SIGL Method: Unified Time Series Feature Extraction

• SIGL method is a unifying framework for feature extraction from multivariate time series using signature transforms and offers order-aware representations.
• It modularizes steps like augmentation, windowing, transformation, and rescaling to create a customizable feature pipeline validated by empirical studies.
• The approach improves classification tasks by subsuming prior signature-based techniques and delivering practical performance gains in time series analysis.

The SIGL method refers to a unifying, generalised framework for extracting features from multivariate time series based on the signature transform from rough path theory. It is designed to systematize and extend the diverse set of signature-based techniques prevalent in both mathematical analysis and machine learning. The SIGL construction modularizes every step—augmentation, windowing, transformation depth, and rescaling—enabling systematic adaptation and providing a canonical pipeline validated by large-scale empirical studies (Morrill et al., 2020). The method is applicable across a wide range of domains, especially in time series classification and analysis.

1. Mathematical Foundation: Signature Transform and Its Properties

Let $X:[0,T]\to\mathbb{R}^d$ be a continuous path, typically realized by piecewise-linear interpolation of a discrete time series $\mathbf{x}$. The truncated signature of depth $N$, denoted $\mathrm{Sig}^N(X)$, is the collection of all iterated integrals up to order $N$: $$\mathrm{Sig}^N(X) = \left\{ S^I(X) \mid I = (i_1,\ldots,i_k),\; 1 \leq k \leq N \right\}, \quad S^I(X) = \int_{0 < t_1 < \ldots < t_k < T} dX_{t_1}^{i_1} \cdots dX_{t_k}^{i_k}.$$ Key theoretical guarantees:

• Universality: Any continuous functional $f(X)$ can be arbitrarily well-approximated linearly on $\mathrm{Sig}(X)$.
• Uniqueness: Under mild hypotheses, the (full) signature determines the path up to translation and time-reparameterization.
• Log-signature: The Lie-series logarithm of $\mathrm{Sig}^N(X)$ removes algebraic redundancies, yielding a lower-dimensional, non-linear representation.

These properties furnish strong theoretical justification for deploying signatures as universal, order-aware features in sequential modeling.

2. Modular Generalised Signature Framework

Traditionally, signature feature extraction is viewed monolithically as $\mathrm{Sig} \circ \mathrm{identity}$. The SIGL method generalizes this via the following composable modules:

(A) Augmentation ($\phi$):

• Time augmentation $\phi_{\mathrm{time}}$: Incorporates time as an extra dimension, breaking parameterization invariance.
• Basepoint augmentation $\phi_{\mathrm{bp}}$: Restores translation visibility; can be implemented as invisibility-reset or coordinate drop to zero.
• Lead–lag transformation $\phi_{\mathrm{ll}}$: Duplicates channels to capture quadratic variation.
• Other: Coordinate projections (random, learned), and trainable stream-preserving functions.

(B) Windowing ($W$):

• Global: Full path.
• Sliding: Local, fixed-length subpaths (start, step, length).
• Expanding: Increasing-size prefixes.
• Hierarchical dyadic: All sub-paths at dyadic scales, yielding $2^q-1$ windows for dyadic depth $q$.

(C) Transform ($S^N$ or $\log S^N$):

• Iterated integrals computed up to order $N$.
• Optionally replaced by log-signature for redundancy reduction.

(D) Rescaling ($\rho$):

• Post-scaling: Each $k$-order term multiplied by $k!$.
• Pre-scaling: Path rescaled so $\alpha^N/N!\approx 1$.

A generic feature vector for augmentation $i$ and window $j$ is: $$z_{i, j} = \rho_{\mathrm{post}} \circ S^N \circ \rho_{\mathrm{pre}} \circ W^j \circ \phi^i (\mathbf{x})$$ The final representation concatenates all $\{z_{i,j}\}$ vectors.

3. Canonical Pipeline and Recommended Defaults

Based on an empirical study across 26 datasets and 4 classifiers, the SIGL framework identifies a robust, domain-agnostic canonical pipeline: $$z_j = S^N \circ W^j_{\mathrm{dyadic},q} \circ \phi_{\mathrm{bp}} \circ \phi_{\mathrm{time}} (\mathbf{x}),\quad j=1,\ldots,2^q-1$$ With the following recommended defaults:

• Signature depth $N=3$.
• Hierarchical dyadic windowing, dyadic depth $q=3$ (i.e., 7 windows).
• Time and basepoint augmentations only.
• No pre-scaling; use simple post-scaling (factorial scaling). Hyperparameters $(N, q)$ are tuned in $\{1,\ldots,6\}\times\{2,3,4\}$.

This canonical pipeline provides strong baseline performance, and is extensible by modifying any of the four principal modules.

4. Connections to Prior Methods and Special Cases

The SIGL framework subsumes earlier signature-based techniques as special cases:

• Levin & Lyons: $\phi_{\mathrm{time}}$, global window, pure $S^N$.
• Yang et al. (“invisibility-reset”): $\phi_{\mathrm{time}} \circ \phi_{\mathrm{inv}}$.
• Lead–lag signature: $\phi_{\mathrm{ll}}$, global window.
• Sliding-window signatures: appropriate $\phi$, sliding window $W_{\mathrm{sliding}}$.

Each is expressible as $z_j = (S^N \circ W^j \circ \phi)(\mathbf{x})$ for appropriate choices of component functions.

5. Practical Implementation Considerations

SIGL offers concrete guidance for practitioners:

• Computational complexity: Calculation of $\mathrm{Sig}^N$ on length-$n$, $d$-channel paths is $\mathcal{O}(n d^N)$, producing $(d^{N+1}-1)/(d-1)$ features.
• Software: Open-source iisignature (C/CUDA) and PyTorch–based Signatory libraries are available.
• Model selection: Depth $N$ and window depth $q$ are cross-validated (typical $N \leq 4$, $q \leq 4$). Downstream predictors include $\ell^2$-regularized linear models, random forests, or small RNN/CNNs.
• Preprocessing: Channel-wise standardization; irregularly sampled time series handled via piecewise-linear interpolation, with no explicit resampling required.

6. Empirical Observations and Performance

Findings from broad evaluation include:

• Time + basepoint augmentations yield marked improvement over unaugmented signals.
• Lead–lag augmentation offers further gains, while coordinate projections and learned heads are particularly helpful in high-dimensional, e.g., EEG data.
• Hierarchical dyadic windowing ($q \geq 2$) outperforms global and sliding windows.
• Conventional signature generally outperforms log-signature (statistically significant, $p \approx 0.01$).
• Optimal extraction depth is typically $N = 3$ or 4; deeper extraction tends to overfit.
• With the canonical SIGL pipeline and a random forest classifier, average method ranking was 4.3 out of 9 strong baselines (MUSE, HIVE-COTE, TapNet, MLCN, etc.) without ever exceeding 32 GB RAM or 30 minutes runtime per dataset on a single GPU.

7. Significance and Flexibility of the SIGL Framework

SIGL is fundamentally a modular, unifying abstraction rather than a fixed method. All prior signature-based feature extraction variants are expressible within its architecture. The framework supports out-of-the-box adoption via its canonical pipeline but permits seamless adaptation to diverse applications simply by exchanging or composing its four principal modules ($\phi, W, S^N, \rho$) (Morrill et al., 2020). This positions SIGL as both a practical standard and an extensible, deeply justified feature generator for multivariate time series in contemporary machine learning settings.