Rough Path Signatures and Data Transformations for Financial Time Series

Summary

This introduction explains path signatures as collections of iterated integrals, organized by tensor order. The first level records path increments; second-level terms encode additional ordering information and obey identities linking them to products of first-level increments. The article presents signatures as a way to represent complex, oscillatory paths for later use in machine learning and time-series analysis.

Because market prices arrive as discrete observations, it describes converting sampled prices into piecewise-linear paths before calculating signatures. It outlines two constructions: a two-dimensional lead-lag path that separates successive observations into leading and lagging components, and a time-joined path that alternates movement along time and price. IBM prices from October to November 2016 illustrate the original series and both transformations. The article is introductory: it motivates the representation and gives transformation procedures, but does not report a trading strategy, predictive test, or performance evidence. Its financial applications are deferred to later installments.

Key ideas

• A path signature consists of iterated integrals that capture information beyond the path's net increment.
• The first signature level equals the path increment, while second-level terms reflect the ordering of movements.
• Discrete price observations can be interpolated into piecewise-linear paths before signature calculation.
• Lead-lag and time-joined transformations offer different ways to encode sampled prices as continuous paths.
• The examples illustrate representation methods rather than demonstrated trading results.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.