Automatic Selection of the Threshold Value $r$ for Approximate Entropy

Calculation of approximate entropy (ApEn) requires
a

priori
determination of two unknown parameters,
m
and
r
. While the recommended values of
r
, in the range of 0.1-0.2 times the standard deviation of the signal, have been shown to be applicable for a wide variety of signals, in certain cases,
r
values within this prescribed range can lead to an incorrect assessment of the complexity of a given signal. To circumvent this limitation, we recently advocated finding the maximum ApEn value by assessing all values of
r
from 0 to 1, and found that maximum ApEn does not always occur within the prescribed range of
r
values. Our results indicate that finding the maximum ApEn leads to the correct interpretation of a signal's complexity. One major limitation, however, is that the calculation of all choices of
r
values is often impractical due to the computational burden. Our new method, based on a heuristic stochastic model, overcomes this computational burden, and leads to the automatic selection of the maximum ApEn value for any given signal. Based on Monte Carlo simulations, we derive general equations that can be used to estimate the maximum ApEn with high accuracy for a given value of
m
. Application to both synthetic and experimental data confirmed the advantages claimed with the proposed approach.