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

Sparse model from optimal nonuniform embedding of time series

An approach to obtaining a parsimonious polynomial model from time series is proposed. An optimal minimal nonuniform time series embedding schema is used to obtain a time delay kernel. This scheme recursively optimizes an objective functional that eliminates a maximum number of false nearest neighbors between successive state space reconstruction cycles. A polynomial basis is then constructed from this time delay kernel. A sparse model from this polynomial basis is obtained by solving a regularized least squares problem. The constraint satisfaction problem is made computationally tractable by keeping the ratio between the number of constraints to the number of variables small by using fewer samples spanning all regions of the reconstructed state space. This helps the structure selection process from an exponentially large combinatorial search space. A forward stagewise algorithm is then used for fast discovery of the optimization path. Results are presented for the Mackey-Glass system.

Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2-H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.

Toward ambulatory balance assessment: estimating variability and stability from short bouts of gait

Stride-to-stride variability and local dynamic stability of gait kinematics are promising measures to identify individuals at increased risk of falling. This study aimed to explore the feasibility of using these metrics in clinical practice and ambulatory assessment, where only a small number of consecutive strides are available. The concurrent validity and reliability were assessed compared to more continuous walking. Twenty young adults walked continuously for 500 m, as well as 36 bouts of 20 m while wearing an accelerometer (DynaPort MiniMod) on the trunk. Within-day reliability was high for stride time variability, mediolateral trunk variability and local dynamic stability, while between-day reliability was low for both variability estimates and moderate for local dynamic stability. Stride time variability and mediolateral trunk variability were increased when walking short bouts and did not correlate well with the longer walking trials. Local dynamic stability did correlate highly between the long and short bouts trials, and 15 bouts of eight strides appeared to be sufficient for valid estimation. These results imply task-specific differences and low reliability of variability estimates rendering them unsuitable for application to short bouts of gait, while local dynamic stability can be readily employed.

A Novel Time Series Prediction Approach Based on a Hybridization of Least Squares Support Vector Regression and Swarm Intelligence

This research aims at establishing a novel hybrid artificial intelligence (AI) approach, named as firefly-tuned least squares support vector regression for time series prediction(FLSVRTSP). The proposed model utilizes the least squares support vector regression (LS-SVR) as a supervised learning technique to generalize the mapping function between input and output of time series data. In order to optimize the LS-SVR’s tuning parameters, theFLSVRTSPincorporates the firefly algorithm (FA) as the search engine. Consequently, the newly construction model can learn from historical data and carry out prediction autonomously without any prior knowledge in parameter setting. Experimental results and comparison have demonstrated that theFLSVRTSPhas achieved a significant improvement in forecasting accuracy when predicting both artificial and real-world time series data. Hence, the proposed hybrid approach is a promising alternative for assisting decision-makers to better cope with time series prediction.

Analysis of dual-task elderly gait using wearable plantar-pressure insoles and accelerometer

Dual-task gait allows assessment of impaired executive function and mobility control in older individuals, which are risk factors of falls. This study investigated gait changes in older individuals due to the addition of a cognitive load, using wearable pressure-sensing insole and tri-axial accelerometer measures. These wearable sensors can be applied at the point-of-care. Eleven elderly (65 years or older) individuals walked 7.62 m with and without a verbal fluency cognitive load task while wearing FScan 3000E pressure-sensing insoles in both shoes and a Gulf Coast X16-1C tri-axial accelerometer at the pelvis. Plantar-pressure derived parameters included center of force (CoF) path and temporal measures. Acceleration derived measures were descriptive statistics, Fast Fourier Transform quartile, ratio of even-to-odd harmonics, and maximum Lyapunov exponent. Stride time, stance time, and swing time all significantly increased during dual-task compared to single-task walking. Minimum, mean, and median CoF stance velocity; cadence; and vertical, anterior-posterior, and medial-lateral harmonic ratio all significantly decreased during dual-task walking. Wearable plantar pressure-sensing insole and lower back accelerometer derived-measures can identify gait differences between single-task and dual-task walking in older individuals and could be used in point-of-care environments to assess for deficits in executive function and mobility impairments.

Predictions of experimentally observed stochastic ground vibrations induced by blasting

In the present paper, we investigate the blast induced ground motion recorded at the limestone quarry “Suva Vrela” near Kosjerić, which is located in the western part of Serbia. We examine the recorded signals by means of surrogate data methods and a determinism test, in order to determine whether the recorded ground velocity is stochastic or deterministic in nature. Longitudinal, transversal and the vertical ground motion component are analyzed at three monitoring points that are located at different distances from the blasting source. The analysis reveals that the recordings belong to a class of stationary linear stochastic processes with Gaussian inputs, which could be distorted by a monotonic, instantaneous, time-independent nonlinear function. Low determinism factors obtained with the determinism test further confirm the stochastic nature of the recordings. Guided by the outcome of time series analysis, we propose an improved prediction model for the peak particle velocity based on a neural network. We show that, while conventional predictors fail to provide acceptable prediction accuracy, the neural network model with four main blast parameters as input, namely total charge, maximum charge per delay, distance from the blasting source to the measuring point, and hole depth, delivers significantly more accurate predictions that may be applicable on site. We also perform a sensitivity analysis, which reveals that the distance from the blasting source has the strongest influence on the final value of the peak particle velocity. This is in full agreement with previous observations and theory, thus additionally validating our methodology and main conclusions.

Chaotic chirped-pulse oscillators

We present results of experimental investigation of the chaotic and quasi-periodic regime in the chirped-pulsed (dissipative soliton) Cr:ZnS and Cr:ZnSe mid-IR oscillators with significant third-order dispersion. The instability develops when the spectrum edge approaches resonance with a linear wave either due to power increase or by dispersion adjustment. In practice, this occurs when the spectrum edge reaches zero dispersion wavelength. The analysis suggests a three-oscillator chaos model, which is confirmed by numerical simulations. The regime is long-term stable and can be easily overlooked in similar systems. We show that chaotic regime is accompanied by a characteristic spectral shape and can be reliably recognized by using wavelength-skewed filters and by second-harmonic or two-photon absorption detectors.

Self-organized topology of recurrence-based complex networks

With the rapid technological advancement, network is almost everywhere in our daily life. Network theory leads to a new way to investigate the dynamics of complex systems. As a result, many methods are proposed to construct a network from nonlinear time series, including the partition of state space, visibility graph, nearest neighbors, and recurrence approaches. However, most previous works focus on deriving the adjacency matrix to represent the complex network and extract new network-theoretic measures. Although the adjacency matrix provides connectivity information of nodes and edges, the network geometry can take variable forms. The research objective of this article is to develop a self-organizing approach to derive the steady geometric structure of a network from the adjacency matrix. We simulate the recurrence network as a physical system by treating the edges as springs and the nodes as electrically charged particles. Then, force-directed algorithms are developed to automatically organize the network geometry by minimizing the system energy. Further, a set of experiments were designed to investigate important factors (i.e., dynamical systems, network construction methods, force-model parameter, nonhomogeneous distribution) affecting this self-organizing process. Interestingly, experimental results show that the self-organized geometry recovers the attractor of a dynamical system that produced the adjacency matrix. This research addresses a question, i.e., "what is the self-organizing geometry of a recurrence network?" and provides a new way to reproduce the attractor or time series from the recurrence plot. As a result, novel network-theoretic measures (e.g., average path length and proximity ratio) can be achieved based on actual node-to-node distances in the self-organized network topology. The paper brings the physical models into the recurrence analysis and discloses the spatial geometry of recurrence networks.

Dynamic stability of running: the effects of speed and leg amputations on the maximal Lyapunov exponent

In this paper, we study dynamic stability during running, focusing on the effects of speed, and the use of a leg prosthesis. We compute and compare the maximal Lyapunov exponents of kinematic time-series data from subjects with and without unilateral transtibial amputations running at a wide range of speeds. We find that the dynamics of the affected leg with the running-specific prosthesis are less stable than the dynamics of the unaffected leg and also less stable than the biological legs of the non-amputee runners. Surprisingly, we find that the center-of-mass dynamics of runners with two intact biological legs are slightly less stable than those of runners with amputations. Our results suggest that while leg asymmetries may be associated with instability, runners may compensate for this effect by increased control of their center-of-mass dynamics.

Effect of correction of aberration dynamics on chaos in human ocular accommodation

We used adaptive optics to determine the effect of monochromatic aberration dynamics on the level of chaos in the accommodation control system. Four participants viewed a stationary target while the dynamics of their aberrations were either left uncorrected, defocus was corrected, or all aberrations except defocus were corrected. Chaos theory analysis was used to discern changes in the accommodative microfluctuations. We found a statistically significant reduction in the chaotic nature of the accommodation microfluctuations during correction of defocus, but not when all aberrations except defocus were corrected. The Lyapunov exponent decreased from 0.71 ± 0.07 D/s (baseline) to 0.55 ± 0.03 D/s (correction of defocus fluctuations). As the reduction of chaos in physiological signals is indicative of stress to the system, the results indicate that for the participants included in this study, fluctuations in defocus have a more profound effect than those of the other aberrations. There were no changes in the power spectrum between experimental conditions. Hence chaos theory analysis is a more subtle marker of changes in the accommodation control system and will be of value in the study of myopia onset and progression.

Generalized synchronization in relay systems with instantaneous coupling

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it.

Phase space reconstruction of an experimental model of cardiac field potential in normal and arrhythmic conditions

Cardiac arrhythmias are one of the most important death causes in the world. Compared to the numerical models, the experimental ones provide a more realistic tool to study the mechanisms of cardiac arrhythmias. The in vitro culture of cardiac cells developed on the Multi-Electrodes Array (MEA) constitutes a suitable model in this context. The extracellular field potential (EFP) acquired from the MEA can be used to measure the electrophysiological parameters of action potential. In this article, the stability of this experimental model is investigated using the phase space reconstruction in normal and in arrhythmia conditions. The results show that the embedding dimension of signal EFP changed slightly in both cases (normal conditions and arrhythmia). The parameter time lag τ in the normal conditions is lower than in the arrhythmia. The shape of attractors remains similar but disturbed in case of arrhythmia compared to the normal conditions.

An approach to absence epileptic seizures detection using Approximate Entropy

Epilepsy is one of the most common chronic neurological diseases and the most common neurological chronic disease of childhood. The electroencephalogram (EEG) signal provides significant information neurologists take into consideration in the investigation and analysis of epileptic seizures. The Approximate Entropy (ApEn) is a formulated statistical parameter commonly used to quantify the regularity of a time series data of physiological signals. In this paper ApEn is used in order to detect the onset of epileptic seizures. The results show that the method provides promising results towards efficient detection of onset and ending of seizures, based on analyzing the corresponding EEG signals. ApEn parameters affect the method's behavior, suggesting that a more detailed study and a consistent methodology of their determination should be established. A preliminary analysis for the proper determination of these parameters is performed towards improving the results.

Comparison of SVM and ANFIS for snore related sounds classification by using the largest Lyapunov exponent and entropy

Snoring, which may be decisive for many diseases, is an important indicator especially for sleep disorders. In recent years, many studies have been performed on the snore related sounds (SRSs) due to producing useful results for detection of sleep apnea/hypopnea syndrome (SAHS). The first important step of these studies is the detection of snore from SRSs by using different time and frequency domain features. The SRSs have a complex nature that is originated from several physiological and physical conditions. The nonlinear characteristics of SRSs can be examined with chaos theory methods which are widely used to evaluate the biomedical signals and systems, recently. The aim of this study is to classify the SRSs as snore/breathing/silence by using the largest Lyapunov exponent (LLE) and entropy with multiclass support vector machines (SVMs) and adaptive network fuzzy inference system (ANFIS). Two different experiments were performed for different training and test data sets. Experimental results show that the multiclass SVMs can produce the better classification results than ANFIS with used nonlinear quantities. Additionally, these nonlinear features are carrying meaningful information for classifying SRSs and are able to be used for diagnosis of sleep disorders such as SAHS.

A model-based method for computation of correlation dimension, Lyapunov exponents and synchronization from depth-EEG signals

In order to predict epileptic seizures many precursory features, extracted from the EEG signals, have been introduced. Before checking out the performance of features in detection of pre-seizure state, it is required to see whether these features are accurately extracted. Evaluation of feature estimation methods has been less considered, mainly due to the lack of a ground truth for the real EEG signals' features. In this paper, some simulated long-term depth-EEG signals, with known state spaces, are generated via a realistic neural mass model with physiological parameters. Thanks to the known ground truth of these synthetic signals, they are suitable for evaluating different algorithms used to extract the features. It is shown that conventional methods of estimating correlation dimension, the largest Lyapunov exponent, and phase coherence have non-negligible errors. Then, a parameter identification-based method is introduced for estimating the features, which leads to better estimation results for synthetic signals. It is shown that the neural mass model is able to reproduce real depth-EEG signals accurately; thus, assuming this model underlying real depth-EEG signals, can improve the accuracy of features' estimation.

Non-linear dynamics of human locomotion: effects of rhythmic auditory cueing on local dynamic stability

It has been observed that times series of gait parameters [stride length (SL), stride time (ST), and stride speed (SS)], exhibit long-term persistence and fractal-like properties. Synchronizing steps with rhythmic auditory stimuli modifies the persistent fluctuation pattern to anti-persistence. Another non-linear method estimates the degree of resilience of gait control to small perturbations, i.e., the local dynamic stability (LDS). The method makes use of the maximal Lyapunov exponent, which estimates how fast a non-linear system embedded in a reconstructed state space (attractor) diverges after an infinitesimal perturbation. We propose to use an instrumented treadmill to simultaneously measure basic gait parameters (time series of SL, ST, and SS from which the statistical persistence among consecutive strides can be assessed), and the trajectory of the center of pressure (from which the LDS can be estimated). In 20 healthy participants, the response to rhythmic auditory cueing (RAC) of LDS and of statistical persistence [assessed with detrended fluctuation analysis (DFA)] was compared. By analyzing the divergence curves, we observed that long-term LDS (computed as the reverse of the average logarithmic rate of divergence between the 4th and the 10th strides downstream from nearest neighbors in the reconstructed attractor) was strongly enhanced (relative change +73%). That is likely the indication of a more dampened dynamics. The change in short-term LDS (divergence over one step) was smaller (+3%). DFA results (scaling exponents) confirmed an anti-persistent pattern in ST, SL, and SS. Long-term LDS (but not short-term LDS) and scaling exponents exhibited a significant correlation between them (r = 0.7). Both phenomena probably result from the more conscious/voluntary gait control that is required by RAC. We suggest that LDS and statistical persistence should be used to evaluate the efficiency of cueing therapy in patients with neurological gait disorders.

Multivariate recurrence network analysis for characterizing horizontal oil-water two-phase flow

Characterizing complex patterns arising from horizontal oil-water two-phase flows is a contemporary and challenging problem of paramount importance. We design a new multisector conductance sensor and systematically carry out horizontal oil-water two-phase flow experiments for measuring multivariate signals of different flow patterns. We then infer multivariate recurrence networks from these experimental data and investigate local cross-network properties for each constructed network. Our results demonstrate that a cross-clustering coefficient from a multivariate recurrence network is very sensitive to transitions among different flow patterns and recovers quantitative insights into the flow behavior underlying horizontal oil-water flows. These properties render multivariate recurrence networks particularly powerful for investigating a horizontal oil-water two-phase flow system and its complex interacting components from a network perspective.

A dynamical systems approach to the control of chaotic dynamics in a spatiotemporal jet flow

We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow that was excited by low level periodic forcing of the primary instability. The control method utilizes unstable periodic orbits (UPO) embedded within the chaotic attractor. Since hydrodynamic instabilities in the open flow system are convective, both monitoring and control can be implemented at a few locations, resulting in a simple and effective control algorithm. Experiments were performed in an incompressible, initially laminar, 4 cm diameter circular air jet, at a Reynolds number of 23,000, housed in a low-noise, large anechoic chamber. Distinct trajectory bundles surrounding the dominant UPOs were found from experimentally derived, time-delayed embedding of the chaotic attractor. Velocity traces from a pair of probes placed at the jet flow exit and farther downstream were used to empirically model the UPOs and compute control perturbations to be applied at the jet nozzle lip. Open loop control was used to sustain several nearly periodic states.

Quantitative performance analysis of four methods of evaluating signal nonlinearity: application to uterine EMG signals

Recently, much attention has been paid to the use of nonlinear analysis techniques for the characterization of biological signals. Several measures have been proposed to detect nonlinear characteristics in time series. The sensitivity of several nonlinear methods to the actual nonlinearity level and their sensitivity to noise have never been evaluated. In this paper we perform this analysis for four methods that are widely used in nonlinearity detection: Time reversibility, Sample Entropy, Lyapunov Exponents and Delay Vector Variance. The evolution of methods with complexity degree (CD) and with different Signal to Noise Ratio was computed for the four methods on nonlinear synthetic signals. The methods were then applied to real uterine EMG signals with the aim of using them to distinguish between pregnancy and labor signals. The results show a clear superiority of the Time reversibility method, in classification of pregnancy and labor signals.

Wireless Wearable Multisensory Suite and Real-Time Prediction of Obstructive Sleep Apnea Episodes

Obstructive sleep apnea (OSA) is a common sleep disorder found in 24% of adult men and 9% of adult women. Although continuous positive airway pressure (CPAP) has emerged as a standard therapy for OSA, a majority of patients are not tolerant to this treatment, largely because of the uncomfortable nasal air delivery during their sleep. Recent advances in wireless communication and advanced ("bigdata") preditive analytics technologies offer radically new point-of-care treatment approaches for OSA episodes with unprecedented comfort and afforadability. We introduce a Dirichlet process-based mixture Gaussian process (DPMG) model to predict the onset of sleep apnea episodes based on analyzing complex cardiorespiratory signals gathered from a custom-designed wireless wearable multisensory suite. Extensive testing with signals from the multisensory suite as well as PhysioNet's OSA database suggests that the accuracy of offline OSA classification is 88%, and accuracy for predicting an OSA episode 1-min ahead is 83% and 3-min ahead is 77%. Such accurate prediction of an impending OSA episode can be used to adaptively adjust CPAP airflow (toward improving the patient's adherence) or the torso posture (e.g., minor chin adjustments to maintain steady levels of the airflow).

NONLINEAR INDICES OF HEART RATE VARIABILITY FOR DIFFERENTIATING ARRHYTHMIAS

Heart rate variability (HRV) is the temporal variation between sequences of consecutive heartbeats. Chaos and fractal-based measurements have been widely used for quantifying the HRV for cardiac risk stratification purposes. In this paper, five different sets of HRVs, viz., normal sinus rhythm (NSR), congestive heart failure (CHF), cardiac arrhythmia suppression trial (CAST), supra ventricular tachyarrhythmia (SVTA) and atrial fibrillation (AF), have been analysed using nonlinear parameters to fix the ranges of each parameter. Data were downloaded from the PhysioNet database with 15 sets in each case. The parameters used for analysis were Poincare plot measures: SD1, SD2 and SD12, largest Lyapunov exponent (LLE), correlation dimension (CD); recurrence plot measures: recurrence rate (REC), determinism (DET), mean diagonal length (L mean ), maximal diagonal length (L max ) and entropy (ENTR); detrended fluctuation analysis measures: scaling exponent (α) and fractal dimension (FD); sample entropy (SampEn); and approximate entropy (ApEn). Analysis of variance (ANOVA) was done for confirming the differences in parameter values between various cases. All parameters except LLE showed a significant statistical difference for different cases.

Characteristics of atrial fibrillation cycle length predict restoration of sinus rhythm by catheter ablation

BACKGROUND

Successful termination of atrial fibrillation (AF) during catheter ablation (CA) is associated with arrhythmia-free follow-up. Preablation factors such as mean atrial fibrillation cycle length (AFCL) predict the likelihood of AF termination during ablation but recurring patterns and AFCL stability have not been evaluated.

OBJECTIVE

To investigate novel predictors of acute and postoperative ablation outcomes from intracardiac electrograms: (1) recurring AFCL patterns and (2) localization index (LI) of the instantaneous fibrillatory rate distribution.

METHODS

Sixty-two patients with AF (32 paroxysmal AF; 45 men; age 57 ± 10 years) referred for CA were enrolled. One-minute electrogram was recorded from coronary sinus (CS; 5 bipoles) and right atrial appendage (HRA; 2 bipoles). Atrial activations were detected automatically to derive the AFCL and instantaneous fibrillatory rate (inverse of AFCL) time series. Recurring AFCL patterns were quantified by using recurrence plot indices (RPIs): percentage determinism, entropy of determinism, and maximum diagonal length. AFCL stability was determined by using the LI. The CA outcome predictivity of individual indices was assessed.

RESULTS

Patients with terminated atrial fibrillation (T-AF) had higher RPI (P < .05 in CS7-8) and LI than did those with nonterminated atrial fibrillation (P < .005 in CS3-4; P < .05 in CS5-6, CS7-8, and HRA). Patients free of arrhythmia after 3-month follow-up had higher RPI and LI (all P < .05 in CS7-8). All indices except percentage determinism predicted T-AF in CS7-8 (area under the curve [AUC] ≥ 0.71; odds ratio [OR] ≥ 4.50; P < .05). The median AFCL and LI predicted T-AF in HRAD (AUC ≥ 0.75; OR ≥ 7.76; P < .05). The RPI and LI predicted 3-month follow-up in CS7-8 (AUC ≥ 0.68; OR ≥ 4.17; P < .05).

CONCLUSIONS

AFCL recurrence and stability indices could be used in selecting patients more likely to benefit from CA.

A comprehensive assessment of gait accelerometry signals in time, frequency and time-frequency domains

Gait accelerometry is a promising tool to assess human walking and reveal deteriorating gait characteristics in patients and can be a rich source of clinically relevant information about functional declines in older adults. Therefore, in this paper, we present a comprehensive set of signal features that may be used to extract clinically valuable information from gait accelerometry signals. To achieve our goal, we collected tri-axial gait accelerometry signals from 35 adults 65 years of age and older. Fourteen subjects were healthy controls, 10 participants were diagnosed with Parkinson's disease, and 11 participants were diagnosed with peripheral neuropathy. The data were collected while the participants walked on a treadmill at a preferred walking speed. Accelerometer signal features in time, frequency and time-frequency domains were extracted. The results of our analysis showed that some of the extracted features were able to differentiate between healthy and clinical populations. Signal features in all three domains were able to emphasize variability among different groups, and also revealed valuable information about variability of the signals between anterior-posterior, mediolateral, and vertical directions within subjects. The current results imply that the proposed signal features can be valuable tools for the analysis of gait accelerometry data and should be utilized in future studies.

A nonlinear analysis of the transport Barkhausen-like noise measured in (Bi,Pb)2Sr2Ca2Cu3O(10+δ) superconductors

We investigated the transport Barkhausen-like noise (TBN) by using nonlinear time series analysis. TBN signals were measured in (Bi,Pb)2Sr2Ca2Cu3O10+δ ceramic samples subjected to different uniaxial compacting pressures (UCP). These samples display similar intragranular properties but different intergranular features. We found positive Lyapunov exponents in all samples, λm≥0.062, indicating the nonlinear dynamics of the experimental TBN signals. It was also observed higher values of the embedding dimension, m>9, and the Kaplan-Yorke dimension, DKY>2.9. Between samples, the behavior of λm and DKY with increasing excitation current is quite different. Such a behavior is explained in terms of changes in the microstructure associated with the UCP. In addition, determinism tests indicated that the TBN masked determinist components, as inferred by |k[over arrow]| values larger than 0.70 in most of the cases. Evidence on the existence of empirical attractors by reconstructing the phase spaces has been also found. All obtained results are useful indicators of the interplay between the uniaxial compacting pressure, differences in the microstructure of the samples, and the TBN signal dynamics.

Event-related desynchronization and synchronization quantification in motor-related EEG by Kolmogorov entropy

OBJECTIVE

Various approaches have been applied for the quantification of event-related desynchronization/synchronization (ERD/ERS) in EEG/MEG data analysis, but most of them are based on band power analysis. In this paper, we sought a novel method using a nonlinear measurement to quantify the ERD/ERS time course of motor-related EEG.

APPROACH

We applied Kolmogorov entropy to quantify the ERD/ERS time course of motor-related EEG in relation to hand movement imagination and execution for the first time. To further test the validity of the Kolmogorov entropy measure, we tested it on five human subjects for feature extraction to classify the left and right hand motor tasks.

MAIN RESULTS

The results show that the relative increase and decrease of Kolmogorov entropy indicates the ERD and ERS respectively. An average classification accuracy of 87.3% was obtained for five subjects.

SIGNIFICANCE

The results prove that Kolmogorov entropy can effectively quantify the dynamic process of event-related EEG, and it also provides a novel method of classifying motor imagery tasks from scalp EEG by Kolmogorov entropy measurement with promising classification accuracy.

Precision of estimates of local stability of repetitive trunk movements

PURPOSE

Local dynamic stability of trunk movements quantified by means of the maximum Lyapunov exponent (λmax) can provide information on trunk motor control and might offer a measure of trunk control in low-back pain patients. It is unknown how many repetitions are necessary to obtain sufficiently precise estimates of λmax and whether fatigue effects on λmax can be avoided while increasing the number of repetitions.

METHOD

Ten healthy subjects performed 100 repetitions of trunk movements in flexion, of trunk rotation and of a task combining these movement directions. λmax was calculated from thorax, pelvis and trunk (thorax relative to pelvis) kinematics. Data series were analyzed using a bootstrap procedure; ICC and coefficient of variation were used to quantify precision as a function of the number of cycles analyzed. ANOVA was used to compare movement tasks and to test for effects of time.

RESULTS

Trunk local stability reached acceptable precision level after 30 repetitions. λmax was higher (indicating lower stability) in flexion, compared to rotation and combined tasks. There was no time effect (fatigue). λmax of trunk movement was lower and less variable than that of thorax and pelvis movements.

CONCLUSIONS

The data provided allow for an informed choice of the number of repetitions in assessing local dynamic stability of trunk movements, weighting the gain in precision against the increase in measurement effort. Within the 100 repetitions tested, fatigue did not affect results. We suggest that increased stability during asymmetric movement may be explained by higher co-activation of trunk muscles.

Research on zheng classification fusing pulse parameters in coronary heart disease

This study was conducted to illustrate that nonlinear dynamic variables of Traditional Chinese Medicine (TCM) pulse can improve the performances of TCM Zheng classification models. Pulse recordings of 334 coronary heart disease (CHD) patients and 117 normal subjects were collected in this study. Recurrence quantification analysis (RQA) was employed to acquire nonlinear dynamic variables of pulse. TCM Zheng models in CHD were constructed, and predictions using a novel multilabel learning algorithm based on different datasets were carried out. Datasets were designed as follows: dataset1, TCM inquiry information including inspection information; dataset2, time-domain variables of pulse and dataset1; dataset3, RQA variables of pulse and dataset1; and dataset4, major principal components of RQA variables and dataset1. The performances of the different models for Zheng differentiation were compared. The model for Zheng differentiation based on RQA variables integrated with inquiry information had the best performance, whereas that based only on inquiry had the worst performance. Meanwhile, the model based on time-domain variables of pulse integrated with inquiry fell between the above two. This result showed that RQA variables of pulse can be used to construct models of TCM Zheng and improve the performance of Zheng differentiation models.

Optimal Selection of Parameters for Nonuniform Embedding of Chaotic Time Series Using Ant Colony Optimization

The optimal selection of parameters for time-delay embedding is crucial to the analysis and the forecasting of chaotic time series. Although various parameter selection techniques have been developed for conventional uniform embedding methods, the study of parameter selection for nonuniform embedding is progressed at a slow pace. In nonuniform embedding, which enables different dimensions to have different time delays, the selection of time delays for different dimensions presents a difficult optimization problem with combinatorial explosion. To solve this problem efficiently, this paper proposes an ant colony optimization (ACO) approach. Taking advantage of the characteristic of incremental solution construction of the ACO, the proposed ACO for nonuniform embedding (ACO-NE) divides the solution construction procedure into two phases, i.e., selection of embedding dimension and selection of time delays. In this way, both the embedding dimension and the time delays can be optimized, along with the search process of the algorithm. To accelerate search speed, we extract useful information from the original time series to define heuristics to guide the search direction of ants. Three geometry- or model-based criteria are used to test the performance of the algorithm. The optimal embeddings found by the algorithm are also applied in time-series forecasting. Experimental results show that the ACO-NE is able to yield good embedding solutions from both the viewpoints of optimization performance and prediction accuracy.

Assessing the stability of human locomotion: a review of current measures

Falling poses a major threat to the steadily growing population of the elderly in modern-day society. A major challenge in the prevention of falls is the identification of individuals who are at risk of falling owing to an unstable gait. At present, several methods are available for estimating gait stability, each with its own advantages and disadvantages. In this paper, we review the currently available measures: the maximum Lyapunov exponent (λS and λL), the maximum Floquet multiplier, variability measures, long-range correlations, extrapolated centre of mass, stabilizing and destabilizing forces, foot placement estimator, gait sensitivity norm and maximum allowable perturbation. We explain what these measures represent and how they are calculated, and we assess their validity, divided up into construct validity, predictive validity in simple models, convergent validity in experimental studies, and predictive validity in observational studies. We conclude that (i) the validity of variability measures and λS is best supported across all levels, (ii) the maximum Floquet multiplier and λL have good construct validity, but negative predictive validity in models, negative convergent validity and (for λL) negative predictive validity in observational studies, (iii) long-range correlations lack construct validity and predictive validity in models and have negative convergent validity, and (iv) measures derived from perturbation experiments have good construct validity, but data are lacking on convergent validity in experimental studies and predictive validity in observational studies. In closing, directions for future research on dynamic gait stability are discussed.

Time Series Analysis Using Geometric Template Matching

We present a novel framework for analyzing univariate time series data. At the heart of the approach is a versatile algorithm for measuring the similarity of two segments of time series called geometric template matching (GeTeM). First, we use GeTeM to compute a similarity measure for clustering and nearest-neighbor classification. Next, we present a semi-supervised learning algorithm that uses the similarity measure with hierarchical clustering in order to improve classification performance when unlabeled training data are available. Finally, we present a boosting framework called TDEBOOST, which uses an ensemble of GeTeM classifiers. TDEBOOST augments the traditional boosting approach with an additional step in which the features used as inputs to the classifier are adapted at each step to improve the training error. We empirically evaluate the proposed approaches on several datasets, such as accelerometer data collected from wearable sensors and ECG data.

Recurrence quantification analysis on pulse morphological changes in patients with coronary heart disease

OBJECTIVE

To show that the pulse diagnosis used in Traditional Chinese Medicine, combined with nonlinear dynamic analysis, can help identify cardiovascular diseases.

METHODS

Recurrence quantification analysis (RQA) was used to study pulse morphological changes in 37 inpatients with coronary heart disease (CHD) and 37 normal subjects (controls). An independent sample t-test detected significant differences in RQA measures of their pulses. A support vector machine (SVM) classified the groups according to their RQA measures. Classic time-domain parameters were used for comparison.

RESULTS

RQA measures can be divided into two groups. One group of measures [ecurrence rate (RR), determinism (DEL), average diagonal line length (L), maximum length of diagonal structures (Lmax), Shannon entropy of the frequency distribution of diagonal line lengths (ENTR), laminarity (LAM), average length of vertical structures (TT), maximum length of vertical structures (Vmax)] showed significantly higher values for patients with CHD than for normal subjects (P < 0.05). The other measures (RR_std, L_std, Lmax_std, TT_std, Vmax_std) showed significantly lower values for the CHD group than for normal subjects (P < 0.05). SVM classification accuracy was higher with RQA measures: With RQA (16 parameters) accuracy was at 88.21%, and with RQA (12 parameters) accuracy was at 84.11%. In contrast, with classic time-domain (15 parameters) accuracy was 75.73%, and with time-domain (7 parameters) accuracy was 74.70%.

CONCLUSION

Nonlinear dynamic methods such as RQA can be used to study functional and structural changes in the pulse noninvasively. Pulse signals of individuals with CHD have greater regularity, determinism, and stability than normal subjects, and their pulse morphology displays less variability. RQA can distinguish the CHD pulse from the healthy pulse with an accuracy of 88.21%, thereby providing an early diagnosis of cardiovascular diseases such as CHD.

Sensitivity vector fields in time-delay coordinate embeddings: theory and experiment

Identifying changes in the parameters of a dynamical system can be vital in many diagnostic and sensing applications. Sensitivity vector fields (SVFs) are one way of identifying such parametric variations by quantifying their effects on the morphology of a dynamical system's attractor. In many cases, SVFs are a more effective means of identification than commonly employed modal methods. Previously, it has only been possible to construct SVFs for a given dynamical system when a full set of state variables is available. This severely restricts SVF applicability because it may be cost prohibitive, or even impossible, to measure the entire state in high-dimensional systems. Thus, the focus of this paper is constructing SVFs with only partial knowledge of the state by using time-delay coordinate embeddings. Local models are employed in which the embedded states of a neighborhood are weighted in a way referred to as embedded point cloud averaging. Application of the presented methodology to both simulated and experimental time series demonstrates its utility and reliability.

Correspondence of chaos in binocular aberration dynamics

We used a binocular Shack-Hartmann sensor to measure the aberration dynamics of six participants at a rate of 21 Hz. Chaos theory analysis was used to determine the Lyapunov exponent for the time evolution of the rms wavefront error, accommodation, and each individual Zernike coefficient up to and including the fifth radial order. In all cases there was no statistically significant difference between the Lyapunov exponents between the two eyes, suggesting that the level of chaos is common between them. The mean Lyapunov exponent averaged across both eyes of all participants was 0.42±0.14 μm/s for the rms wavefront error, 0.37±0.06 D/s for accommodation, and 0.32±0.09 μm/s averaged across Zernike coefficients. We found no statistically significant correlation per se between the eyes, except for horizontal coma. The correlation may be masked by the impact of differing tear film dynamics. Understanding the nature of aberration dynamics has utility in optimizing the performance of adaptive optics systems for the human eye.

Optimal state-space reconstruction using derivatives on projected manifold

A paradigm for optimal state-space reconstruction with nonuniform time delays is proposed. A comparison based on a diffeomorphic measure and a smoothness cost function shows that the proposed methodology achieves a better reconstruction compared to a reconstruction based on time delays that are multiples of the first minimum of mutual information. It is also shown how the proposed methodology is a more reliable approach to determining the embedding dimension.

Local dynamic stability of lower extremity joints in lower limb amputees during slope walking

Lower limb amputees have a higher fall risk during slope walking compared with non-amputees. However, studies on amputees' slope walking were not well addressed. The aim of this study was to identify the difference of slope walking between amputees and non-amputees. Lyapunov exponents λS was used to estimate the local dynamic stability of 7 transtibial amputees' and 7 controls' lower extremity joint kinematics during uphill and downhill walking. Compared with the controls, amputees exhibited significantly lower λS in hip (P=0.04) and ankle (P=0.01) joints of the sound limb, and hip joints (P=0.01) of the prosthetic limb during uphill walking, while they exhibited significantly lower λS in knee (P=0.02) and ankle (P=0.03) joints of the sound limb, and hip joints (P=0.03) of the prosthetic limb during downhill walking. Compared with amputees level walking, they exhibited significantly lower λS in ankle joints of the sound limb during both uphill (P=0.01) and downhill walking (P=0.01). We hypothesized that the better local dynamic stability of amputees was caused by compensation strategy during slope walking.