Asymptotic Behavior of Approximate Entropy

Abstract

In this chapter, a new concept of approximate entropy is modified and applied to the problem of testing the randomness of a string of binary bits. This concept has been introduced in a series of papers by Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is evaluated through incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomess based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now, this distribution remained unknown and was thought to be difficult to obtain. The key step leading to the limiting distribution of approximate entropy is a modification of its definition based on the frequencies of different patterns in the augmented or circular version of the original sequence. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ ²-random variable.