Thin and Thick Instability in Physiological Systems: Why Local Instability Measures Are Not Risk Measures

This is the authors' abstract. We don't add key points for this paper.

This paper argues that local instability measures—Lyapunov exponents, entropies, fractal exponents, variability indices, and indicators of critical slowing down—describe how a signal varies, not how likely it is to reach a clinically defined failure state. The author models a monitored patient as a partially observed, controlled, nonstationary stochastic system with an endpoint-specific failure region, and defines reach as the probability of entering that region within a horizon. The central theorem shows that every measure invariant under translation of the signal, a class that includes Lyapunov exponents, sample and permutation entropy, detrended fluctuation exponents, and autocorrelation, takes the same value on systems whose reach covers the entire interval from zero to one. The same holds for order-invariant measures of oxygen saturation, which cannot distinguish operating points on the dissociation curve. In populations, the share of reach variance such measures can explain has an explicit ceiling, and their calibration changes whenever the relation between complexity and location changes. Four further results show how local measures mislead: positive exponents can coexist with zero reach and negative exponents with certain reach; divergence estimators return positive exponents for stable noisy systems, with more data making the bias larger; escape depends on barrier height relative to noise rather than local curvature, and noise-driven escape gives no local warning; slowing indicators and reach can move independently; and support changes local measures without changing the intrinsic system. What does govern reach is identified in three settings: a thinness ratio and scaled horizon in the canonical stochastic model, the exponent minus the entropy in open chaotic maps, and a drift-fluctuation bound in general. The paper assembles a minimal valid diagnostic with an estimator, a minimum window, and identification conditions for five routes to failure, and proposes a reporting standard. Synthetic simulations check each result; no clinical data are used.