Before revision:  The proposed procedure applies a dimensionality reduction technique via principal component analysis to the point cloud of the Takens' embedding from the observed time series and then evaluates the persistence landscape and silhouettes based on the corresponding Rips complex . We define a new notion of Rips distance function that is especially suited for persistence homologies built on Rips complexes and prove stability theorems for it. We use these results to demonstrate in turn some stability properties of the topological features extracted using our procedure with respect to additive noise and sampling .
After revision:  The proposed procedure applies a dimensionality reduction technique via principal component analysis to the point cloud of the Takens' embedding from the observed time series . The proposed methodology relies on the persistent landscapes and silhouette of the Rips complex obtained after a de-noising step based on principal components applied to a time-delayed embedding of a noisy, discrete time series sample. We analyze the stability properties of the topological features extracted using our procedure with respect to additive noise and sampling .