Randomness and degrees of irregularity.

TLDR

The paper tackles the fundamental question of whether sequential data are random, a problem that arises in many settings with limited data length and motivates a computable framework for quantifying regularity. The authors aim to delineate how close a sequence is to randomness in order to assess, for example, the effectiveness of medical therapies. They employ approximate entropy (ApEn) to measure maximal randomness for short sequences (N = 5) and extend it to infinite sequences, linking independence and normality to limit theorems and using ApEn to quantify rates of convergence.

Abstract

The fundamental question "Are sequential data random?" arises in myriad contexts, often with severe data length constraints. Furthermore, there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness--e.g., to evaluate the efficacy of therapy in medicine. We address both these issues from a computable framework via a quantification of regularity. ApEn (approximate entropy), defining maximal randomness for sequences of arbitrary length, indicating the applicability to sequences as short as N = 5 points. An infinite sequence formulation of randomness is introduced that retains the operational (and computable) features of the finite case. In the infinite sequence setting, we indicate how the "foundational" definition of independence in probability theory, and the definition of normality in number theory, reduce to limit theorems without rates of convergence, from which we utilize ApEn to address rates of convergence (of a deficit from maximal randomness), refining the aforementioned concepts in a computationally essential manner. Representative applications among many are indicated to assess (i) random number generation output; (ii) well-shuffled arrangements; and (iii) (the quality of) bootstrap replicates.