Using Takens Embedding and Topological Data Analysis on Price Series
Summary
The article introduces a pipeline for applying topological data analysis to financial time series. Takens time-delay embedding converts a one-dimensional series into vectors in a higher-dimensional point cloud, controlled by embedding dimension and delay. The article explains how persistent homology can track connected components and loops across distance scales, representing each feature by its birth and death scales. Longer-lived features are presented as more likely to reflect structure than short-lived features.
The document focuses on foundational software components for creating embedded point clouds and computing pairwise distances under Euclidean, Manhattan, or Chebyshev norms; it does not yet calculate persistence diagrams or generate trades. A conceptual comparison suggests that trending and ranging windows can form different point-cloud shapes even when common summary statistics are similar. The author cautions that market prices are not clean dynamical systems, so Takens' theorem does not provide deterministic guarantees for them. The embedding parameters also require choices, and TDA is framed as a complement to conventional indicators rather than a complete description of price behavior.
Key ideas
- Time-delay embedding maps successive observations into vectors that form a phase-space point cloud.
- Embedding dimension and delay determine how the series is unfolded into higher dimensions.
- Persistent homology tracks topological features across scales using birth and death values.
- The article builds point-cloud and pairwise-distance components but stops before persistence analysis or trading signals.
- Market applications are statistical, and embedding choices affect the resulting cloud.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.