Clarification of the method in topological data analysis

External question — Data Science Stack Exchange Author: Sillyasker Original post: https://datascience.stackexchange.com/questions/131031 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I want to detect the stock price crashes using topological data analysis. For example I have taken an excel file named tesla with columns date,open,high,low,close and volume. I want the time series of the Close column. Now first I apply takens embedding theorem , then apply the slinging window. Then use rips complex to detect holes. I have used some research papers on tda for gaining this information, but unable to understand why the thing s are implemented . Can anyone explain what is going on ?

External answer — Data Science Stack Exchange Author: M. M. Original post: https://datascience.stackexchange.com/a/131032 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. The general idea behind using topological data analysis on time series data is that -- according to Taken's embedding theorem -- if there is an attractor in a dynamical system, then it can be embedded in high-dimensional space by a sliding-window embedding. This embedding is smooth, and topological -- that is, the overall connectivity of the attractor is maintained (i.e. is it a cycle? is it a torus?). Applying topological data analysis methods to the resulting object allow you to get measurements of the topological properties of the embedding that you have constructed. For example, a cyclical attractor (stable oscillator) will simply look like a one-dimensional hole ( $\beta_1 = 1$ ). If you have two fairly independent oscillatory behaviors you may expect a torus ( $\beta_1 = 2, \beta_2 = 1$ ). For example, Suppose there is a simple seasonal cycle in your time series and nothing else, you would expect to see a clear $\beta_1 = 1$ .