The Extremum Stack is a Minimal Sufficient Statistic for Rate-Independent Functionals: A Kolmogorov Complexity Characterisation

TL;DR

This paper characterizes the extremum stack as a minimal sufficient statistic for rate-independent functionals using Kolmogorov complexity, establishing optimal compression bounds for hysteresis-driven streams.

Contribution

It proves the extremum stack's minimal sufficiency for all computable, causal, rate-independent functionals and introduces a Kolmogorov optimal compression method.

Findings

The extremum stack is a minimal sufficient statistic with Kolmogorov complexity bounds.

Any lossless compression preserving the class R requires at least K(Pi_n) - O(1) bits.

The proposed stack-based compression algorithm is Kolmogorov optimal.

Abstract

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) - O(1) <= K_R(u_{0:n}) <= K(Pi_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in the class R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Pi_n) - O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee that none of the standard time-series compression methods provide.