Using Path Signatures and Levy Area to Measure Lead–Lag
Summary
The document presents the second-level path signature, especially its antisymmetric component called Levy area, as a rolling measure of ordering between two time series. Rather than scanning candidate lags as cross-correlation does, it treats paired observations as a path and measures its signed area. The sign indicates which channel tends to lead within the window, while the magnitude reflects the path’s shape. The article derives an incremental update based on Chen’s identity and describes time augmentation and normalization for preparing market data.
It compares the method with lagged correlation using shifted sine waves, where the signed area distinguishes lead direction without selecting a lag. It also describes an MQL5 indicator, a threshold-rule Expert Advisor, and ground-truth checks using signature identities and independent cross-checking. The examples illustrate mathematical behavior, not predictive performance in live markets. The author cautions that noisy path shape affects the reading, recommends calibrating thresholds, and treats the EA as a baseline; Levy area is offered as a complementary statistic rather than a universal replacement for correlation.
Key ideas
- A path signature encodes the ordered movement of multiple data channels in a fixed-length set of values.
- The antisymmetric second-level term, Levy area, provides a signed, lag-free reading of channel ordering within a window.
- An incremental Chen update computes the level-two signature without repeatedly evaluating a double integral.
- Time augmentation and normalization help turn market observations into a path suitable for signature analysis.
- Noisy path shape can affect the lead–lag reading, so the measure should be validated and its trading threshold calibrated.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.