Entropy and compression: two measures of complexity

Abstract

Rationale, aims and objectives: Traditional complexity measures are used to capture the amount of structured information present in a certain phenomenon. Several approaches developed to facilitate the characterization of complexity have been described in the related literature. Fetal heart rate (FHR) monitoring has been used and improved during the last decades. The importance of these studies lies on an attempt to predict the fetus outcome, but complexity measures are not yet established in clinical practice. In this study, we have focused on two conceptually different measures: Shannon entropy, a probabilistic approach, and Kolmogorov complexity, an algorithmic approach. The main aim of the current investigation was to show that approximation to Kolmogorov complexity through different compressors, although applied to a lesser extent, may be as useful as Shannon entropy calculated by approximation through different entropies, which has been successfully applied to different scientific areas.

Methods: To illustrate the applicability of both approaches, two entropy measures, approximate and sample entropy, and two compressors, paq8l and bzip2, were considered. These indices were applied to FHR tracings pertaining to a dataset composed of 48 delivered fetuses with umbilical artery blood (UAB) pH in the normal range (pH ≥ 7.20), 10 delivered mildly acidemic fetuses and 10 moderate-to-severe acidemic fetuses. The complexity indices were computed on the initial and final segments of the last hour of labour, considering 5- and 10-minute segments.

Results: In our sample set, both entropies and compressors were successfully utilized to distinguish fetuses at risk of hypoxia from healthy ones. Fetuses with lower UAB pH presented significantly lower entropy and compression indices, more markedly in the final segments.

Conclusions: The combination of these conceptually different measures appeared to present an improved approach in the characterization of different pathophysiological states, reinforcing the theory that entropies and compressors measure different complexity features. In view of these findings, we recommend a combination of the two approaches.

Keywords: medical informatics.