Compression Scaling Law (CSL) – Detecting hidden structure in time series

What is CSL?
A simple, interpretable way to flag instability windows in a time series. It looks for periods where multiple properties change together — variance ↑, short-term persistence ↑, and spectral entropy ↓ — which are exactly the windows where forecasts and controls tend to fail.

Two flavours you can adopt immediately

• Canonical CSL (publishable): quantize → lossless compress → compare against matched surrogates (IAAFT) across window sizes; fit a power law to extra compressibility; the slope gives α (hidden order index).
• Lightweight proxy (ops-ready): compute an instability proxy (e.g., variance ratio or coherence tilt), form CSL(t) = Level × Slope, and trigger alerts with a rolling quantile (budgeted), e.g., ~12 alerts / 5 years for monthly data.

CSL is best used as an early‑warning guardrail, not as a point-event predictor.

1. Pick a regularly sampled series (monthly works well).
2. Compute an instability proxy α(t):
   • Variance‑ratio: α₁(t) = Var(6) / Var(24), ranked or z‑scored, or
   • Coherence‑tilt (proxy for canonical): estimate scale tilt inside a 48‑sample window.
3. Build Level and Slope on α(t):
   • Level: (L(t) = [\alpha(t) - \text{median}{13m}(\alpha)]+)
   • Slope: (S(t) = [\alpha(t) - \alpha(t-12)]_+)
4. CSL score: (\text{CSL}(t) = L(t) \times S(t))
5. Budgeted threshold: alert if CSL(t) > rolling (1 − p) quantile of past CSL values
   (monthly: p ≈ 12/60 → about 12 alerts per 5 years; warm‑up ~36 months).
6. Record interpretable probes for each alert: Var(6), VarRatio(6/24), AC(1), Ljung–Box proxy, spectral entropy, band powers.

See docs/CSL_Compact_Explainer.pdf for a 2–3 page explanation and figures.

• Model governance / risk gating: widen CIs, require human sign‑off, or use robust settings under alerts.
• Adaptive retraining: only retrain when CSL stays elevated.
• Observability triage: quick “why now?” via probes.
• Scientific change‑hunting: highlight regime transitions (ENSO, sunspots, hydrology).

What it is not: an exact trough/top predictor.

1. Window the series (L ∈ {48, 64, …, 192}), 2) µ‑law 8‑bit quantize, 3) lossless compress (DEFLATE/bzip2/LZMA),
2. build surrogates (IAAFT; preserves spectrum/marginal), 5) compress surrogates identically,
3. for each L compute average code‑length contrast ∆(L), define ( \kappa(L) = -L\cdot\Delta(L) ),
4. fit ( \log \kappa(L) = a + b \log L ) and set α = 1 − b.

Interpretation: α ≈ 1 (near‑null), α < 1 (scale‑reinforcing hidden order), α > 1 (rare/divergent).

• docs/CSL_Compact_Explainer.pdf — 2–3 page explainer (everything you need to adopt CSL).
• examples/synthetic_series.csv — a tiny toy series with a variance‑burst regime change.
• LICENSE — MIT (broad reuse).
• CITATION.cff — How to cite.
• CONTRIBUTING.md — Short guidance for issues and small additions (keep it compact).

MIT — see LICENSE.

If you use CSL in a report or post, please cite this repository and the explainer PDF.