Determining embedding dimension for phase-space reconstruction using a geometrical construction

Using curvature to select the time lag for delay reconstruction

We propose a curvature-based approach for choosing a good value for the time-delay parameter τ in delay reconstructions. The idea is based on the effects of the delay on the geometry of the reconstructions. If the delay is too small, the reconstructed dynamics are flattened along the main diagonal of the embedding space; too-large delays, on the other hand, can overfold the dynamics. Calculating the curvature of a two-dimensional delay reconstruction is an effective way to identify these extremes and to find a middle ground between them: both the sharp reversals at the extremes of an insufficiently unfolded reconstruction and the bends in an overfolded one create spikes in the curvature. We operationalize this observation by computing the mean Menger curvature of a trajectory segment on 2D reconstructions as a function of time delay. We show that the minimum of these values gives an effective heuristic for choosing the time delay. In addition, we show that this curvature-based heuristic is useful even in cases where the customary approach, which uses average mutual information, fails-e.g., noisy or filtered data.

DISCOVERING CAUSALITIES FROM CARDIOTOCOGRAPHY SIGNALS USING IMPROVED CONVERGENT CROSS MAPPING WITH GAUSSIAN PROCESSES

Convergent cross mapping (CCM) is designed for causal discovery in coupled time series, where Granger causality may not be applicable because of a separability assumption. However, CCM is not robust to observation noise which limits its applicability on signals that are known to be noisy. Moreover, the parameters for state space reconstruction need to be selected using grid search methods. In this paper, we propose a novel improved version of CCM using Gaussian processes for discovery of causality from noisy time series. Specifically, we adopt the concept of CCM and carry out the key steps using Gaussian processes within a non-parametric Bayesian probabilistic framework in a principled manner. The proposed approach is first validated on simulated data, and then used for understanding the interaction between fetal heart rate and uterine activity in the last two hours before delivery and of interest in obstetrics. Our results indicate that uterine activity affects the fetal heart rate, which agrees with recent clinical studies.

Towards the Development of Nonlinear Approaches to Discriminate AF from NSR Using a Single-Lead ECG

Paroxysmal atrial fibrillation (Paro. AF) is challenging to identify at the right moment. This disease is often undiagnosed using currently existing methods. Nonlinear analysis is gaining importance due to its capability to provide more insight into complex heart dynamics. The aim of this study is to use several recently developed nonlinear techniques to discriminate persistent AF (Pers. AF) from normal sinus rhythm (NSR), and more importantly, Paro. AF from NSR, using short-term single-lead electrocardiogram (ECG) signals. Specifically, we adapted and modified the time-delayed embedding method to minimize incorrect embedding parameter selection and further support to reconstruct proper phase plots of NSR and AF heart dynamics, from MIT-BIH databases. We also examine information-based methods, such as multiscale entropy (MSE) and kurtosis (Kt) for the same purposes. Our results demonstrate that embedding parameter time delay (τ), as well as MSE and Kt values can be successfully used to discriminate between Pers. AF and NSR. Moreover, we demonstrate that τ and Kt can successfully discriminate Paro. AF from NSR. Our results suggest that nonlinear time-delayed embedding method and information-based methods provide robust discriminating features to distinguish both Pers. AF and Paro. AF from NSR, thus offering effective treatment before suffering chaotic Pers. AF.

Filtering affects the calculation of the largest Lyapunov exponent

The calculation of the largest Lyapunov exponent (LyE) requires the reconstruction of the time series in an N-dimensional state space. For this, the time delay (Tau) and embedding dimension (EmD) are estimated using the Average Mutual Information and False Nearest Neighbor algorithms. However, the estimation of these variables (LyE, Tau, EmD) could be compromised by prior filtering of the time series evaluated. Therefore, we investigated the effect of filtering kinematic marker data on the calculation of Tau, EmD and LyE using several different computational codes. Kinematic marker data were recorded from 37 subjects during treadmill walking and filtered using a low pass digital filter with a range of cut-off frequencies (23.5-2Hz). Subsequently, the Tau, EmD and LyE were calculated from all cut-off frequencies. Our results demonstrated that the level of filtering affected the outcome of the Tau, EmD and LyE calculations for all computational codes used. However, there was a more consistent outcome for cut-off frequencies above 10 Hz which corresponded to the optimal cut-off frequency that could be used with this data. This suggested that kinematic data should remain unfiltered or filtered conservatively before calculating Tau, EmD and LyE.

Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer

The structure of turbulent dynamics in a stable atmospheric boundary layer was studied by means of a phase-space description. Data from the CASES-99 experiment, decomposed in local modes (with increasing time scale) using empirical mode decomposition, were analyzed in order to extract the proper time lag and the embedding dimension of the phase-space manifold, and subsequently to estimate their scale-dependent correlation dimension. Results show that the dynamics are low-dimensional and anisotropic for a large scale, where the flow is dominated by the bulk motion. Then, they become progressively more high-dimensional while transiting into the inertial sub-range. Finally, they reach three-dimensionality in the range of scales compatible with the center of the inertial sub-range, where the phase-space-filling turbulent fluctuations dominate the dynamics.

Tranquilizer effect on the Lyapunov exponents of lame horses

Tranquilization of horses with acepromazine has been used to suppress erratic head movements and increase the accuracy of a lameness examination. Some equine clinicians believe that tranquilization with acepromazine will make lameness more evident by causing the horse to focus on adjusting its gait to avoid limb pain rather than its surroundings. The aim of this study was to investigate the effect of acepromazine on the Lyapunov exponents of lame horses. Ten lame horses were trotted in a straight line for a minimum of 25 strides. Kinematic data created by head movement were analyzed. Nonlinear analysis methods were applied to lame horse locomotion. The effect of acepromazine on the largest Lyapunov exponents of the lame horses were investigated. There was no statistically significant effect of acepromazine on the maximum value of Lyapunov exponents. The nonlinear dynamic methods can be used to analyze the gait in horses. Local stability of horse gait remains unchanged after the administration of acepromazine.

Temporal evolution of sample entropy in thermal lens system

The photothermal phenomenon resulting in thermal lens formation in liquid media involves complex molecular dynamics responsible for temperature and refractive index variation. As a thermodynamic system, the entropy of the medium also changes. In this paper, the time series and phase portrait analysis of the thermal lens signal is carried out to understand the molecular dynamics. The study reveals the increase in complexity, disorder, and antipersistance nature through fractal dimension, sample entropy, and Hurst exponent, respectively. The analysis of the signal on segmentation reveals the evolution of sample entropy and the stochastic nature of the system with time. The phase portrait analysis also is in support of these observations. Thus, the study suggests that the temporal evolution of sample entropy is similar to the temperature-dependent refractive index.

Application of recurrence quantification analysis for early detection of lean blowout in a swirl-stabilized dump combustor

Lean blowout (LBO) is a serious issue in modern gas turbine engines that operate in a lean (premixed) mode to follow the stringent emission norms. When an engine operates with a lean fuel-air mixture, the flame becomes unstable and is at times carried out of the combustion chamber by the unburnt flow. Thus, the sudden loss of the flame, known as lean blowout, leads to fatal accidents in aircrafts and loss of production in power plants. Therefore, an in-depth analysis of lean blowout is necessary as the phenomenon involves complex interactions between flow dynamics and chemical kinetics. For understanding the complex dynamics of this phenomenon, recurrence analysis can be a very useful method. In the current study, we observe a transition to LBO as the global fuel-air ratio is reduced from stoichiometric condition and perform recurrence quantification analysis (RQA) with the CH^∗ chemiluminescence data obtained experimentally. The extent of fuel-air mixing is varied with an objective of developing some robust early predictors of LBO that would work over a wide range of premixing. We find some RQA measures, such as determinism, laminarity, and trapping time, which show distinctive signature toward LBO and thereby can be used as early predictors of LBO for both premixed and partially premixed flames. Our analysis shows that the computational time for laminarity and trapping time is relatively less. However, computational time for those measures depends upon the dynamics of the combustor, size of the data taken, and choice of recurrence threshold.

Electrocortical dynamics differentiate athletes exhibiting low- and high- ACL injury risk biomechanics

Anterior cruciate ligament (ACL) injuries are physically and emotionally debilitating for athletes,while motor and biomechanical deficits that contribute to ACL injury have been identified, limited knowledge about the relationship between the central nervous system (CNS) and biomechanical patterns of motion has impeded approaches to optimize ACL injury risk reduction strategies. In the current study it was hypothesized that high-risk athletes would exhibit altered temporal dynamics in their resting state electrocortical activity when compared to low-risk athletes. Thirty-eight female athletes performed a drop vertical jump (DVJ) to assess their biomechanical risk factors related to an ACL injury. The athletes' electrocortical activity was also recorded during resting state in the same visit as the DVJ assessment. Athletes were divided into low- and high-risk groups based on their performance of the DVJ. Recurrence quantification analysis was used to quantify the temporal dynamics of two frequency bands previously shown to relate to sensorimotor and attentional control. Results revealed that high-risk participants showed more deterministic electrocortical behavior than the low-risk group in the frontal theta and central/parietal alpha-2 frequency bands. The more deterministic resting state electrocortical dynamics for the high-risk group may reflect maladaptive neural behavior-excessively stable deterministic patterning that makes transitioning among functional task-specific networks more difficult-related to attentional control and sensorimotor processing neural regions.

Local Dynamic Stability of Self-Paced Treadmill Walking Versus Fixed-Speed Treadmill Walking

The purpose of this study was to assess the influence of gait stability induced by treadmill accelerations during self-paced treadmill walking (SPW). Local dynamic stability of three-dimensional (3D) upper body accelerations and hip angles were quantified. The results demonstrated that SPW was more unstable and had higher risk of falling than fixed-speed treadmill walking (FSW) under the impact of treadmill accelerations. The frequency domain analysis of treadmill speed indicated that intrastride treadmill speed variation was the dominating cause of the instability, and self-paced control strategies which can reduce the intrastride variation may achieve higher gait stability during SPW.

Detecting Causality using Deep Gaussian Processes

Convergent cross mapping (CCM) is a state space reconstruction (SSR)-based method designed for causal discovery in coupled time series, where Granger causality may not be applicable due to a separability assumption. However, CCM requires a large number of observations and is not robust to observation noise which limits its applicability. Moreover, in CCM and its variants, the SSR step is mostly implemented with delay embedding where the parameters for reconstruction usually need to be selected using grid search-based methods. In this paper, we propose a Bayesian version of CCM using deep Gaussian processes (DGPs), which are naturally connected with deep neural networks. In particular, we adopt the framework of SSR-based causal discovery and carry out the key steps using DGPs within a non-parametric Bayesian probabilistic framework in a principled manner. The proposed approach is first validated on simulated data and then tested on data used in obstetrics for monitoring the well-being of fetuses, i.e., fetal heart rate (FHR) and uterine activity (UA) signals in the last two hours before delivery. Our results indicate that UA affects the FHR, which agrees with recent clinical studies.

Nonlinear Methods Most Applied to Heart-Rate Time Series: A Review

The heart-rate dynamics are one of the most analyzed physiological interactions. Many mathematical methods were proposed to evaluate heart-rate variability. These methods have been successfully applied in research to expand knowledge concerning the cardiovascular dynamics in healthy as well as in pathological conditions. Notwithstanding, they are still far from clinical practice. In this paper, we aim to review the nonlinear methods most used to assess heart-rate dynamics. We focused on methods based on concepts of chaos, fractality, and complexity: Poincaré plot, recurrence plot analysis, fractal dimension (and the correlation dimension), detrended fluctuation analysis, Hurst exponent, Lyapunov exponent entropies (Shannon, conditional, approximate, sample entropy, and multiscale entropy), and symbolic dynamics. We present the description of the methods along with their most notable applications.

Walking barefoot vs. with minimalist footwear - influence on gait in younger and older adults

Background

In recent years, minimalist footwear has been increasingly promoted for its use in sportive and recreational activities. These shoes are considered to function naturally like barefoot walking while providing a protective surface. Despite a growing popularity of these shoes in the older population, little is known about the influence of minimalist footwear on gait patterns. This study investigated whether overground walking with minimalist shoes is comparable to barefoot walking regarding gait stability and variability parameters.

Methods

In a randomized within-subject study design, 31 healthy younger (29 ± 4 years) and 33 healthy community-dwelling older adults (71 ± 4 years) volunteered. Participants walked on flat ground, once barefoot and once with minimalist shoes. Gait variability of minimum toe clearance (MTC), stride length, stride time, and local dynamic gait stability were analysed.

Results

The results for both age groups showed significant condition effects (minimalist shoes vs. barefoot walking) for the outcomes of local dynamic stability (p = .013), MTC variability (p = .018), and stride length variability (p < .001) indicating increased local dynamic stability and decreased gait variability during the minimalist shoe condition. Group effects (young vs. older adults) were detected in all gait outcomes.

Conclusion

Walking with minimalist shoes appeared to be associated with better gait performance than walking barefoot in both age groups. Thus, walking with minimalist shoes is not similar to barefoot walking. With respect to reducing the risk of falling, we suggest that minimalist shoes could be an alternative to barefoot walking or a transition option between shoes to barefoot for older adults.

Audience spontaneous entrainment during the collective enjoyment of live performances: physiological and behavioral measurements

Cardiac synchrony is a crucial component of shared experiences, considered as an objective measure of emotional processes accompanying empathic interactions. No study has investigated whether cardiac synchrony among people engaged in collective situations links to the individual emotional evaluation of the shared experience. We investigated theatrical live performances as collective experiences evoking strong emotional engagement in the audience. Cross Recurrence Quantification Analysis was applied to obtain the cardiac synchrony of twelve spectators’ quartets attending to two live acting performances. This physiological measure was then correlated with spectators’ emotional intensity ratings. Results showed an expected increment in synchrony among people belonging to the same quartet during both performances attendance and rest periods. Furthermore, participants’ cardiac synchrony was found to be correlated with audience’s convergence in the explicit emotional evaluation of the performances they attended to. These findings demonstrate that the mere co-presence of other people sharing a common experience is enough for cardiac synchrony to occur spontaneously and that it increases in function of a shared and coherent explicit emotional experience.

On the automatic parameter selection for permutation entropy

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

Accuracy of image data stream of a markerless motion capture system in determining the local dynamic stability and joint kinematics of human gait

Assessment of gait parameters is commonly performed through the high-end motion tracking systems, which limits the measurement to sophisticated laboratory settings due to its excessive cost. Recently, Microsoft Kinect (v2) sensor has become popular in clinical gait analysis due to its low-cost. But, determining the accuracy of its RGB-D image data stream in measuring the joint kinematics and local dynamic stability remains an unsolved problem. This study examined the suitability of Kinect(v2) RGB-D image data stream in assessing those gait parameters. Fifteen healthy participants walked on a treadmill during which lower body kinematics were measured by a Kinect(v2) sensor and a optophotogrametric tracking system, simultaneously. Extended Kalman filter was used to extract the lower extremity joint angles from Kinect, while inverse kinematics was used for the gold standard system. For both systems, local dynamic stability was assessed using maximal Lyapunov exponent. Sprague's validation metrics, root mean square error (RMSE) and normalized RMSE were computed to confirm the difference between the joint angles time series of the two systems while relative agreement between them was investigated through Pearson's correlation coefficient (p_r). Fisher's Exact Test was performed on maximal Lyapunov exponent to investigate the data independence while reliability was assessed using intraclass correlation coefficients. This study concludes that the RGB-D data stream of Kinect sensor is efficient in estimating joint kinematics, but not suitable for measuring the local dynamic stability.

Algebraic Method for the Reconstruction of Partially Observed Nonlinear Systems Using Differential and Integral Embedding

The identification of partially observed continuous nonlinear systems from noisy and incomplete data series is an actual problem in many branches of science, for example, biology, chemistry, physics, and others. Two stages are needed to reconstruct a partially observed dynamical system. First, one should reconstruct the entire phase space to restore unobserved state variables. For this purpose, the integration or differentiation of the observed data series can be performed. Then, a fast-algebraic method can be used to obtain a nonlinear system in the form of a polynomial dynamical system. In this paper, we extend the algebraic method proposed by Kera and Hasegawa to Laurent polynomials which contain negative powers of variables, unlike ordinary polynomials. We provide a theoretical basis and experimental evidence that the integration of a data series can give more accurate results than the widely used differentiation. With this technique, we reconstruct Lorenz attractor from a one-dimensional data series and B. Muthuswamy’s circuit equations from a three-dimensional data series.

The influence of attractor stability of intrinsic coordination patterns on the adaptation to new constraints

In most human movement tasks, the same goal can be achieved by a diversity of coordination patterns. For instance, when learning to juggle, individuals adopt their own unique coordination patterns in the early stages of acquiring the fundamental skills of juggling. These individual differences in the learning paths lead to differences in adaptability to new constraints. However, the reason for these differences in adaptability is still unknown. To address this problem, we quantified these differences in terms of attractor stability of the coordination patterns of expert jugglers using Recurrence Quantification Analysis. Furthermore, we quantified the attractor stability of intermediate jugglers and examined adaptability in a sensorimotor synchronization task. We found differences in attractor stability among coordination patterns of expert jugglers, as well as a difference in attractor stability between intrinsic coordination patterns of intermediate jugglers. Whereas, almost no significant direct correlation between attractor stability and adaptability of intermediate jugglers was found, suggesting a difference in both attractor stability and adaptability between intrinsic coordination patterns such that the difference in attractor stability might affect adaptability to new constraints. We submit that the learning path selected by each learner in the early stages of learning plays an important role in the subsequent development of expertise.

Hip extensor fatigue alters hip and knee coupling dynamics during single-limb step-downs: A randomized controlled trial

Impaired hip muscle function may cause movement coordination deficits that increase lower extremity injury risks. We examined whether hip and knee coordination is altered during single-limb step-downs following a hip extensor fatigue protocol. Forty participants in this randomized controlled trial performed 20 single-limb step-downs before and after completing either a fatigue protocol or a sham fatigue protocol. Means and SDs of sagittal and frontal plane hip and knee kinematics were measured. Nonlinear measures of coupled hip and knee coordination were examined with cross recurrence quantification analyses. Pre- to post-fatigue change scores were analyzed inferentially (α = 0.05). The fatigue protocol induced 22.2% strength reduction in the fatigue group, versus 4% reduction in the sham group. Neither the magnitude nor variability in sagittal and frontal plane hip and knee kinematics changed following the fatigue protocol (P > .05, small effect sizes). Participants in the fatigue group, however, performed post-fatigue step-downs with greater cross determinism (P = .046, effect size = 0.71) and mean line (P = .038, effect size = 0.74) in sagittal plane hip and frontal plane knee coupling, whereas participants in the sham-control group performed step-downs with reductions in those measures. In the fatigued state, participants performed repeated step-downs with subtle increases in cross determinism and cross mean line, which implies they performed the task with greater predictability of hip and knee coupling and less adaptability. The findings may provide insight into coupled movement patterns and their reflection of motor control.

Quantitative coordination evaluation for screening children with Duchenne muscular dystrophy

As the potential for a treatment of Duchenne muscular dystrophy (DMD) grows, the need for methods for the early diagnosis of DMD becomes more and more important. Clinical experiences suggest that children with DMD will show some lack of motor ability in the early stage when compared with children at the same age, especially in balance and coordination abilities. Is it possible to quantify the coordination differences between DMD and typically developing (TD) children to achieve the goal of screening for DMD diseases? In this study, we introduced a Local Manifold Structure Mapping approach in phase space and extracted a novel index, relative coupling coefficient (RCC), from gait pattern signals, which were acquired by wearable accelerometers to evaluate the coordination of children with DMD during a walking task. Furthermore, we compared the RCC of 100 children with DMD and 100 TD children in four different age groups and verified the feasibility and reliability of the proposed indices to distinguish children with TD from DMD. T-test results show that, for all age groups, children of the same age with DMD and TD show significant differences in RCC (p < 0.001). Moreover, RCC comprehensively reflects that the coordination ability of DMD patients under walking tasks gradually decreases with age, which is consistent with clinical experience. As a functional biomarker extracted in the phase space of the gait data, the proposed coupling degree index RCC could sensitively distinguish between DMD and TD children at the same age and provide alternative insights and potentially valuable tools for the screening of DMD.

Wireless multichannel vibroarthrographic recordings for the assessment of knee osteoarthritis during three activities of daily living

BACKGROUND

Variations in the internal pressure distribution applied to cartilage and synovial fluid explain the spatial dependencies of the knee vibroarthrographic signals. These spatial dependencies were assessed by multi-channel recordings during activities of daily living in patients with painful knee osteoarthrosis.

METHODS

Knee vibroarthrographic signals were detected using eight miniature accelerometers, and vibroarthrographic maps were calculated for the most affected knee of 20 osteoarthritis patients and 20 asymptomatic participants during three activities: (i) sit to stand, (ii) stairs descent, and (iii) stairs ascent in real life conditions. Vibroarthrographic maps of average rectified value, variance of means squared, form factor, mean power frequency, % of recurrence and, % of determinism were obtained from the eight VAG recordings.

FINDINGS

Higher average rectified value and lower % of recurrence were found in knee osteoarthritis patients compared with asymptomatic participants. All vibroarthrographic parameters, except for % of recurrence, differentiated the type of activity. Average rectified value, variance of means squared, form factor, and % of determinism were lowest while mean power frequency was highest during sit-to-stand compared with stairs ascent and descent.

INTERPRETATION

Distinct topographical vibroarthrographic maps underlined that the computed parameters represent unique features. The present study demonstrated that wireless multichannel vibroarthrographic recordings and the associated topographical maps highlighted differences between (i) knee osteoarthritis patients and asymptomatic participants, (ii) sit to stand, stairs descent and ascent and (iii) knee locations. The technique offers new perspectives for biomechanical assessments of physical functions of the knee joint in ecological environment.

Application of Time-Scale Decomposition of Entropy for Eye Movement Analysis

The methods for nonlinear time series analysis were used in the presented research to reveal eye movement signal characteristics. Three measures were used: approximate entropy, fuzzy entropy, and the Largest Lyapunov Exponent, for which the multilevel maps (MMs), being their time-scale decomposition, were defined. To check whether the estimated characteristics might be useful in eye movement events detection, these structures were applied in the classification process conducted with the usage of the kNN method. The elements of three MMs were used to define feature vectors for this process. They consisted of differently combined MM segments, belonging either to one or several selected levels, as well as included values either of one or all the analysed measures. Such a classification produced an improvement in the accuracy for saccadic latency and saccade, when compared with the previously conducted studies using eye movement dynamics.

Planetary-Gearbox Fault Classification by Convolutional Neural Network and Recurrence Plot

Recurrence-plot (RP) analysis is a graphical tool to visualize and analyze the recurrence of nonlinear dynamic systems. By combining the advantages of the RP and a convolutional neural network (CNN), a fault-classification scheme for planetary gear sets is proposed in this paper. In the proposed approach, a vibration is first picked up from the planetary-gear test rig and converted into an angular-domain quasistationary signal through computed order tracking to eliminate the frequency blur caused by speed fluctuations. Then, the signal in the angular domain is divided into several segments, and each segment is processed by the RP to constitute the training sample. Moreover, a two-dimensional CNN model was developed to adaptively extract faulty features. Experiments on a planetary-gear test rig with four conditions under three operating speeds were carried out. The results of measured vibration demonstrated the validity of CNN and recurrence plot analysis for the fault classification of planetary-gear sets.

Diagnosing Automotive Damper Defects Using Convolutional Neural Networks and Electronic Stability Control Sensor Signals

Chassis system components such as dampers have a significant impact on vehicle stability, driving safety, and driving comfort. Therefore, monitoring and diagnosing the defects of these components is necessary. Currently, this task is based on the driver’s perception of component defects in series production vehicles, even though model-based approaches in the literature exist. As we observe an increased availability of data in modern vehicles and advances in the field of deep learning, this paper deals with the analysis of the performance of Convolutional Neural Networks (CNN) for the diagnosis of automotive damper defects. To ensure a broad applicability of the generated diagnosis system, only signals of a classic Electronic Stability Control (ESC) system, such as wheel speeds, longitudinal and lateral vehicle acceleration, and yaw rate, were used. A structured analysis of data pre-processing and CNN configuration parameters were investigated in terms of the defect detection result. The results show that simple Fast Fourier Transformation (FFT) pre-processing and configuration parameters resulting in small networks are sufficient for a high defect detection rate.

Nonlinear Analysis of Eye-Tracking Information for Motor Imagery Assessments

This study investigates the assessment of motor imagery (MI) ability in humans. Commonly, MI ability is measured through two methodologies: a self-administered questionnaire (MIQ-3) and the mental chronometry (MC), which measures the temporal discrepancy between the actual and the imagined motor tasks. However, both measures rely on subjects' self-assessment and do not use physiological measures. In this study, we propose a novel set of features extracted from the nonlinear dynamics of the eye gaze signal to discriminate between good and bad imagers. To this aim, we designed an experiment where twenty volunteers, categorized as good or bad imagers according to MC, performed three tasks: a motor task (MT), a visual Imagery task (VI), and a kinaesthetic Imagery task (KI). Throughout the experiment, the subjects' eye gaze was continuously monitored using an eye-tracking system. Eye gaze time series were analyzed through recurrence quantification analysis of the reconstructed phase space and compared between the two groups. Statistical results have shown how nonlinear eye behavior can express an inner dynamics of imagery mental process and may be used as a more objective and physiological-based measure of MI ability.

Dynamical interaction between heart rate and blood pressure of end-stage renal disease patients evaluated by cross recurrence plot diagonal analysis

The assessment of spontaneous variability of blood pressure and heart rate is based on specific physiological hypotheses about dynamic features, for example, the baroreflex modulation of heart rate over time in daily life. Usually, arterial baroreflex control of heart rate is explored without delays between blood pressure and heart rate data points, within a narrow range of values, excluding the analysis of saturation regions or low-threshold changes. In this work, we examine the dynamic interactions between systolic blood pressure (SBP) and interbeat interval (IBI), in 15-min length time series and for the first time using the analysis of diagonals derived from a cross-recurrence plots in healthy persons and end-stage renal disease (ESRD) patients. We found that ESRD patients have stronger intermittent dynamical interactions between IBI and SBP, but they lose most of the dynamical interactions. Although healthy subjects exhibit a continuously changing order of precedence between IBI and SBP at different lags, ESRD patients preserve this changing order of precedence only for lags >0 beats.NEW & NOTEWORTHY This study is the first to compare the time-variant pattern of systolic blood pressure (SBP) and interbeat interval (IBI) coupling between ESRD patients and healthy volunteers through the analysis of diagonal in cross-recurrence plots, and in the face of an orthostatic challenge. Our results demonstrated alternant interactions on the order of precedence (IBI → SBP or SBP→ IBI) at different time delays. This pattern is different in resting position and during active standing for the two groups studied, and interestingly, some association patterns are lost in ESRD patients. These patterns of alternant interactions on the order of precedence could be related to autonomic neural activities and cardiovascular synchronization at different scales both in time and space. This could reflect physiological adaptive flexibility of cardiovascular regulation. Losing some association patterns in ESRD may be the result of chronic adjustments of many physiological mechanisms (including chronic sympathetic hyperactivity), which could increase cardiovascular vulnerability to hemodynamic challenges.

Dimension of reservoir computers

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance dimension, and Kaplan-Yorke dimension, are used to estimate the dimension of the reservoir dynamical system. It is shown that the signals in the reservoir system exist on a relatively low dimensional surface. Changing the spectral radius of the reservoir network can increase the fractal dimension of the reservoir signals, leading to an increase in a testing error.

Usefulness of recurrence plots from airflow recordings to aid in paediatric sleep apnoea diagnosis

BACKGROUND AND OBJECTIVE

In-laboratory overnight polysomnography (PSG) is the gold standard method to diagnose the Sleep Apnoea-Hypopnoea Syndrome (SAHS). PSG is a complex, expensive, labour-intensive and time-consuming test. Consequently, simplified diagnostic methods are desirable. We propose the analysis of the airflow (AF) signal by means of recurrence plots (RP) features. The main goal of our study was to evaluate the utility of the information from RPs of the AF signals to detect paediatric SAHS at different levels of severity. In addition, we also evaluated the complementarity with the 3% oxygen desaturation index (ODI₃).

METHODS

946 AF and blood oxygen saturation (SpO₂) recordings from children ages 0-13 years were used. The population under study was randomly split into training (60%) and test (40%) sets. RP was computed and 9 RP features were extracted from each AF recording. ODI₃ was also calculated from each SpO₂ recording. A feature selection stage was conducted in the training group by means of the fast correlation-based filter (FCBF) methodology to obtain a relevant and non-redundant optimum feature subset. A multi-layer perceptron neural network with Bayesian approach (BY-MLP), trained with these optimum features, was used to estimate the apnoea-hypopnoea index (AHI).

RESULTS

8 of the RP features showed statistically significant differences (p-value <0.01) among the SAHS severity groups. FCBF selected the maximum length of the diagonal lines from RP, as well as the ODI₃. Using these optimum features, the BY-MLP model achieved 83.2%, 78.5%, and 91.0% accuracy in the test group for the AHI thresholds 1, 5, and 10 events/h, respectively. Moreover, this model reached a negative likelihood ratio of 0.1 for 1 event/h and a positive likelihood ratio of 13.7 for 10 events/h.

CONCLUSIONS

RP analysis enables extraction of useful SAHS-related information from overnight AF paediatric recordings. Moreover, it provides complementary information to the widely-used clinical variable ODI₃. Thus, RP applied to AF signals can be used along with ODI₃ to help in paediatric SAHS diagnosis, particularly to either confirm the absence of SAHS or the presence of severe SAHS.

On a topological criterion to select a recurrence threshold

In this work, a topological criterion is proposed for selecting a recurrence threshold for constructing a recurrence plot of a time series. It is based on a metric structure of the set of the recurrence plots that is defined by the recurrence plot deviation distance among recurrence plots, introduced in a previous paper by the authors. In this process for a range of threshold values, the corresponding recurrence plots are constructed. Then, a value of the threshold is considered to be optimal when the image of its recurrence plot remains close to images of the other recurrence plots constructed for close values of thresholds based on the topological criterion introduced in the present work. The results are applied both to time series emanating from the Lorenz dynamical system and to Molecular Dynamic simulations.

Biometrics Using Electroencephalograms Stimulated by Personal Ultrasound and Multidimensional Nonlinear Features

Biometrics such as fingerprints and iris scans has been used in authentication. However, conventional biometrics is vulnerable to identity theft, especially in user-management systems. As a new biometrics without this vulnerability, brain waves have been a focus. In this paper, brain waves (electroencephalograms (EEGs)) were measured from ten experiment subjects. Individual features were extracted from the log power spectra of the EEGs using principal component analysis, and verification was achieved using a support vector machine. It was found that, for the proposed authentication method, the equal error rate (EER) for a single electrode was about 22–32%, and that, for a multiple electrodes, was 4.4% by using the majority decision rule. Furthermore, nonlinear features based on chaos analysis were introduced for feature extraction and then extended to multidimensional ones. By fusing the results of all electrodes when using the proposed multidimensional nonlinear features and the spectral feature, an EER of 0% was achieved. As a result, it was confirmed that individuals can be authenticated using induced brain waves when they are subjected to ultrasounds.

The impact of obesity on gait stability in older adults

Obesity increases fall risk, and fall-related injuries in older adults. While prior work suggests obesity influences postural stability during standing, little is known about how obesity affects walking stability. Therefore, this study compared walking stability in older adults with and without obesity. Exploratory analyses were also conducted to evaluate the associations between measures of body habitus and gait stability as well as the association between prospective stumbles and falls and gait stability. A total of 34 older adults (17 with obesity, 17 with normal weight) walked on a treadmill at a self-selected speed. Walking stability was quantified as the local dynamic stability of the trunk in all three planes of motion. Participants also performed a series of functional tests, and were followed for a one-year period during which they reported falls and stumbles. Although participants with obesity performed significantly worse than participants without obesity on most functional tests, there were no differences in stability between groups in any direction (p = 0.18-0.78; η² = 0.003-0.056), nor between those with and without a prospective fall or stumble (p = 0.18-0.93; η² = 0.003-0.054). There were significant, albeit weak, correlations between BMI, waist circumference, and waist-to-height ratio and walking instability (p = 0.027-0.042; ρ = 0.36-0.39). Increased body mass, in absence of other obesity-related comorbidities, may have minimum impact on walking stability and in turn fall risk in older adults.

A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots

Nonlinear time series analysis gained prominence from the late 1980s on, primarily because of its ability to characterize, analyze, and predict nontrivial features in data sets that stem from a wide range of fields such as finance, music, human physiology, cognitive science, astrophysics, climate, and engineering. More recently, recurrence plots, initially proposed as a visual tool for the analysis of complex systems, have proven to be a powerful framework to quantify and reveal nontrivial dynamical features in time series data. This tutorial review provides a brief introduction to the fundamentals of nonlinear time series analysis, before discussing in greater detail a few (out of the many existing) approaches of recurrence plot-based analysis of time series. In particular, it focusses on recurrence plot-based measures which characterize dynamical features such as determinism, synchronization, and regime changes. The concept of surrogate-based hypothesis testing, which is crucial to drawing any inference from data analyses, is also discussed. Finally, the presented recurrence plot approaches are applied to two climatic indices related to the equatorial and North Pacific regions, and their dynamical behavior and their interrelations are investigated.

Spatiotemporal dynamics of turbulent coaxial jet analyzed by symbolic information-theory quantifiers and complex-network approach

We numerically study the spatiotemporal dynamics of a turbulent coaxial jet in a model rocket engine combustor from the viewpoints of symbolic information-theory quantifiers and complex networks. The dynamic behavior of flow velocity undergoes a significant transition from a stochastic to chaotic state as the turbulent jet moves downstream. The small-world nature exists in the near field forming a stochastic state, whereas it disappears by the formation of a chaotic state in the far field. The dynamic behavior of hydrogen and oxygen concentrations in the far field also represents deterministic chaos. The simultaneous dynamic behavior with chaotic mixing forms the phase-synchronization state.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Quantifying the auto-oscillation complexity following water spraying with interest for phonation

Human voiced sound production or phonation is the result of a fluid-structure instability in the larynx leading to vocal folds auto-oscillation. In this paper, the effect of surface hydration following water spraying (0 up to 5 ml) on an ongoing auto-oscillation is studied experimentally using different mechanical deformable vocal folds replicas. The complexity of the oscillation is quantified on the upstream pressure by a phase space recurrence and complexity analysis. It is shown that: (1) the ratio of the degree of determinism to the recurrence rate of the phase space states γ and (2) estimated correlation dimension D_{2} are suitable parameters to grasp the effect of hydration on the oscillation pattern. The oscillation regime after hydration can either remain deterministic or approach a chaotic regime depending on initial conditions prior to water spraying, such as elasticity, glottal aperture, as well as oscillation complexity.

Synergies reciprocally relate end-effector and joint-angles in rhythmic pointing movements

During rhythmic pointing movements, degrees of freedom (DOF) in the human action system-such as joint-angles in the arm-are assumed to covary to stabilise end-effector movement, e.g. index finger. In this paper, it is suggested that the end-effector movement and the coordination of DOF are reciprocally related in synergies that link DOF so as to produce the end-effector movement. The coordination of DOF in synergies and the relation between end-effector movement and DOF coordination received little attention, though essential to understand the principles of synergy formation. Therefore, the current study assessed how the end-effector movement related to the coordination of joint-angles during rhythmic pointing across target widths and distances. Results demonstrated that joint-angles were linked in different synergies when end-effector movements differed across conditions. Furthermore, in every condition, three joint-angles (shoulder plane of elevation, shoulder inward-outward rotation, elbow flexion-extension) largely drove the end-effector, and all joint-angles contributed to covariation that stabilised the end-effector. Together, results demonstrated synergies that produced the end-effector movement, constrained joint-angles so that they covaried to stabilise the end-effector, and differed when end-effector movement differed. Hence, end-effector and joint-angles were reciprocally related in synergies-indicating that the action system was organised as a complex dynamical system.

Subjects With COPD Walk With Less Consistent Organization of Movement Patterns of the Lower Extremity

BACKGROUND

The inherent stride-to-stride fluctuations during walking are altered in the aging population and could provide insight into gait impairments and falls in patients with COPD. Stride-to-stride fluctuations are quantified two ways: variability of the fluctuations (eg, standard deviation), and movement patterns within the fluctuations. Our objective was to investigate stride-to-stride fluctuations by evaluating the variability and movement patterns of lower limb joints in subjects with COPD compared to subjects without COPD as control subjects.

METHODS

In this cross-sectional study, 22 subjects with COPD (age 63 ± 9 y; FEV₁ 54 ± 19% predicted) and 22 control subjects (age 62 ± 9 y; FEV₁ 95 ± 18% predicted) walked for 3 min on a treadmill while their gait was recorded. The amount of variability (ie, standard deviation and coefficient of variation) and movement patterns (ie, predictability and consistency in organization) were quantified for the range of motion and joint angle of the hip, knee, and ankle, at 3 walking speeds (ie, self-selected, fast, and slow). General linear mixed models were used for analysis.

RESULTS

Control subjects had more consistent organization of the hip and knee joint movement patterns compared to subjects with COPD (P = .02 and P = .02, respectively). Further, control subjects adapted to speed changes by demonstrating more consistent organization of movement patterns with faster speeds, whereas subjects with COPD did not. At the fast walking speed, subjects with COPD demonstrated less consistent organization of knee and hip joint movement patterns as compared to control subjects without COPD (P = .03 and P = .005, respectively). The amount of variability did not differ between groups.

CONCLUSIONS

Although subjects with COPD did not demonstrate decreased amount of variability, their hip and knee joint movement patterns were less consistent in organization during walking. Reduced consistency in organization of movement patterns may be a contributing factor to falls and mobility problems experienced by patients with COPD.

On the role of ageing and musculoskeletal pain on dynamic balance in manual workers

The purpose of this study was to explore the interacting effects of age and musculoskeletal pain on balance in manual workers. Ninety male manual workers aged 51-72 yr were recruited and stratified according to lower extremity musculoskeletal pain intensity (pain/no pain) and work status (working/retired). The five-repetition sit-to-stand (STS) test was used to assess lower extremity function including completion time, stand time, sit time and dynamic rate of force development both in the upwards (RFD_up) and downwards moving phase (RFD_down). Dynamic balance was expressed as the range, velocity, standard deviation (SD), maximum Lyapunov Exponent and sample entropy of centre of pressure displacement in the anterior-posterior and medial-lateral direction, as well as free moment during the STS test. Except for higher age, no marked differences were seen between working and retired participants. Both age and musculoskeletal pain were negatively associated with motor function. Age × pain interactions showed that completion time, stand time, RFD_up and RFD_down were negatively associated with age for participants without pain, but positively for those with pain. Similar findings were seen for dynamic balance. These findings indicate that the effects of lower extremity musculoskeletal pain on lower extremity function and dynamic balance are age dependent.

Dopamine receptor antagonists effects on low-dimensional attractors of local field potentials in optogenetic mice

The goal of this study was to investigate the effects of acute cocaine injection or dopamine (DA) receptor antagonists on the medial prefrontal cortex (mPFC) gamma oscillations and their relationship to short term neuroadaptation that may mediate addiction. For this purpose, optogenetically evoked local field potentials (LFPs) in response to a brief 10 ms laser light pulse were recorded from 17 mice. D1-like receptor antagonist SCH 23390 or D2-like receptor antagonist sulpiride, or both, were administered either before or after cocaine. A Euclidian distance-based dendrogram classifier separated the 100 trials for each animal in disjoint clusters. When baseline and DA receptor antagonists trials were combined in a single trial, a minimum of 20% overlap occurred in some dendrogram clusters, which suggests a possible common, invariant, dynamic mechanism shared by both baseline and DA receptor antagonists data. The delay-embedding method of neural activity reconstruction was performed using the correlation time and mutual information to determine the lag/correlation time of LFPs and false nearest neighbors to determine the embedding dimension. We found that DA receptor antagonists applied before cocaine cancels out the effect of cocaine and leaves the lag time distributions at baseline values. On the other hand, cocaine applied after DA receptor antagonists shifts the lag time distributions to longer durations, i.e. increase the correlation time of LFPs. Fourier analysis showed that a reasonable accurate decomposition of the LFP data can be obtained with a relatively small (less than ten) Fourier coefficients.

Quantification of Contractile Dynamic Complexities Exhibited by Human Stem Cell-Derived Cardiomyocytes Using Nonlinear Dimensional Analysis

Understanding the complexity of biological signals has been gaining widespread attention due to increasing knowledge on the nonlinearity that exists in these systems. Cardiac signals are known to exhibit highly complex dynamics, consisting of high degrees of interdependency that regulate the cardiac contractile functions. These regulatory mechanisms are important to understand for the development of novel in vitro cardiac systems, especially with the exponential growth in deriving cardiac tissue directly from human induced pluripotent stem cells (hiPSCs). This work describes a unique analytical approach that integrates linear amplitude and frequency analysis of physical cardiac contraction, with nonlinear analysis of the contraction signals to measure the signals’ complexity. We generated contraction motion waveforms reflecting the physical contraction of hiPSC-derived cardiomyocytes (hiPSC-CMs) and implemented these signals to nonlinear analysis to compute the capacity and correlation dimensions. These parameters allowed us to characterize the dynamics of the cardiac signals when reconstructed into a phase space and provided a measure of signal complexity to supplement contractile physiology data. Thus, we applied this approach to evaluate drug response and observed that relationships between contractile physiology and dynamic complexity were unique to each tested drug. This illustrated the applicability of this approach in not only characterization of cardiac signals, but also monitoring and diagnostics of cardiac health in response to external stress.

Characterizing Local Dynamic Stability of Lumbar Spine Sub-regions During Repetitive Trunk Flexion-Extension Movements

Using a technique of tracking intersegmental spine kinematics via skin surface markers, this study aimed to estimate local dynamic spine stability across smaller sub-regions (or segments) of the lumbar spine while also considering the impact of an external pelvic constraint during repetitive movements. Sixteen participants (10 males) performed two trials [Free Motion (FM), Pelvis Constrained (PC)] each consisting of 65 repetitive trunk flexion-extension movements to assess dynamic spine stability using maximum Lyapunov exponents (LyE). First, results indicated that LyE obtained from analysis of 30 repetitive flexion-extension movements did not differ from those obtained from analysis of greater numbers of repetitive movements, which aligns with results from a previous study for the whole lumbar spine. Next, for both males and females, and FM and PC trials, the most caudal region of the lumbar spine behaved the most dynamically stable, while upper lumbar regions behaved the most dynamically unstable. Finally, females demonstrated greater lumbar and intersegmental stability (lower LyE) during PC trials compared to FM, while males demonstrated slightly decreased lumbar and intersegmental stability (higher LyE) during PC trials compared to FM; this resulted in PC trials, but not FM trials, being different between sexes. Altogether, these data show that dynamic stability of lumbar spine sub-regions may be related to the proximity of the motion segment to rigid skeletal structures, and that consideration is needed when deciding whether to constrain the pelvis during analyses of dynamic spine stability.

Persistent homology of complex networks for dynamic state detection

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Deciphering Dynamical Nonlinearities in Short Time Series Using Recurrent Neural Networks

Surrogate testing techniques have been used widely to investigate the presence of dynamical nonlinearities, an essential ingredient of deterministic chaotic processes. Traditional surrogate testing subscribes to statistical hypothesis testing and investigates potential differences in discriminant statistics between the given empirical sample and its surrogate counterparts. The choice and estimation of the discriminant statistics can be challenging across short time series. Also, conclusion based on a single empirical sample is an inherent limitation. The present study proposes a recurrent neural network classification framework that uses the raw time series obviating the need for discriminant statistic while accommodating multiple time series realizations for enhanced generalizability of the findings. The results are demonstrated on short time series with lengths (L = 32, 64, 128) from continuous and discrete dynamical systems in chaotic regimes, nonlinear transform of linearly correlated noise and experimental data. Accuracy of the classifier is shown to be markedly higher than ≫50% for the processes in chaotic regimes whereas those of nonlinearly correlated noise were around ~50% similar to that of random guess from a one-sample binomial test. These results are promising and elucidate the usefulness of the proposed framework in identifying potential dynamical nonlinearities from short experimental time series.

An Effective Bearing Fault Diagnosis Technique via Local Robust Principal Component Analysis and Multi-Scale Permutation Entropy

The acquired bearing fault signal usually reveals nonlinear and non-stationary nature. Moreover, in the actual environment, some other interference components and strong background noise are unavoidable, which lead to the fault feature signal being weak. Considering the above issues, an effective bearing fault diagnosis technique via local robust principal component analysis (LRPCA) and multi-scale permutation entropy (MSPE) was introduced in this paper. Robust principal component analysis (RPCA) has proven to be a powerful de-noising method, which can extract a low-dimensional submanifold structure representing signal feature from the signal trajectory matrix. However, RPCA can only handle single-component signal. Therefore, in order to suppress background noise, an improved RPCA method named LRPCA is proposed to decompose the signal into several single-components. Since MSPE can efficiently evaluate the dynamic complexity and randomness of the signals under different scales, the fault-related single-components can be identified according the MPSE characteristic of the signals. Thereafter, these identified components are combined into a one-dimensional signal to represent the fault feature component for further diagnosis. The numerical simulation experimentation and the analysis of bearing outer race fault data both verified the effectiveness of the proposed technique.