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.