Consistency in approximate entropy given by a volumetric estimate

Non-linear methods for estimating variability in time-series are currently of widespread use. Among such methods are approximate entropy (ApEn) and sample approximate entropy (SampEn). The applicability of ApEn and SampEn in analyzing data is evident and their use is increasing. However, consistency is a point of concern in these tools, i.e., the classification of the temporal organization of a data set might indicate a relative less ordered series in relation to another when the opposite is true. As highlighted by their proponents themselves, ApEn and SampEn might present incorrect results due to this lack of consistency. In this study, we present a method which gains consistency by using ApEn repeatedly in a wide range of combinations of window lengths and matching error tolerance. The tool is called volumetric approximate entropy, vApEn. We analyze nine artificially generated prototypical time-series with different degrees of temporal order (combinations of sine waves, logistic maps with different control parameter values, random noises). While ApEn/SampEn clearly fail to consistently identify the temporal order of the sequences, vApEn correctly do. In order to validate the tool we performed shuffled and surrogate data analysis. Statistical analysis confirmed the consistency of the method. (C) 2008 Elsevier Ltd. All rights reserved.

Prediction of hypertensive crisis based on average, variability and approximate entropy of 24-h ambulatory blood pressure monitoring

Approximate entropy (ApEn) of blood pressure (BP) can be easily measured based on software analysing 24-h ambulatory BP monitoring (ABPM), but the clinical value of this measure is unknown. In a prospective study we investigated whether ApEn of BP predicts, in addition to average and variability of BP, the risk of hypertensive crisis. In 57 patients with known hypertension we measured ApEn, average and variability of systolic and diastolic BP based on 24-h ABPM. Eight of these fifty-seven patients developed hypertensive crisis during follow-up (mean follow-up duration 726 days). In bivariate regression analysis, ApEn of systolic BP (P<0.01), average of systolic BP (P=0.02) and average of diastolic BP (P=0.03) were significant predictors of hypertensive crisis. The incidence rate ratio of hypertensive crisis was 14.0 (95% confidence interval (CI) 1.8, 631.5; P<0.01) for high ApEn of systolic BP as compared to low values. In multivariable regression analysis, ApEn of systolic (P=0.01) and average of diastolic BP (P<0.01) were independent predictors of hypertensive crisis. A combination of these two measures had a positive predictive value of 75%, and a negative predictive value of 91%, respectively. ApEn, combined with other measures of 24-h ABPM, is a potentially powerful predictor of hypertensive crisis. If confirmed in independent samples, these findings have major clinical implications since measures predicting the risk of hypertensive crisis define patients requiring intensive follow-up and intensified therapy.

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

Evaluation of physiologic complexity in time series using generalized sample entropy and surrogate data analysis

Complexity in time series is an intriguing feature of living dynamical systems, with potential use for identification of system state. Although various methods have been proposed for measuring physiologic complexity, uncorrelated time series are often assigned high values of complexity, errouneously classifying them as a complex physiological signals. Here, we propose and discuss a method for complex system analysis based on generalized statistical formalism and surrogate time series. Sample entropy (SampEn) was rewritten inspired in Tsallis generalized entropy, as function of q parameter (qSampEn). qSDiff curves were calculated, which consist of differences between original and surrogate series qSampEn. We evaluated qSDiff for 125 real heart rate variability (HRV) dynamics, divided into groups of 70 healthy, 44 congestive heart failure (CHF), and 11 atrial fibrillation (AF) subjects, and for simulated series of stochastic and chaotic process. The evaluations showed that, for nonperiodic signals, qSDiff curves have a maximum point (qSDiff(max)) for q not equal 1. Values of q where the maximum point occurs and where qSDiff is zero were also evaluated. Only qSDiff(max) values were capable of distinguish HRV groups (p-values 5.10 x 10(-3); 1.11 x 10(-7), and 5.50 x 10(-7) for healthy vs. CHF, healthy vs. AF, and CHF vs. AF, respectively), consistently with the concept of physiologic complexity, and suggests a potential use for chaotic system analysis. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4758815]

A new entropy measure based on the Renyi entropy rate using Gaussian kernels

The concept of entropy rate is well defined in dynamical systems theory but is impossible to apply it directly to finite real world data sets. With this in mind, Pincus developed Approximate Entropy (ApEn), which uses ideas from Eckmann and Ruelle to create a regularity measure based on entropy rate that can be used to determine the influence of chaotic behaviour in a real world signal. However, this measure was found not to be robust and so an improved formulation known as the Sample Entropy (SampEn) was created by Richman and Moorman to address these issues. We have developed a new, related, regularity measure which is not based on the theory provided by Eckmann and Ruelle and proves a more well-behaved measure of complexity than the previous measures whilst still retaining a low computational cost.

Entropy feature extraction of EEG signals for automatic person identification

The use of human brain electroencephalography (EEG) signals for automatic person identi cation has been investigated for a decade. It has been found that the performance of an EEG-based person identication system highly depends on what feature to be extracted from multi-channel EEG signals. Linear methods such as Power Spectral Density and Autoregressive Model have been used to extract EEG features. However these methods assumed that EEG signals are stationary. In fact, EEG signals are complex, non-linear, non-stationary, and random in nature. In addition, other factors such as brain condition or human characteristics may have impacts on the performance, however these factors have not been investigated and evaluated in previous studies. It has been found in the literature that entropy is used to measure the randomness of non-linear time series data. Entropy is also used to measure the level of chaos of braincomputer interface systems. Therefore, this thesis proposes to study the role of entropy in non-linear analysis of EEG signals to discover new features for EEG-based person identi- cation. Five dierent entropy methods including Shannon Entropy, Approximate Entropy, Sample Entropy, Spectral Entropy, and Conditional Entropy have been proposed to extract entropy features that are used to evaluate the performance of EEG-based person identication systems and the impacts of epilepsy, alcohol, age and gender characteristics on these systems. Experiments were performed on the Australian EEG and Alcoholism datasets. Experimental results have shown that, in most cases, the proposed entropy features yield very fast person identication, yet with compatible accuracy because the feature dimension is low. In real life security operation, timely response is critical. The experimental results have also shown that epilepsy, alcohol, age and gender characteristics have impacts on the EEG-based person identication systems.

Variability, regularity and coupling measures distinguish PD tremor from voluntary 5Hz tremor

A characteristic of Parkinson's disease (PD) is the development of tremor within the 4–6 Hz range. One method used to better understand pathological tremor is to compare the responses to tremor-type actions generated intentionally in healthy adults. This study was designed to investigate the similarities and differences between voluntarily generated 4–6 Hz tremor and PD tremor in regards to their amplitude, frequency and coupling characteristics. Tremor responses for 8 PD individuals (on- and off-medication) and 12 healthy adults were assessed under postural and resting conditions. Results showed that the voluntary and PD tremor were essentially identical with regards to the amplitude and peak frequency. However, differences between the groups were found for the variability (SD of peak frequency, proportional power) and regularity (Approximate Entropy, ApEn) of the tremor signal. Additionally, coherence analysis revealed strong inter-limb coupling during voluntary conditions while no bilateral coupling was seen for the PD persons. Overall, healthy participants were able to produce a 5 Hz tremulous motion indistinguishable to that of PD patients in terms of peak frequency and amplitude. However, differences in the structure of variability and level of inter-limb coupling were found for the tremor responses of the PD and healthy adults. These differences were preserved irrespective of the medication state of the PD persons. The results illustrate the importance of assessing the pattern of signal structure/variability to discriminate between different tremor forms, especially where no differences emerge in standard measures of mean amplitude as traditionally defined.

Interpersonal coordination tendencies shape 1-vs-1 sub-phase performance outcomes in youth soccer

This study investigated the influence of interpersonal coordination tendencies on performance outcomes of 1-vs-1 subphases in youth soccer. Eight male developing soccer players (age: 11.8+0.4 years; training experience: 3.6+1.1 years) performed an in situ simulation of a 1-vs-1 sub-phase of soccer. Data from 82 trials were obtained with motion-analysis techniques, and relative phase used to measure the space-time coordination tendencies of attacker-defender dyads. Approximate entropy (ApEn) was then used to quantify the unpredictability of interpersonal interactions over trials. Results revealed how different modes of interpersonal coordination emerging from attacker-defender dyads influenced the 1-vs-1 performance outcomes. High levels of space-time synchronisation (47%) and unpredictability in interpersonal coordination processes (ApEn: 0.91+0.34) were identified as key features of an attacking player’s success. A lead-lag relation attributed to a defending player (34% around 7308 values) and a more predictable coordination mode (ApEn: 0.65+0.27, P50.001), demonstrated the coordination tendencies underlying the success of defending players in 1-vs-1 sub-phases. These findings revealed how the mutual influence of each player on the behaviour of dyadic systems shaped emergent performance outcomes. More specifically, the findings showed that attacking players should be constrained to exploit the space-time synchrony with defenders in an unpredictable and creative way, while defenders should be encouraged to adopt postures and behaviours that actively constrain the attacker’s actions.

Capturing complex, non-linear team behaviours during competitive football performance

This study investigated changes in the complexity (magnitude and structure of variability) of the collective behaviours of association football teams during competitive performance. Raw positional data from an entire competitive match between two professional teams were obtained with the ProZone® tracking system. Five compound positional variables were used to investigate the collective patterns of performance of each team including: surface area, stretch index, team length, team width, and geometrical centre. Analyses involve the coefficient of variation (%CV) and approximate entropy (ApEn), as well as the linear association between both parameters. Collective measures successfully captured the idiosyncratic behaviours of each team and their variations across the six time periods of the match. Key events such as goals scored and game breaks (such as half time and full time) seemed to influence the collective patterns of performance. While ApEn values significantly decreased during each half, the %CV increased. Teams seem to become more regular and predictable, but with increased magnitudes of variation in their organisational shape over the natural course of a match.

Nonlinear measures of respiration: Respiratory irregularity and increased chaos of respiration in patients with panic disorder

Background. Respiratory irregularity has been previously reported in patients with panic disorder using time domain measures. However, the respiratory signal is not entirely linear and a few previous studies used approximate entropy (APEN), a measure of regularity of time series. We have been studying APEN and other nonlinear measures including a measure of chaos, the largest Lyapunov exponent (LLE) of heart rate time series, in some detail. In this study, we used these measures of respiration to compare normal controls (n = 18) and patients with panic disorder (n = 22) in addition to the traditional time domain measures of respiratory rate and tidal volume. Methods: Respiratory signal was obtained by the Respitrace system using a thoracic and an abdominal belt, which was digitized at 500 Hz. Later, the time series were constructed at 4 Hz, as the highest frequency in this signal is limited to 0.5 Hz. We used 256 s of data (1,024 points) during supine and standing postures under normal breathing and controlled breathing at 12 breaths/min. Results: APEN was significantly higher in patients in standing posture during normal as well as controlled breathing (p = 0.002 and 0.02, respectively). LLE was also significantly higher in standing posture during normal breathing (p = 0.009). Similarly, the time domain measures of standard deviations and the coefficient of variation (COV) of tidal volume (TV) were significantly higher in the patient group (p = 0.02 and 0.004, respectively). The frequency of sighs was also higher in the patient group in standing posture (p = 0.02). In standing posture, LLE (p < 0.05) as well as APEN (p < 0.01) contributed significantly toward the separation of the two groups over and beyond the linear measure, i.e. the COV of TV. Conclusion: These findings support the previously described respiratory irregularity in patients with panic disorder and also illustrate the utility of nonlinear measures such as APEN and LLE as additional measures toward a better understanding of the abnormalities of respiratory physiology in similar patient populations as the correlation between LLE, APEN and some of the time domain measures only explained up to 50-60% of the variation. Copyright (C) 2002 S. Karger AG, Basel.

Computerized identification of motor complications in Parkinson's disease

Objective: To investigate whether spirography-based objective measures are able to effectively characterize the severity of unwanted symptom states (Off and dyskinesia) and discriminate them from motor state of healthy elderly subjects. Background: Sixty-five patients with advanced Parkinson’s disease (PD) and 10 healthy elderly (HE) subjects performed repeated assessments of spirography, using a touch screen telemetry device in their home environments. On inclusion, the patients were either treated with levodopa-carbidopa intestinal gel or were candidates for switching to this treatment. On each test occasion, the subjects were asked trace a pre-drawn Archimedes spiral shown on the screen, using an ergonomic pen stylus. The test was repeated three times and was performed using dominant hand. A clinician used a web interface which animated the spiral drawings, allowing him to observe different kinematic features, like accelerations and spatial changes, during the drawing process and to rate different motor impairments. Initially, the motor impairments of drawing speed, irregularity and hesitation were rated on a 0 (normal) to 4 (extremely severe) scales followed by marking the momentary motor state of the patient into 2 categories that is Off and Dyskinesia. A sample of spirals drawn by HE subjects was randomly selected and used in subsequent analysis. Methods: The raw spiral data, consisting of stylus position and timestamp, were processed using time series analysis techniques like discrete wavelet transform, approximate entropy and dynamic time warping in order to extract 13 quantitative measures for representing meaningful motor impairment information. A principal component analysis (PCA) was used to reduce the dimensions of the quantitative measures into 4 principal components (PC). In order to classify the motor states into 3 categories that is Off, HE and dyskinesia, a logistic regression model was used as a classifier to map the 4 PCs to the corresponding clinically assigned motor state categories. A stratified 10-fold cross-validation (also known as rotation estimation) was applied to assess the generalization ability of the logistic regression classifier to future independent data sets. To investigate mean differences of the 4 PCs across the three categories, a one-way ANOVA test followed by Tukey multiple comparisons was used. Results: The agreements between computed and clinician ratings were very good with a weighted area under the receiver operating characteristic curve (AUC) coefficient of 0.91. The mean PC scores were different across the three motor state categories, only at different levels. The first 2 PCs were good at discriminating between the motor states whereas the PC3 was good at discriminating between HE subjects and PD patients. The mean scores of PC4 showed a trend across the three states but without significant differences. The Spearman’s rank correlations between the first 2 PCs and clinically assessed motor impairments were as follows: drawing speed (PC1, 0.34; PC2, 0.83), irregularity (PC1, 0.17; PC2, 0.17), and hesitation (PC1, 0.27; PC2, 0.77). Conclusions: These findings suggest that spirography-based objective measures are valid measures of spatial- and time-dependent deficits and can be used to distinguish drug-related motor dysfunctions between Off and dyskinesia in PD. These measures can be potentially useful during clinical evaluation of individualized drug-related complications such as over- and under-medications thus maximizing the amount of time the patients spend in the On state.

Digital spiral analysis for objective assessment of fine motor timing variability in Parkinson's disease

OBJECTIVES: To develop a method for objective assessment of fine motor timing variability in Parkinson’s disease (PD) patients, using digital spiral data gathered by a touch screen device. BACKGROUND: A retrospective analysis was conducted on data from 105 subjects including65 patients with advanced PD (group A), 15 intermediate patients experiencing motor fluctuations (group I), 15 early stage patients (group S), and 10 healthy elderly subjects (HE) were examined. The subjects were asked to perform repeated upper limb motor tasks by tracing a pre-drawn Archimedes spiral as shown on the screen of the device. The spiral tracing test was performed using an ergonomic pen stylus, using dominant hand. The test was repeated three times per test occasion and the subjects were instructed to complete it within 10 seconds. Digital spiral data including stylus position (x-ycoordinates) and timestamps (milliseconds) were collected and used in subsequent analysis. The total number of observations with the test battery were as follows: Swedish group (n=10079), Italian I group (n=822), Italian S group (n = 811), and HE (n=299). METHODS: The raw spiral data were processed with three data processing methods. To quantify motor timing variability during spiral drawing tasks Approximate Entropy (APEN) method was applied on digitized spiral data. APEN is designed to capture the amount of irregularity or complexity in time series. APEN requires determination of two parameters, namely, the window size and similarity measure. In our work and after experimentation, window size was set to 4 and similarity measure to 0.2 (20% of the standard deviation of the time series). The final score obtained by APEN was normalized by total drawing completion time and used in subsequent analysis. The score generated by this method is hence on denoted APEN. In addition, two more methods were applied on digital spiral data and their scores were used in subsequent analysis. The first method was based on Digital Wavelet Transform and Principal Component Analysis and generated a score representing spiral drawing impairment. The score generated by this method is hence on denoted WAV. The second method was based on standard deviation of frequency filtered drawing velocity. The score generated by this method is hence on denoted SDDV. Linear mixed-effects (LME) models were used to evaluate mean differences of the spiral scores of the three methods across the four subject groups. Test-retest reliability of the three scores was assessed after taking mean of the three possible correlations (Spearman’s rank coefficients) between the three test trials. Internal consistency of the methods was assessed by calculating correlations between their scores. RESULTS: When comparing mean spiral scores between the four subject groups, the APEN scores were different between HE subjects and three patient groups (P=0.626 for S group with 9.9% mean value difference, P=0.089 for I group with 30.2%, and P=0.0019 for A group with 44.1%). However, there were no significant differences in mean scores of the other two methods, except for the WAV between the HE and A groups (P<0.001). WAV and SDDV were highly and significantly correlated to each other with a coefficient of 0.69. However, APEN was not correlated to neither WAV nor SDDV with coefficients of 0.11 and 0.12, respectively. Test-retest reliability coefficients of the three scores were as follows: APEN (0.9), WAV(0.83) and SD-DV (0.55). CONCLUSIONS: The results show that the digital spiral analysis-based objective APEN measure is able to significantly differentiate the healthy subjects from patients at advanced level. In contrast to the other two methods (WAV and SDDV) that are designed to quantify dyskinesias (over-medications), this method can be useful for characterizing Off symptoms in PD. The APEN was not correlated to none of the other two methods indicating that it measures a different construct of upper limb motor function in PD patients than WAV and SDDV. The APEN also had a better test-retest reliability indicating that it is more stable and consistent over time than WAV and SDDV.

Sensitivity, specificity and predictive values of linear and nonlinear indices of heart rate variability in stable angina patients

Background: Decreased heart rate variability (HRV) is related to higher morbidity and mortality. In this study we evaluated the linear and nonlinear indices of the HRV in stable angina patients submitted to coronary angiography. Methods. We studied 77 unselected patients for elective coronary angiography, which were divided into two groups: coronary artery disease (CAD) and non-CAD groups. For analysis of HRV indices, HRV was recorded beat by beat with the volunteers in the supine position for 40 minutes. We analyzed the linear indices in the time (SDNN [standard deviation of normal to normal], NN50 [total number of adjacent RR intervals with a difference of duration greater than 50ms] and RMSSD [root-mean square of differences]) and frequency domains ultra-low frequency (ULF) ≤ 0,003 Hz, very low frequency (VLF) 0,003 - 0,04 Hz, low frequency (LF) (0.04-0.15 Hz), and high frequency (HF) (0.15-0.40 Hz) as well as the ratio between LF and HF components (LF/HF). In relation to the nonlinear indices we evaluated SD1, SD2, SD1/SD2, approximate entropy (-ApEn), α1, α2, Lyapunov Exponent, Hurst Exponent, autocorrelation and dimension correlation. The definition of the cutoff point of the variables for predictive tests was obtained by the Receiver Operating Characteristic curve (ROC). The area under the ROC curve was calculated by the extended trapezoidal rule, assuming as relevant areas under the curve ≥ 0.650. Results: Coronary arterial disease patients presented reduced values of SDNN, RMSSD, NN50, HF, SD1, SD2 and -ApEn. HF ≤ 66 ms§ssup§2§esup§, RMSSD ≤ 23.9 ms, ApEn ≤-0.296 and NN50 ≤ 16 presented the best discriminatory power for the presence of significant coronary obstruction. Conclusion: We suggest the use of Heart Rate Variability Analysis in linear and nonlinear domains, for prognostic purposes in patients with stable angina pectoris, in view of their overall impairment. © 2012 Pivatelli et al.; licensee BioMed Central Ltd.

Contribuições da conectância de rede e complexidade da dinâmica do sistema de trocas gasosas para a estabilidade na utilização de luz por espécies florestais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)