rollsig · working notes

rollsig — Working Notes

The mathematics and the build of a causal, rolling-window path-signature feature library, recorded chapter by chapter as it was written rather than distilled afterwards.

What this is

rollsig turns financial time series into causal, rolling-window path-signature features, and slides the window at O(1) steady-state cost per tick using the group structure of the tensor algebra — Chen's identity and the group inverse. These three documents are the full record of why it is built the way it is, including the parts that did not work.

The three parts

Part I · Chapters 1–5 The algebra, worked from a live example The signature as an element of the tensor algebra; Chen's identity and the shuffle identity checked numerically on a real path; lead-lag and quadratic variation; log-signatures and the free Lie algebra; and the transformer built on top of it, with five mathematical correctness oracles. Part II · Chapters 6–9 The benchmark: Optiver realized volatility Whether signature features forecast realized volatility better than reproduced order-book baselines — the arms, the grouped cross-validation, what the competition's own data does and does not permit, and a negative result reported as one. Part III · Chapters 10–12 The streaming engine Sliding a window by cancelling the departing increment with a group inverse: the identity, the implementation, the measured flatness in window length, the crossover against a compiled backend, the floating-point drift and what it costs to pin it — and one retracted prediction from Part II.

The two results

streaming — positive

Steady-state per-tick cost is flat across a 120× change in window length, which makes it 129× faster than RoughPy at window 1200. Against compiled iisignature it wins only past a measured crossover, and the library's method="auto" encodes that rather than guessing. Numerical drift is measured, bounded, and priced.

benchmark — negative

On the Optiver Realized Volatility Prediction data, signature features did not improve on reproduced order-book baselines — on any of three pre-registered stock subsets, and not after the order-book state was handed to them as extra path channels. That line of enquiry is closed.