing (TSP), has gained much attention in recent
years, largely through a standard pipeline that computes the persistent
homology of the point cloud generated by the Takens' embedding. However, this
procedure is limited by computation time since the simplicial complex generated
in this case is large, but also has a great deal of redundant data. For this
reason, we turn to a more recent method for encoding the structure of the
attractor, which constructs an ordinal partition network (OPN) representing
information about when the dynamical system has passed between certain regions
of state space. The result is a weighted graph whose structure encodes
information about the underlying attractor. Our previous work began to find
ways to package the information of the OPN in a manner that is amenable to TDA;
however, that work only used the network structure and did nothing to encode
the additional weighting information. In this paper, we take the next step:
building a pipeline to analyze the weighted OPN with TDA and showing that this
framework provides more resilience to noise or perturbations in the system and
improves the accuracy of the dynamic state detection.
