Persistent Homology: Column Reduction and Persistence Diagrams for Market Data
Summary
The document explains how standard column reduction turns a boundary matrix over the two-element field into a persistence diagram. Each diagram pair records when a topological feature appears and disappears as the Vietoris-Rips scale grows. The article reviews connected components, loops, and voids, and interprets long-lived features as potentially meaningful structure while short-lived features may represent noise. Its MQL5 implementation processes simplex columns in filtration order, combines columns by symmetric difference, and uses pivot ownership to find pairings efficiently.
The software organizes reduction, diagram analytics, and a full pipeline behind reusable classes. The article demonstrates the output on a sampled circle and reports checks against known examples and the independent Ripser implementation, including a live market case. Those checks support agreement of the computed diagrams, but do not show that topological features forecast returns or improve a trading strategy. Because the described complex includes simplices only up to triangles, it reliably targets H0 and H1; apparent higher-dimensional features can be artifacts of that truncation, and true H2 analysis requires tetrahedra.
Key ideas
- Column reduction pairs creator simplices with the simplices that eliminate their features.
- Over Z/2, column addition is the symmetric difference of sorted face lists.
- Persistence diagrams encode feature births and deaths across filtration scales.
- Longer-lived features are highlighted as potentially meaningful, while short-lived features may be noise.
- The implementation is checked against known cases and Ripser, but this validates computation rather than trading value.
- Limiting the complex to triangles supports H0 and H1 while restricting genuine H2 detection.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.