Entropy of difference works similarly to permutation entropy for the assessment of anesthesia and sleep EEG despite the lower computational effort

Abstract

EEG monitoring during anesthesia or for diagnosing sleep disorders is a common standard. Different approaches for measuring the important information of this biosignal are used. The most often and efficient one for entropic parameters is permutation entropy as it can distinguish the vigilance states in the different settings. Due to high calculation times, it has mostly been used for low orders, although it shows good results even for higher orders. Entropy of difference has a similar way of extracting information from the EEG as permutation entropy. Both parameters and different algorithms for encoding the associated patterns in the signal are described. The runtimes of both entropic measures are compared, not only for the needed encoding but also for calculating the value itself. The mutual information that both parameters extract is measured with the AUC for a linear discriminant analysis classifier. Entropy of difference shows a smaller calculation time than permutation entropy. The reduction is much larger for higher orders, some of them can even only be computed with the entropy of difference. The distinguishing of the vigilance states between both measures is similar as the AUC values for the classification do not differ significantly. As the runtimes for the entropy of difference are smaller than for the permutation entropy, even though the performance stays the same, we state the entropy of difference could be a useful method for analyzing EEG data. Higher orders of entropic features may also be investigated better and more easily.

Keywords: Anesthesia; Electroencephalogram; Entropy of difference; Monitoring; Permutation entropy.

Fig. 1

The PeEn and the EoD break the EEG time series down in specific patterns to analyze the signal further in the time domain. In the middle you can see an example time series encoded once for the PeEn on the left side and once for the EoD on the right side. The numbers in the left box present the rank order of amplitudes, with the highest number being the highest amplitude in the pattern, that are used for PeEn calculation. The signs in the right box represent the sign pattern reflecting the difference in amplitude (higher of lower) of the next amplitude value of the pattern. These sign patterns are used for EoD calculation

Fig. 2

Number of possible patterns for the PeEn and the EoD for orders up to $m=12$

Algorithm 1

Plain Algorithm for Difference Patterns

Algorithm 2

Iterative Algorithm for Difference Patterns

Fig. 3

Runtimes for encoding the difference or ordinal patterns of a white noise signal of length $14.4\times {10}^6$. The figure shows the mean of 20 runs. Ties were not masked, i.e. a tie $x_i=x_j$ with $i<j$ was treated as if $x_i<x_j$

Fig. 4

Sum of the runtimes for pattern encoding, counting the number of occurring patterns, and calculating the entropy values of 30 s windows over all 105 considered EEG recordings from the CAP Sleep Database. As before ties in the data were treated as ascending data points. The points connected by the blue and red lines indicate the total time not spent for encoding

Fig. 5

EoD and PeEn values on 30 s windows of single channel EEG for orders $m=3$ and $m=7$ respectively for all patients. The value ranges are similar for EoD and PeEn when comparing sleep phases, for higher orders the gap between the two entropic parameters gets bigger

Fig. 6

EoD and PeEn values of order $m=7$ for the patient labeled "n2" (control group, no pathology). The correlation between EoD and PeEn values of this patient is 99.53%. A similar high correlation was found across patients across orders with a light decrease for higher orders

Fig. 7

EoD and PeEn values on 10 s windows of single channel EEG for orders $m=3$ and $m=7$ respectively for all EEG recordings. The value ranges are similar for EoD and PeEn when comparing anesthesia levels, for higher orders the gap between the two entropic parameters gets bigger. Burst Supp. indicates the burst suppression level