Regularity Of Time Series Research Articles

Analysis of magnetoencephalography recordings from Alzheimer's disease patients using embedding entropies.

The aim of this study was to examine the magnetoencephalography (MEG) background activity in Alzheimer's disease (AD) using three embedding entropies: approximate entropy (ApEn), sample entropy (SampEn), and fuzzy entropy (FuzzyEn). These three methods measure the time series regularity. Five minutes of recording were acquired with a 148-channel whole-head magnetometer from 36 AD patients and 24 elderly control subjects. Our results showed that MEG activity was more regular in AD patients than in controls. Additionally, FuzzyEn revealed statistically significant differences between the two groups (p <; 0.01, Bonferroni-corrected Mann-Whitney U-test), while ApEn and SampEn did not. The better discriminating results of FuzzyEn in comparison with the other entropy algorithms suggest that it is more efficient for the characterization of MEG activity in AD.

Time-Series Characteristics of Wind Power and Its Impact on Jilin Power Grid

With the rapid development of wind power, the large-scale wind power integration brings a new range of issues in dispatching operation. In order to gain a better grasp of the influence caused by wind power combined to the grid, the paper first establishes the impact characteristic indexes, and then analyzes the regularity of wind power time series in different spatial and temporal scales. At last, according to the analysis results, this paper assesses the impact of time-series characteristics of wind power on power grid, such as the frequency regulation, peak load regulation, which can provide the reference for wind power optimal dispatching of Jilin Power Grid.

Assessing skin blood flow dynamics in older adults using a modified sample entropy approach.

The aging process may result in attenuated microvascular reactivity in response to environmental stimuli, which can be evaluated by analyzing skin blood flow (SBF) signals. Among various methods for analyzing physiological signals, sample entropy (SE) is commonly used to quantify the degree of regularity of time series. However, we found that for temporally correlated data, SE value depends on the sampling rate. When data are oversampled, SE may give misleading results. To address this problem, we propose to modify the definition of SE by using time-lagged vectors in the calculation of the conditional probability that any two vectors of successive data points are within a tolerance r for m points remain within the tolerance at the next point. The lag could be chosen as the first minimum of the auto mutual information function. We tested the performance of modified SE using simulated signals and SBF data. The results showed that modified SE is able to quantify the degree of regularity of the signals regardless of sampling rate. Using this approach, we observed a more regular behavior of blood flow oscillations (BFO) during local heating-induced maximal vasodilation period compared to the baseline in young and older adults and a more regular behavior of BFO in older adults compared to young adults. These results suggest that modified SE may be useful in the study of SBF dynamics.

Evaluating direct and indirect evidence of climatic change by Hölder regularity and order pattern in time series

The several competing reconstructions of Northern Hemisphere temperature variability exists with partly conflicting evidence. Here we considered a specific problem of choosing “the best” reconstruction, in a frame of assumptions, that they are produced by smooth dynamical system as well as instrumental time series. Using some sophisticated measures of similarity, we could not match to all existing hemispheric paleoclimate reconstructions with hypothesis about any deterministic system which can produced its as typical projection of their trajectories. The results cast doubt on the conventional techniques frequently used to reconstruct past climate variability.

Sample entropy characteristics of movement for four foot types based on plantar centre of pressure during stance phase

BackgroundMotion characteristics of CoP (Centre of Pressure, the point of application of the resultant ground reaction force acting on the plate) are useful for foot type characteristics detection. To date, only few studies have investigated the nonlinear characteristics of CoP velocity and acceleration during the stance phase. The aim of this study is to investigate whether CoP regularity is different among four foot types (normal foot, pes valgus, hallux valgus and pes cavus); this might be useful for classification and diagnosis of foot injuries and diseases. To meet this goal, sample entropy, a measure of time-series regularity, was used to quantify the CoP regularity of four foot types.MethodsOne hundred and sixty five subjects that had the same foot type bilaterally (48 subjects with healthy feet, 22 with pes valgus, 47 with hallux valgus, and 48 with pes cavus) were recruited for this study. A Footscan® system was used to collect CoP data when each subject walked at normal and steady speed. The velocity and acceleration in medial-lateral (ML) and anterior-posterior (AP) directions, and resultant velocity and acceleration were derived from CoP. The sample entropy is the negative natural logarithm of the conditional probability that a subseries of length m that matches pointwise within a tolerance r also matches at the next point. This was used to quantify variables of CoP velocity and acceleration of four foot types. The parameters r (the tolerance) and m (the matching length) for sample entropy calculation have been determined by an optimal method.ResultsIt has been found that in order to analyze all CoP parameters of velocity and acceleration during the stance phase of walking gait, for each variable there is a different optimal r value. On the contrary, the value m=4 is optimal for all variables.Sample entropies of both velocity and acceleration in AP direction were highly correlated with their corresponding resultant variables for r>0.91. The sample entropy of the velocity in AP direction was moderately correlated with the one of the acceleration in the same direction (r≥0.673), as well as with the resultant acceleration (r≥0.660). The sample entropy of resultant velocity was moderately correlated with the one of the acceleration in AP direction, as well as with the resultant acceleration (for the both r≥0.689). Moderate correlations were found between variables for the left foot and their corresponding variables for the right foot.Sample entropies of AP velocity, resultant velocity, AP acceleration, and resultant acceleration of the right foot as well as AP velocity and resultant velocity of the left foot were, respectively, significantly different among the four foot types.ConclusionsIt can be concluded that the sample entropy of AP velocity (or the resultant velocity) of the left foot, ML velocity, resultant velocity, ML acceleration and resultant acceleration could serve for evaluation of foot types or selection of appropriate footwear.

Entropy in the Analysis of Gait Complexity: A State of the Art

Human gait is a non-linear complex process requiring appropriate mathematical measuring tools. Entropy is a measure that quantifies regularity in time series: the more predictable a series is the lower the entropy value. The mathematical methods used to estimate entropy have evolved over time. At present, three algorithms are the most used to study human gait complexity: the approximate entropy (ApEn), the sample entropy (SampEn), and the multi-scale entropy (MSE). Most studies on human gait complexity have been conducted on elderly subjects or subjects with specific disorders affecting gait patterns and they used ApEn; but, because of a set of conceptual errors, the ApEn is not the most appropriate algorithm for the analysis of biological signals. Very few studies use SampEn or MSE to analyze human gait variability, but they agree that these algorithms might contribute new perspectives in the analysis of human gait and that MSE seems to be the most sensitive algorithm to changes in gait in healthy subjects.

Finding the Disappearing Dividends: Stock Repurchaser Characteristics and Their Increasing Propensity to Pay

The total propensity to repay capital has risen not fallen as suggested by Fama and French 2001. What has fallen is the time series regularity, in respect to annual capital repayment as firms have shifted from dividend payments to stock repurchases. To understand capital repayment policy I focus my attention on the repayment participation decision of the marginal firm by classifying firms into discrete payment-types and weighting by firms not dollars. Weighting by dollars or using a continuous payment-type variable only answers the question: why do the largest most successful firms pay dividends? That is an interesting question but not the only one. I show that the first firms to employ stock repurchases in the 1980s look like an average of firms that did not repay capital and firms that paid dividends. Over time stock repurchasers have encroached on the territory of dividend payers, supporting the substitution hypothesis and explaining the disappearing dividends of Fama and French 2001. Stock repurchasers have also encroached on the territory of non-payers, helping to increase overall capital repayment rates at the bottom of the firm distribution. After showing the spread of stock repurchases, I document the time series behavior of stock repurchases shows regular patterns despite not occurring annually as dividend payments do. The lower frequency of stock repurchases causes the standard cross-sectional classification of firms that worked well when dividend payments dominated, to undercount stock repurchasers by as much as 50%. To correct this bias I propose an alternative measure: classify firms by their lifetime payment behavior rather than annual payment behavior.

Optimal parameters estimation and input subset for grey model based on chaotic particle swarm optimization algorithm

Optimum prediction is a difficult problem, because there are no optimal models for all forecasting problems. In this paper, the authors attempt to find the high precision prediction for grey forecasting model (GM). Considering that chaotic particle swarm optimization algorithm (CPSO) will not get into local optimum and is easy to implement, the paper develops an approach for grey forecasting model, which is particularly suitable for small sample forecasting, based on chaotic particle swarm optimization and optimal input subset which is a new concept. The input subset of traditional time series consists of the whole original data, but the whole original does not always reflect the internal regularity of time series, so the new optimal subset method is proposed to better reflect the internal characters of time series and improve the prediction precision. The numerical simulation result of financial revenue demonstrates that developed algorithm provides very remarkable results compared to traditional grey forecasting model for small dataset forecasting.

Motor variability in sports: A non-linear analysis of race walking

This aim of this study was to analyse the nature of movement variability and to assess whether entropy measures may represent a valuable synthetic index of neuromuscular organization. The regularity of kinematic/kinetic time series during race walking, the changes in the structure of intra-individual variability over the test session, and the influence of athletic skill in (inter)national rank athletes were investigated. Motion analysis techniques were used. Sample entropy (SampEn) was adopted to examine fluctuations in lower limb angles and ground reaction forces. The regularity of both original and surrogate time series was assessed and compared, by estimating SampEn, to verify the presence of non-linear features in movement variability. SampEn was statistically lower in the original data than in surrogates. In contrast, the regularity of time series did not change significantly throughout the subsequent intra-individual repetitions. Hip and ankle joint angles and vertical ground reaction force manifested increased entropy for skilled athletes. Results suggest that race walking variability was not only the product of random noise but also contained information about the inherent propriety of the neuro-musculo-skeletal system. Furthermore, they provide some indications about neuromuscular control of the lower limb joints during race walking gait, and about the differences between more and less skilled individuals.

An algorithm for analysis of multidimensional time series with smoothly varying regularities and its application

A new approach to revealing regularities in nonstationary k-valued multidimensional time series is proposed. It allows one to discover regularities that are subject to gentle structural changes with time. A measure of similarity between regularities is proposed to describe such changes, and its application in the form of weight in the graph of regularities is discussed. The discovered regularities can be used to predict the subsequent elements in multidimensional time series, to analyze the phenomenon described by this series, and to model the phenomenon. This allows one to use the proposed algorithm in a wide variety of problems concerning prediction of time series and for examining and describing the processes that can be represented by multidimensional time series. Means for direct practical application of the proposed methods of the analysis and prediction of time series are described, and the use of these methods for short-term prediction of model series and a real-life multidimensional time series consisting of the stock prices of companies operating in similar fields is discussed.

Permuted Sample Entropy

Sample entropy (SaEn) was recently developed to quantify the amount of regularity in data. However, the computation of this feature in an online application is infeasible. In this work, we examine a heuristic approach using permuted limited number of samples to estimate SaEn and discuss estimation variability in this context. We conclude that the computation of permuted SaEn provides a fair estimate of time series regularity that can be used in online applications.

Influence of Age on Linear and Nonlinear Measures of Autonomic Cardiovascular Modulation

Age has been identified as an independent risk factor for cardiovascular diseases. In addition, autonomic imbalance toward sympathetic preponderance has been shown to facilitate the occurrence of heart disease. Here, we aimed to assess autonomic modulation of cardiovascular parameters during normal ageing applying well-established linear and novel nonlinear parameters. Linear and nonlinear measures of heart rate variability and complexity as well as measures of QT interval variability and baroreflex sensitivity were obtained from a total of 131 healthy, medication-free participants from a continuous age range between 20 and 90 years, who were allocated to three different age groups. Heart rate variability and complexity significantly decreased with age, while regularity of heart rate time series increased. In addition, QT interval variability linearly increased with age, while baroreflex sensitivity showed a pronounced decrease. Overall, concerning effects of ageing, linear and nonlinear parameters showed equal differentiation between groups. These data indicate a shift of autonomic balance toward sympathetic predominance in higher age groups, limiting the reactiveness of the cardiovascular system to adjust to different demands and increasing the risk for developing tachyarrhythmias.

Time and frequency series combination for non-invasive regularity analysis of atrial fibrillation

In this work, we present a new method based on electrocardiogram signal processing to distinguish between the atrial fibrillation (AF) episodes that terminate immediately and those that sustain. The spectrogram of the atrial activity is computed and 12 numerical series of spectral parameters are constructed. The sample entropy (SampEn) of six series are relevant in the characterization of AF termination (p < 0.05). Furthermore, a combined discriminant analysis in both time and frequency domains is performed, which improves the univariant time-frequency analysis. The discriminant analysis achieves optimal combination of parameters so that the percentage of correctly classified recordings reaches 100% for the learning set and 93.33% for the test set. The main conclusion is that the combined analysis of time and frequency series regularity might be used to predict spontaneous termination of paroxysmal AF and could provide information about the organization of atrial activation in AF.

Measuring time series regularity using nonlinear similarity-based sample entropy

Sampe Entropy (SampEn), a measure quantifying regularity and complexity, is believed to be an effective analyzing method of diverse settings that include both deterministic chaotic and stochastic processes, particularly operative in the analysis of physiological signals that involve relatively small amount of data. However, the similarity definition of vectors is based on Heaviside function, of which the boundary is discontinuous and hard, may cause some problems in the validity and accuracy of SampEn. Sigmoid function is a smoothed and continuous version of Heaviside function. To overcome the problems SampEn encountered, a modified SampEn (mSampEn) based on nonlinear Sigmoid function was proposed. The performance of mSampEn was tested on the independent identically distributed (i.i.d.) uniform random numbers, the MIX stochastic model, the Rossler map, and the Hennon map. The results showed that mSampEn was superior to SampEn in several aspects, including giving entropy definition in case of small parameters, better relative consistency, robust to noise, and more independence on record length when characterizing time series generated from either deterministic or stochastic system with different regularities.

Approximate entropy and auto mutual information analysis of the electroencephalogram in Alzheimer’s disease patients

We analysed the electroencephalogram (EEG) from Alzheimer's disease (AD) patients with two nonlinear methods: approximate entropy (ApEn) and auto mutual information (AMI). ApEn quantifies regularity in data, while AMI detects linear and nonlinear dependencies in time series. EEGs from 11 AD patients and 11 age-matched controls were analysed. ApEn was significantly lower in AD patients at electrodes O1, O2, P3 and P4 (p < 0.01). The EEG AMI decreased more slowly with time delays in patients than in controls, with significant differences at electrodes T5, T6, O1, O2, P3 and P4 (p < 0.01). The strong correlation between results from both methods shows that the AMI rate of decrease can be used to estimate the regularity in time series. Our work suggests that nonlinear EEG analysis may contribute to increase the insight into brain dysfunction in AD, especially when different time scales are inspected, as is the case with AMI.

New Insights into Driving Using Recurrence Quantification Analysis

Traditional measures of central tendency and dispersion, such as the mean and standard deviation, ignore ordering effects in time-series data. Hidden within the ordered regularity of time series' may lie unique human performance characteristics. Recurrence quantification analysis (RQA), a contemporary tool designed for the investigation of nonlinear-time-series data, is used to explore lateral driving movement in a simulated car-following task. This investigation assesses a previously published data set that contrasts baseline driving performance, with performance while legally intoxicated, and hands-free/hand-held cell phone conversation. A number of distinguishing lateral movement characteristics were found using RQA. Free from the constraints imposed by discrete driving measures, RQA has the potential to provide real-time measures of driver workload under a variety of conditions.

Coemergence of regularity and complexity during neural network development

With the growing recognition that rhythmic and oscillatory patterns are widespread in the brain and play important roles in all aspects of the function of our nervous system, there has been a resurgence of interest in neuronal synchronized bursting activity. Here, we were interested in understanding the development of synchronized bursts as information-bearing neuronal activity patterns. For that, we have monitored the morphological organization and spontaneous activity of neuronal networks cultured on multielectrode-arrays during their self-executed evolvement from a mixture of dissociated cells into an active network. Complex collective network electrical activity evolved from sporadic firing patterns of the single neurons. On the system (network) level, the activity was marked by bursting events with interneuronal synchronization and nonarbitrary temporal ordering. We quantified these individual-to-collective activity transitions using newly-developed system level quantitative measures of time series regularity and complexity. We found that individual neuronal activity before synchronization was characterized by high regularity and low complexity. During neuronal wiring, there was a transient period of reorganization marked by low regularity, which then leads to coemergence of elevated regularity and functional (nonstochastic) complexity. We further investigated the morphology-activity interplay by modeling artificial neuronal networks with different topological organizations and connectivity schemes. The simulations support our experimental results by showing increased levels of complexity of neuronal activity patterns when neurons are wired up and organized in clusters (similar to mature real networks), as well as network-level activity regulation once collective activity forms.

Characterization of Surface EMG Signal Based on Fuzzy Entropy

Fuzzy entropy (FuzzyEn), a new measure of time series regularity, was proposed and applied to the characterization of surface electromyography (EMG) signals. Similar to the two existing related measures ApEn and SampEn, FuzzyEn is the negative natural logarithm of the conditional probability that two vectors similar for m points remain similar for the next m + 1 points. Importing the concept of fuzzy sets, vectors' similarity is fuzzily defined in FuzzyEn on the basis of exponential function and their shapes. Besides possessing the good properties of SampEn superior to ApEn, FuzzyEn also succeeds in giving the entropy definition in the case of small parameters. Its performance on characterizing surface EMG signals, as well as independent, identically distributed (i.i.d.) random numbers and periodical sinusoidal signals, shows that FuzzyEn can more efficiently measure the regularity of time series. The method introduced here can also be applied to other noisy physiological signals with relatively short datasets.

The "Chaos Theory" and nonlinear dynamics in heart rate variability analysis: does it work in short-time series in patients with coronary heart disease?

Dynamic analysis techniques may quantify abnormalities in heart rate variability (HRV) based on nonlinear and fractal analysis (chaos theory). The article emphasizes clinical and prognostic significance of dynamic changes in short-time series applied on patients with coronary heart disease (CHD) during the exercise electrocardiograph (ECG) test. The subjects were included in the series after complete cardiovascular diagnostic data. Series of R-R and ST-T intervals were obtained from exercise ECG data after sampling digitally. The range rescaled analysis method determined the fractal dimension of the intervals. To quantify fractal long-range correlation's properties of heart rate variability, the detrended fluctuation analysis technique was used. Approximate entropy (ApEn) was applied to quantify the regularity and complexity of time series, as well as unpredictability of fluctuations in time series. It was found that the short-term fractal scaling exponent (alpha(1)) is significantly lower in patients with CHD (0.93 +/- 0.07 vs 1.09 +/- 0.04; P < 0.001). The patients with CHD had higher fractal dimension in each exercise test program separately, as well as in exercise program at all. ApEn was significant lower in CHD group in both RR and ST-T ECG intervals (P < 0.001). The nonlinear dynamic methods could have clinical and prognostic applicability also in short-time ECG series. Dynamic analysis based on chaos theory during the exercise ECG test point out the multifractal time series in CHD patients who loss normal fractal characteristics and regularity in HRV. Nonlinear analysis technique may complement traditional ECG analysis.

Utility of Approximate Entropy From Overnight Pulse Oximetry Data in the Diagnosis of the Obstructive Sleep Apnea Syndrome

Approximate entropy (ApEn) is a family of statistics introduced as a quantification of regularity in time series without any a priori knowledge about the system generating them. The aim of this preliminary study was to assess whether a time series analysis of arterial oxygen saturation (SaO2) signals from overnight pulse oximetry by means of ApEn could yield essential information on the diagnosis of obstructive sleep apnea (OSA) syndrome. We analyzed SaO2 signals from 187 subjects: 111 with a positive diagnosis of OSA and 76 with a negative diagnosis of OSA. We divided our data in a training set (44 patients with OSA Positive and 30 patients with OSA Negative) and a test set (67 patients with OSA Positive and 46 patients with OSA Negative). The training set was used for algorithm development and optimum threshold selection. Results showed that recurrence of apnea events in patients with OSA determined a significant increase in ApEn values. This method was assessed prospectively using the test dataset, where we obtained 82.09% sensitivity and 86.96% specificity. We conclude that ApEn analysis of SaO2 from pulse oximetric recording could be useful in the study of OSA.