Healthy systems exhibit complex dynamics on the changing of information embedded in physiologic signals on multiple time scales that can be quantified by employing multiscale entropy (MSE) analysis. Here, we propose a measure of complexity, called entropy of entropy (EoE) analysis. The analysis combines the features of MSE and an alternate measure of information, called superinformation, useful for DNA sequences. In this work, we apply the hybrid analysis to the cardiac interbeat interval time series. We find that the EoE value is significantly higher for the healthy than the pathologic groups. Particularly, short time series of 70 heart beats is sufficient for EoE analysis with an accuracy of 81% and longer series of 500 beats results in an accuracy of 90%. In addition, the EoE versus Shannon entropy plot of heart rate time series exhibits an inverted U relationship with the maximal EoE value appearing in the middle of extreme order and disorder.

Mode mixing is a limitation of the empirical mode decomposition (EMD) method appropriate for physiological signal analysis. In 2008, Boundary Condition Map presented by Rilling and Flandrin provided the efficiency of separating the two components of a two-tone signal as a function of their amplitude and frequency ratios. Till 2019, their findings were still applied. However, their maps only give an uncertainty-like efficiency of mode mixing separation for two-tone signals. In this paper, we propose a criterion for mode mixing separation in EMD, which provides a binary judgement on mode mixing separation instead of the above efficiency. By comparing the slopes of the two components, we found that the phenomenon of mode mixing occurs as the extrema of the high-tone component are suppressed by the lowtone component. Under this condition, the criterion shows the relation among their amplitude ratio, frequency ratio, and relative phase between the two components. Given with values of the three parameters, one can affirm whether the two components are mixed according to the criterion. Accordingly, we derive a black/white three-dimensional (3D) map which plots the binary result of mode mixing in black or white as a function of the three parameters. Our map agrees with Rilling’s map and the results obtained from our gait analysis. Among the 23 sets of center-of-mass trajectory signals, six sets encountered the mode mixing problem and their coordinates of the three parameters were found in the black region of the map while the other 17 sets were in the white region.

Multiscale entropy (MSE) was used to analyze electroencephalography (EEG) signals to differentiate patients with Alzheimer’s disease (AD) from healthy subjects. It was found that the MSE values of the EEG signals from the healthy subjects are higher than those of the AD ones at small time scale factors in the MSE algorithm, while lower than those of the AD patients at large time scale factors. Based on the finding, we applied the linear discriminant analysis (LDA) to optimize the differentiating performance by comparing the resulting weighted sum of the MSE values under some specific time scales of each subject. The EEG data from 15 healthy subjects, 69 patients with mild AD, and 15 patients with moderate to severe AD were recorded. As a result, the weighted sum values are significantly higher for the healthy than the patients with moderate to severe AD groups. The optimal testing accuracy under five specific scales is 100% based on the EEG signals acquired from the T4 electrode. The resulting weighted sum value for the mild AD group is in the middle of those for the healthy and the moderate to severe AD groups. Therefore, the MSE-based weighted sum value can potentially be an index of severity of Alzheimer’s disease.

Gait stability has been measured by using many entropy-based methods. However, the relation between the entropy values and gait stability is worth further investigation. A research reported that average entropy (AE), a measure of disorder, could measure the static standing postural stability better than multiscale entropy and entropy of entropy (EoE), two measures of complexity. This study tested the validity of AE in gait stability measurement from the viewpoint of the disorder. For comparison, another five disorders, the EoE, and two traditional metrics methods were, respectively, used to measure the degrees of disorder and complexity of 10 step interval (SPI) and 79 stride interval (SI) time series, individually. As a result, every one of the 10 participants exhibited a relatively high AE value of the SPI when walking with eyes closed and a relatively low AE value when walking with eyes open. Most of the AE values of the SI of the 53 diseased subjects were greater than those of the 26 healthy subjects. A maximal overall accuracy of AE in differentiating the healthy from the diseased was 91.1%. Similar features also exists on those 5 disorder measurements but do not exist on the EoE values. Nevertheless, the EoE versus AE plot of the SI also exhibits an inverted U relation, consistent with the hypothesis for physiologic signals.

Multiscale entropy (MSE) is widely used to analyze heartbeat signals. Even though cardiologists do not use MSE to diagnose heart failure at present, these studies are of importance and have potential clinical applications. In previous studies, MSE discrimination between old congestive heart failure (CHF) and healthy individuals has remained controversial. Few studies have been published on the discrimination between them, using only MSE with machine learning for automatic multidimensional analysis, with reported testing accuracies of less than 86%. In this study, we determined the optimal MSE scales for discrimination by using a low-dimensional exhaustive search along with three classifiers—linear discriminant analysis (LDA), support vector machine (SVM), and k-nearest neighbor (KNN). In younger people (<55 years), the results showed an accuracy of up to 95.5% with two optimal MSE scales (2D) and up to 97.7% with four optimal MSE scales (4D) in discriminating between young CHF and healthy participants. In older people (≥55 years), the discrimination accuracy reached 90.1% using LDA in 2D, SVM in 3D (three optimal MSE scales), and KNN in 5D (five optimal MSE scales). LDA with a 3D exhaustive search also achieved 94.4% accuracy in older people. Therefore, the results indicate that MSE analysis can differentiate between CHF and healthy individuals of any age.