Signature Methods

Signature Methods are a mathematical approach to time series analysis based on the path signature — an infinite sequence of iterated integrals that provides a universal, order-invariant feature set capturing all essential information about a path's geometry and order of events.

What Is the Path Signature?

• Definition: $S(X)_{t_0,t_1} = (1, \int dX_t, intint dX_s dX_t, \ldots)$ — iterated integrals of all orders.
• Truncation: In practice, truncate to depth $k$ (e.g., $k=3-5$), giving a finite-dimensional feature vector.
• Properties: Invariant to reparameterization, captures order of events, universally nonlinear.
• Log-Signature: A more compact representation using the log map, with the same information content.

Why It Matters

• Feature Extraction: The signature provides a principled, hand-crafted-free feature set for any time series.
• Kernel Methods: Signature kernels enable Gaussian processes and SVMs on time series data.
• Neural Integration: Signature features can be combined with neural networks (Deep Signature Networks).

Signatures are the DNA of a time series — capturing all essential geometric and sequential information in a mathematically principled feature set.