Published October 27, 2026
| Version v1
Preprint
Open
Sampling Rate as an Influence Quantity: Invariance Orders and a Definition-Level Audit of Physiological Feature Sets
Description
In the vocabulary of metrology, a feature computed from a sampled signal is a measurement result, and the
sampling rate is an influence quantity — a quantity that is not the measurand but affects the measured value.
The literature treats hand-crafted features as if they were free of this influence and compares them across
devices, datasets and rates on that assumption. We show that the assumption fails in a way that is predictable,
quantifiable, and — for one large family of features — impossible to repair by sampling faster. We introduce
the invariance order 𝑝 of a feature, defined by 𝐹(𝑥𝑟) = 𝐹(𝑥∞)(1 + 𝑂(𝑟 −𝑝 )) with 𝑟 = 𝑓𝑠/𝐵 the ratio of
sampling rate to content bandwidth, and prove four propositions that assign 𝑝 from the algebraic form of a
definition alone: sample-indexed quantities have 𝑝 = 1 and a relative bias that does not depend on 𝑟; finite
difference and crossing-count functionals have 𝑝 = 2; bin-normalised spectral entropy decays only as 1/log𝑟;
moment-based features have 𝑝 = ∞. A fifth proposition links the unobservable bias to the observable rank
statistic through AUC = Φ(Δ/√2𝑠𝑎 2 + 𝑠𝑒 2), which explains why the AUC exponent and the bias exponent
diverge once the bias saturates. In the terms of the GUM the influence produces a systematic effect that belongs
in the uncertainty budget but is almost never entered there.
The bias law is verified on synthetic signals with known ground truth (fitted slopes −2.27, −2.17, −2.02;
𝑅 2 ≥ 0.998) and on five signal families — 500 Hz ECG, 360 Hz ECG, 256 Hz photoplethysmography, 333 Hz
radar displacement and 200 kHz acoustic emission from additive manufacturing — where the measured
exponents are 2.03/1.86/1.95, 2.29/2.06/1.94, 2.41/2.37/1.96, and 1.03/1.11/0.93 for the sample-indexed
controls. Fitted on 𝑟 ≤ 25 and extrapolated to 𝑟 = 100, the law predicts held-out AUCs to within 0.002. The
link is verified on 211 cells across three datasets with 𝑅 2 = 0.9988, 0.9988 and 0.9987 and no free parameters.
Auditing sixteen feature definitions drawn from the literature, twelve leak at 𝑟 = 2.5 with |AUC − 0.5| > 0.1.
The effect sizes separate by two orders of magnitude between orders: 𝑑 = 15–7035 for sample-indexed
features against 𝑑 = 0.45–1.32 for difference-based ones. A paired design across seven conditions (four ECG
leads, two window lengths, two modalities) reproduces the taxonomy in 164 of 168 cells at FDR < 0.05. We
give a design rule (r at least 10, i.e. 440–520 Hz for the 40 Hz diagnostic band), a static procedure that assigns
an order before any experiment is run (14 of 16 definitions within 0.1), and evidence that p-values alone would
misclassify a zero-effect feature as leaking.
The contribution is a controlled, statically executable diagnostic, not a simulation of any particular instrument.
Analog effects are excluded by design and are additive, so every number reported here is a strict lower bound
on the device-level discrepancy: the audit ranks definitions by their structural vulnerability to the sampling grid,
a property of the formula that holds whatever hardware is attached.
Files
manuscript.pdf
Additional details
Software
- Repository URL
- https://github.com/Xbotsd/sampling-rate-audit
- Programming language
- Python