Sleep staging from a fingertip oximeter's three signals: a pre-registered ceiling, a sequence control, and a stack of published formulas scored blind

Michal Planicka  ·  corresponding author — Tepna Project

OxyDex node (SpO₂ · pulse rate · motion), Tepna physiological-signal suite

Draft v0 · October 2026 · Skeleton written 2026-10-04 while v21/v22 run · Inputs: ten open and NSRR cohorts, 9 100 nights, feature set oxy1 · bands pre-stated in briefs/STAGE-LAB-2026-09-27-BRIEF.md §§5–20 before each run · nothing here is a published rate until the owner opens the P5 gate

Keywords: sleep staging · pulse oximetry · pulse rate · actigraphy · pre-registration · knowledge distillation · conditional random field · auto-context · blind validation

1. Introduction

Two human scorers reading the same polysomnogram agree at κ ≈ 0.83 (measured here on ISRUC, Khalighi et al., 2016). A wearable has none of the channels they read. The question of this paper is therefore not whether a ring equals a lab, but how much of the night a ring's three 1-Hz signals contain, and whether that information can be expressed in formulas rather than in a black box.

Comparators. A proprietary stager for the same ring family reports κ 0.367 live / 0.390 offline on a sealed 134-night Challenge-2018 holdout (published figures only; its model and outputs were never used, per its licence §3g). Sridhar et al. (2020) reached κ 0.66 from instantaneous heart rate alone with a whole-night CNN on SHHS/MESA. Vaquerizo-Villar et al. (2025) reached κ 0.73–0.78 from 1-Hz pulse rate and SpO₂ on 17 303 PSGs and published the frequency bands their network attends to, which this paper adopts as formula terms.

Design. Three learners on one feature matrix and identical folds: trees as the floor, a sequence network as the ceiling and teacher ("control", never shipped), and a stack of published formulas as the candidate for release. The owner's rule "train on everything in-house, publish only what the formulas carry" is what the design implements.

2. Methods

2.1 Cohorts and licences

Nights with a present-but-empty SpO₂ channel (< 50 % valid) are skipped and counted; a cohort without a signal yields NaN in that signal's columns, never a filled value. Aggregate statistics only are reported; no record identifiers leave the corpus (NSRR DAUA §5).

2.2 Harmonisation to the ring

SpO₂ is reduced to an 8-s moving mean at 1 Hz in whole percent, which is what the ring writes (one byte per second per signal). Pulse rate is the oximeter's own 1-Hz rate where the PSG has one; otherwise 60/RR from ECG beats found by the suite's Pan–Tompkins detector (Pan & Tompkins, 1985; validated at Se/PPV ≥ 99 % on MIT-BIH, SLPDB and NSRDB, Table S1) with a 4-s mean. Motion is the per-second standard deviation of wrist acceleration magnitude; a 1-bit "moved this second" flag (> 0.02 g) is carried beside it, as the poorest signal the ring could report. Every signal is standardised per night (median/IQR).

2.3 Feature matrix (feature set oxy1, 466 columns)

Multi-scale statistics (1–10 min), desaturation counts and depths, SpO₂ and pulse-rate band powers including the 0.004–0.020 Hz and 0.18–0.40 Hz bands of Vaquerizo-Villar et al. (2025), pulse-rate variability terms after Redmond & Heneghan (2006), actigraphy terms after Cole et al. (1992) and Sadeh et al. (1994), each with ±2/±10/±30-epoch context. The complete list with citations, where each term enters and whether its constants are pinned or fitted, is docs/STAGE-LAB-FORMULAS.md.

2.4 Learners

• Trees: histogram gradient boosting (Friedman, 2001), NaN-aware, 200 rounds, lr 0.1, 31 leaves.
• Control: dilated temporal convolutions (receptive field ≈ 2 h) with a bidirectional GRU over the whole night, class-weighted cross-entropy, epoch selected on an inner 10 % split; never shipped, used as ceiling and teacher.
• Literature stack: softmax over named terms (line + hinge), fitted by L-BFGS; auto-context (Tu & Bai, 2010) on out-of-fold posteriors; a linear-chain CRF (Lafferty et al., 2001 — the one-dimensional Potts chain, Potts, 1952) over out-of-fold unaries with the unary scale fixed at 1. Variants: physiology-fixed coefficient signs; distillation from the control (Hinton et al., 2015); a bridge of long-range context terms, cumulative stage mix and an ultradian phase by matched filter; symbolic regression on the residual to the control.
• Decoding: HMM Viterbi over scaled likelihoods (Rabiner, 1989) for trees and control; the CRF's own Viterbi for the stack.

2.5 Pre-registration and plants

Each unit's expected range, pass line and reading rule were written in briefs/STAGE-LAB-2026-09-27-BRIEF.md before the unit ran (§§5–20); measured numbers were appended beside them, including every failed expectation. The gate for all units: pooled κ ≥ 0.415, half the human inter-scorer κ (0.829). Plant P1: labels shifted by half a night must collapse κ below half its value, or the run is void. Folds are a hash of the subject key (SHHS1/SHHS2 visits of one person share a fold); Challenge 2018 and ISRUC are never in any fit.

3. Results

PENDING. Each table below is filled from the brief section named in its caption when that run lands; the values are copied from the bench summaries under lab-cache/bench-*/, never typed from memory.

4. Discussion

PENDING. Stubs to be written from the measured sections: (i) where the gap between formulas and the sequence control lives (deep and REM recall; long-range structure the oracle bound shows is present); (ii) why distillation into a small closed-form student lost to label fitting both times; (iii) what the ring's motion byte is worth and why a 1-bit flag equals the full accelerometer for trees; (iv) the data curve against the 17 303-night benchmark; (v) what can ship, under which licence, and the role of the blind cohorts.

5. Reproducibility

# features (one tag per feature set; a changed layout never mixes with an older cache)
node tools/stage-lab.mjs features --harmonize --blocks spo2,pulse,motion --tag oxy1 --cohort <c> --dir <dir> --cache <cache> --workers 4
# trees and control on identical subject-grouped folds
node tools/stage-lab.mjs bench --cache <cache> --cohorts <ten> --features oxy1 --group-subjects --model gbdt --fold <f>
python seqnet.py --cache <cache> --cohorts <ten> --features oxy1 --group-subjects --fold <f> --save-proba <dir>
# literature stack (label-fitted, signed, distilled, bridge) and the residual search
python litstack.py --cache <cache> --cohorts <ten> --features oxy1 --group-subjects --fold <f> [--signed] [--distill <dir>] [--bridge]
python symreg.py --cache <cache> --control <dir> --stack <dir> --out <dir>

• Every job runs under an 8 GB cgroup cap and logs its full-size peak memory; trainers read one night at a time.
• Fold assignment: 32-bit hash of the subject key modulo 5; deterministic, so added cohorts never move existing nights.
• Human ceiling: ISRUC scorer 1 vs scorer 2, κ 0.829, computed by stage-lab.mjs on the same epochs.
• Verdicts are emitted as tepna.verdict/1 with the pre-stated criterion embedded.

6. Sample size & statistical power

With 9 100 nights (≈ 9 million epochs) the night-bootstrap 95 % CI on pooled κ is ±0.004 (measured: 0.608 [0.604, 0.612] for the control, brief §17), so any difference ≥ 0.01 between learners on the same folds is resolvable; the pre-stated reading rules use 0.01 as the smallest difference worth a claim. The blind cohorts (994 and 127 nights) resolve ≈ ±0.015 and ±0.04 respectively.

References

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Supplement S1 (beat-detector validation on MIT-BIH, SLPDB, NSRDB) is gated by the P5 ruling and is not reproduced here. Every figure in this draft is a pooled aggregate over the cohorts in Table 1; no per-record value and no record identifier appears here, and the external blind run has not been carried out.