Correlation dimension and entropy in the assessment of sex differences based on human gait data

Introduction

It is proved that there are differences between gait performed by females and males, which appear in movements of selected body parts. Despite numerous state-of-the-art studies related to the discriminative analysis of motion capture data, the question of whether measures of signal complexity and uncertainty can extract valuable features for the problem of sex distinction still remains open. It is the subject of the paper.

Methods

Correlation dimension, as well as approximate and sample entropies, are selected to describe motion data. In the numerical experiments, the collected dataset with 884 samples of 25 females and 30 males was used. The measurements took place in the Human Motion Laboratory (HML), equipped with a highly precise motion capture system. Two variants of data representation were investigated-time series that contain joint rotations of taken skeleton model as well as positions of the markers attached to the human body. Finally, a comparative analysis between the populations of females and males using descriptive statistics, non-parametric estimation, and statistical hypotheses verification was carried out.

Results

There are statistically significant sex differences extracted by the taken measures. In general, the movements of lower limbs result in greater values of correlation dimension and entropies for females, while selected upper body parts play a similar role for males. The dissimilarities are mainly observed in hip, ankle, shoulder, and head movements.

Discussion

Correlation dimension and entropy measures provide robust and explainable features of motion capture data with a valuable description of the human locomotion system. Thus, beyond the importance of discovered differences between females and males, their interpretation and understanding are also known.

Effects of Motor Task Difficulty on Postural Control Complexity during Dual Tasks in Young Adults: A Nonlinear Approach

Few studies have evaluated the effect of a secondary motor task on the standing posture based on nonlinear analysis. However, it is helpful to extract information related to the complexity, stability, and adaptability to the environment of the human postural system. This study aimed to analyze the effect of two motor tasks with different difficulty levels in motor performance complexity on the static standing posture in healthy young adults. Thirty-five healthy participants (23.08 ± 3.92 years) performed a postural single task (ST: keep a quiet standing posture) and two motor dual tasks (DT). i.e., mot-DT(A)—perform the ST while performing simultaneously an easy motor task (taking a smartphone out of a bag, bringing it to the ear, and putting it back in the bag)—and mot-DT(T)—perform the ST while performing a concurrent difficult motor task (typing on the smartphone keyboard). The approximate entropy (ApEn), Lyapunov exponent (LyE), correlation dimension (CoDim), and fractal dimension (detrending fluctuation analysis, DFA) for the mediolateral (ML) and anterior-posterior (AP) center-of-pressure (CoP) displacement were measured with a force plate while performing the tasks. A significant difference was found between the two motor dual tasks in ApEn, DFA, and CoDim-AP (p < 0.05). For the ML CoP direction, all nonlinear variables in the study were significantly different (p < 0.05) between ST and mot-DT(T), showing impairment in postural control during mot-DT(T) compared to ST. Differences were found across ST and mot-DT(A) in ApEn-AP and DFA (p < 0.05). The mot-DT(T) was associated with less effectiveness in postural control, a lower number of degrees of freedom, less complexity and adaptability of the dynamic system than the postural single task and the mot-DT(A).

Impact of Joint Fixation on Postural Dynamics during Single-Leg Stance

We investigated the relationship between the mechanical degrees of freedom (DoF) and its postural dynamics. The joint DoF was fixed to constrain the mechanical DoF. Nine participants were required to perform a single-leg stance task. The center of pressure trajectory data was analyzed. Ankle fixation induced a larger amount of variability in the anteroposterior direction, and less dimensionality and complexity in the mediolateral direction. These results suggest that the ankle joint fixation caused limited postural sway in the mediolateral direction; therefore, functional DoF and complexity decreased. In contrast, it increased the amount of postural sway variability in the anteroposterior direction. Our findings imply a direct relationship between the mechanical DoF of the human movement system and its postural dynamics.

Attractor dynamics of elite performance: the high bar longswing

Combining biomechanics and motor control, the aim of this study was to investigate the limit cycle dynamics during the high bar longswing across the UK elite gymnastics pathway age groupings. Senior, junior and development gymnasts (N = 30) performed three sets of eight consecutive longswings on the high bar. The centre of mass motion was examined through Poincaré plots and recurrence quantification analysis exploring the limit cycle dynamics of the longswing. Close to one-dimensional limit cycles were displayed for the senior (correlation dimension (CD) = 1.17 ± .08), junior (CD = 1.26 ± .08) and development gymnasts (CD = 1.33 ± .14). Senior elite gymnasts displayed increased recurrence characteristics in addition to longer longswing duration (p < .01) and lower radial angular velocity of the mass centre (p < .01). All groups of gymnasts had highly recurrent and predictable limit cycle characteristics. The findings of this research support the postulation that the further practice, experience and individual development associated with the senior gymnasts contribute to the refinement of the longswing from a nonlinear dynamics perspective. These findings support the idea of functional task decomposition informing the understanding of skill and influencing coaches' decisions around skill development and physical preparation.

An Extended Correlation Dimension of Complex Networks

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.

Nonlinear analysis of scalp EEGs from normal and brain tumour subjects

Measurement of features from the chaos theory or as popularly known, the concept of nonlinear dynamics, as indicatives of several pathological conditions and cognition states using the electroencephalography (EEG) signal is very popular. In this paper, the analysis of scalp EEG signals of normal subjects and brain tumour patients using the nonlinear dynamic features has been presented. The nonlinear dynamic features that represent the dimensional and waveform complexities of the signal being analyzed have been considered. The statistical analysis of the selected nonlinear dynamic features has been presented. The results show that the nonlinear dynamic features significantly discriminate the brain tumour group from the normal group.

Estimating a Sleep Apnea Hypopnea Index Based on the ERB Correlation Dimension of Snore Sounds

This paper proposes a new perspective of analyzing non-linear acoustic characteristics of the snore sounds. According to the ERB (Equivalent Rectangular Bandwidth) scale used in psychoacoustics, the ERB correlation dimension (ECD) of the snore sound was computed to feature different severity levels of sleep apnea hypopnea syndrome (SAHS). For the training group of 93 subjects, snore episodes were manually segmented and the ECD parameters of the snores were extracted, which established the gaussian mixture models (GMM). The nocturnal snore sound of the testing group of another 120 subjects was tested to detect SAHS snores, thus estimating the apnea hypopnea index (AHI), which is called AHIECD. Compared to the AHIPSG value of the gold standard polysomnography (PSG) diagnosis, the estimated AHIECD achieved an accuracy of 87.5% in diagnosis the SAHS severity levels. The results suggest that the ECD vectors can be effective parameters for screening SAHS.

A Comprehensive Framework for Uncovering Non-Linearity and Chaos in Financial Markets: Empirical Evidence for Four Major Stock Market Indices

The presence of chaos in the financial markets has been the subject of a great number of studies, but the results have been contradictory and inconclusive. This research tests for the existence of nonlinear patterns and chaotic nature in four major stock market indices: namely Dow Jones Industrial Average, Ibex 35, Nasdaq-100 and Nikkei 225. To this end, a comprehensive framework has been adopted encompassing a wide range of techniques and the most suitable methods for the analysis of noisy time series. By using daily closing values from January 1992 to July 2013, this study employs twelve techniques and tools of which five are specific to detecting chaos. The findings show no clear evidence of chaos, suggesting that the behavior of financial markets is nonlinear and stochastic.

Complexity Analysis of Surface Electromyography for Assessing the Myoelectric Manifestation of Muscle Fatigue: A Review

The surface electromyography (sEMG) records the electrical activity of muscle fibers during contraction: one of its uses is to assess changes taking place within muscles in the course of a fatiguing contraction to provide insights into our understanding of muscle fatigue in training protocols and rehabilitation medicine. Until recently, these myoelectric manifestations of muscle fatigue (MMF) have been assessed essentially by linear sEMG analyses. However, sEMG shows a complex behavior, due to many concurrent factors. Therefore, in the last years, complexity-based methods have been tentatively applied to the sEMG signal to better individuate the MMF onset during sustained contractions. In this review, after describing concisely the traditional linear methods employed to assess MMF we present the complexity methods used for sEMG analysis based on an extensive literature search. We show that some of these indices, like those derived from recurrence plots, from entropy or fractal analysis, can detect MMF efficiently. However, we also show that more work remains to be done to compare the complexity indices in terms of reliability and sensibility; to optimize the choice of embedding dimension, time delay and threshold distance in reconstructing the phase space; and to elucidate the relationship between complexity estimators and the physiologic phenomena underlying the onset of MMF in exercising muscles.

On Geometry of Information Flow for Causal Inference

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore, contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write GeoCy→x. This avoids some of the boundedness issues that we show exist for the transfer entropy, Ty→x. We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the Hénon map example, and finally a real physiological example relating breathing and heart rate function.

Classification of Pain Event Related Potential for Evaluation of Pain Perception Induced by Electrical Stimulation

Variability in individual pain sensitivity is a major problem in pain assessment. There have been studies reported using pain-event related potential (pain-ERP) for evaluating pain perception. However, none of them has achieved high accuracy in estimating multiple pain perception levels. A major reason lies in the lack of investigation of feature extraction. The goal of this study is to assess four different pain perception levels through classification of pain-ERP, elicited by transcutaneous electrical stimulation on healthy subjects. Nonlinear methods: Higuchi's fractal dimension, Grassberger-Procaccia correlation dimension, with auto-correlation, and moving variance functions were introduced into the feature extraction. Fisher score was used to select the most discriminative channels and features. As a result, the correlation dimension with a moving variance without channel selection achieved the best accuracies of 100% for both the two-level and the three-level classification but degraded to 75% for the four-level classification. The best combined feature group is the variance-based one, which achieved accuracy of 87.5% and 100% for the four-level and three-level classification, respectively. Moreover, the features extracted from less than 20 trials could not achieve sensible accuracy, which makes it difficult for an instantaneous pain perception levels evaluation. These results show strong evidence on the possibility of objective pain assessment using nonlinear feature-based classification of pain-ERP.

Chaos-Based Analysis of Heart Rate Variability Time Series in Obstructive Sleep Apnea Subjects

Obstructive sleep apnea (OSA) is a common disorder which can cause periodic fluctuations in heart rate. To diagnose sleep apnea, some studies analyze electrocardiogram (ECG) signals by adopting chaos-based analysis. This research is going to specifically focus on whether it is possible to use chaos-based analysis of heart rate variability (HRV) signals rather than using chaotic analysis of ECG signals to diagnose OSA. While conventional studies mostly use chaos-based analysis of ECG signals to detect OSA, here, we apply correlation dimension (CD) as a chaotic index to analyze HRV data in OSA patients. For this purpose, 17 patients with OSA and 9 healthy individuals referred to a sleep clinic in Isfahan/Iran are studied, and their HRV time series were extracted from 1-h ECG signals recorded overnight. The preliminary step to calculate CD is phase-space reconstruction of the system based on HRV time series. Corresponding parameters, including embedding dimension and lag time, are estimated optimally using enhanced related methods, and then CD is calculated using Grassberger–Procaccia algorithm. Moreover, to evaluate our results, detrended fluctuation analysis (DFA), one of the well-known nonlinear methods in HRV analysis to detect OSA, is also applied to our data and the result is compared with those obtained from CD analysis of HRV. CD index with P < 0.005 indicates a significant difference in nonlinear dynamics of HRV signals detected from OSA patients and healthy individuals.

Correlation Dimension Detects Causal Links in Coupled Dynamical Systems

It is becoming increasingly clear that causal analysis of dynamical systems requires different approaches than, for example, causal analysis of interconnected autoregressive processes. In this study, a correlation dimension estimated in reconstructed state spaces is used to detect causality. If deterministic dynamics plays a dominant role in data then the method based on the correlation dimension can serve as a fast and reliable way to reveal causal relationships between and within the systems. This study demonstrates that the method, unlike most other causal approaches, detects causality well, even for very weak links. It can also identify cases of uncoupled systems that are causally affected by a hidden common driver.

Mental Workload Alters Heart Rate Variability, Lowering Non-linear Dynamics

Mental workload is known to alter cardiovascular function leading to increased cardiovascular risk. Nevertheless, there is no clear autonomic nervous system unbalance to be quantified during mental stress. We aimed to characterize the mental workload impact on the cardiovascular function with a focus on heart rate variability (HRV) non-linear indexes. A 1-h computerized switching task (letter recognition) was performed by 24 subjects while monitoring their performance (accuracy, response time), electrocardiogram and blood pressure waveform (finger volume clamp method). The HRV was evaluated from the beat-to-beat RR intervals (RRI) in time-, frequency-, and informational- domains, before (Control) and during the task. The task induced a significant mental workload (visual analog scale of fatigue from 27 ± 26 to 50 ± 31 mm, p < 0.001, and NASA-TLX score of 56 ± 17). The heart rate, blood pressure and baroreflex function were unchanged, whereas most of the HRV parameters markedly decreased. The maximum decrease occurred during the first 15 min of the task (P1), before starting to return to the baseline values reached at the end of the task (P4). The RRI dimension correlation (D2) decrease was the most significant (P1 vs. Control: 1.42 ± 0.85 vs. 2.21 ± 0.8, p < 0.001) and only D2 lasted until the task ended (P4 vs. Control: 1.96 ± 0.9 vs. 2.21 ± 0.9, p < 0.05). D2 was identified as the most robust cardiovascular variable impacted by the mental workload as determined by posterior predictive simulations (p = 0.9). The Spearman correlation matrix highlighted that D2 could be a marker of the generated frustration (R = -0.61, p < 0.01) induced by a mental task, as well as the myocardial oxygen consumption changes assessed by the double product (R = -0.53, p < 0.05). In conclusion, we showed that mental workload sharply lowered the non-linear RRI dynamics, particularly the RRI correlation dimension.

[Study on the property of correlation dimension of sleep apnea syndrome electroencephalogram]

Sleep apnea syndrome (SAS) is a kind of common and harmful systemic sleep disorder. SAS patients have significant iconography changes in brain structure and function, and electroencephalogram (EEG) is the most intuitive parameter to describe the sleep process which can reflect the electrical activity and function of brain tissues. Based on the non-stationary and nonlinear characteristics of EEG, this paper analyzes the correlation dimension of sleep EEG in patients with SAS. Six SAS patients were classed as SAS group and six healthy persons were classified into a control group. The results showed that the correlation dimension of sleep EEG in the SAS group and the control group decreased gradually with the deepening of sleep, and then increased to the level of awake and light sleep stage with rapid eye movement (REM). The correlation dimension of SAS group was significantly lower than that of control group ( P<0.01) throughout all the stages. The results suggested that there were significant nonlinear dynamic differences between the EEG signals of SAS patients and of healthy people, which provided a new direction for the study of the physiological mechanism and automatic detection of SAS.

Complexity and schizophrenia

Complexity estimators have been broadly utilized in schizophrenia investigation. Early studies reported increased complexity in schizophrenia patients, associated with a higher variability or "irregularity" of their brain signals. However, further investigations showed reduced complexities, thus introducing a clear divergence. Nowadays, both increased and reduced complexity values are reported. The explanation of such divergence is a critical issue to understand the role of complexity measures in schizophrenia research. Considering previous arguments a complementary hypothesis is advanced: if the increased irregularity of schizophrenia patients' neurophysiological activity is assumed, a "natural" tendency to increased complexity in EEG and MEG scans should be expected, probably reflecting an abnormal neuronal firing pattern in some critical regions such as the frontal lobes. This "natural" tendency to increased complexity might be modulated by the interaction of three main factors: medication effects, symptomatology, and age effects. Therefore, young, medication-naïve, and highly symptomatic (positive symptoms) patients are expected to exhibit increased complexities. More importantly, the investigation of these interacting factors by means of complexity estimators might help to elucidate some of the neuropathological processes involved in schizophrenia.

Complexity of spontaneous brain activity in mental disorders

Recent reports of functional and anatomical studies have provided evidence that aberrant neural connectivity lies at the heart of many mental disorders. Information related to neural networks has elucidated the nonlinear dynamical complexity in brain signals over a range of temporal scales. The recent advent of nonlinear analytic methods, which have served for the quantitative description of the brain signal complexity, has provided new insights into aberrant neural connectivity in many mental disorders. Although many studies have underpinned aberrant neural connectivity, findings related to complexity behavior are still inconsistent. This inconsistency might result from (i) heterogeneity in mental disorders, (ii) analytical issues, (iii) interference of typical development and aging. First, most mental disorders are heterogeneous in their clinical feature or intrinsic pathological mechanisms. Second, neurophysiologic output signals from complex brain connectivity might be characterized with multiple time scales or frequencies. Finally, age-related brain complexity changes must be considered when investigating pathological brain because typical brain complexity is not constant across generations. Future systematic studies addressing these issues will greatly expand our knowledge of neural connections and dynamics related to mental disorders.

Development of the navigational system in homing pigeons: increase in complexity of the navigational map

In the present study we analysed GPS-recorded tracks from pigeons of different ages from 11 sites between 3.6 and 22.1 km from the home loft, which revealed changes in the navigational system as the birds grew older and became more experienced. The efficiency of juveniles in their first year of life, at only 0.27, was rather low, indicating that the young birds covered more than three times the direct distance home. In the second year, after a standard training programme, the efficiency of the same birds increased to 0.80 and was no longer different from that of older pigeons. The short-term correlation dimension, a variable that reflects the number of factors involved in the navigational process, also increased with age. In juveniles, it was markedly lower than in the other two groups, but even in yearlings it was still significantly lower than that of old pigeons, indicating that the navigational map of yearlings is still developing. Our results indicate that the map system, although functional in the first year of life, continues to become more complex - older pigeons seem to either consider more navigational factors than younger ones or at least weigh the same factors differently.

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Acoustic analysis of the tremulous voice: assessing the utility of the correlation dimension and perturbation parameters

Acoustic analysis may provide a useful means to quantitatively characterize the tremulous voice. Signals were obtained from 25 subjects with diagnoses of either Parkinson's disease or vocal polyps exhibiting vocal tremor. These were compared to signals from 24 subjects with normal voices. Signals were analyzed via correlation dimension and several parameters from the Multi-Dimensional Voice Program (MDVP): percent jitter, percent shimmer, amplitude tremor intensity index (ATRI), frequency tremor intensity index (FTRI), amplitude tremor frequency (Fatr), and fundamental frequency tremor frequency (Fftr). No significant difference was found between the tremor and control groups for ATRI and Fatr. Percent jitter, percent shimmer, FTRI, Fftr, and correlation dimension values were found to be significantly higher in the tremor group than in the control group. We conclude that these parameters may have utility for the clinical quantification of tremor severity and treatment effects.

LEARNING OUTCOMES

The reader will understand the utility of applying select perturbation parameters and the nonlinear measure of correlation dimension for the characterization of the tremulous voice.

Circadian rhythmic fractal scaling of heart rate variability in health and coronary artery disease

BACKGROUND

In clinical cardiology, heart rate variability is a putative index of autonomic cardiovascular function. Signs of reduced vagal activity are not only associated with an enhanced risk of sudden cardiac death, but such impaired heart rate variability became a new predictor of sudden cardiac death and other mortality in patients with a variety of diseased states.

HYPOTHESIS

It is postulated (1) that the time structure (chronome) of heart rate variability in clinical health includes a circadian rhythm and deterministic chaos, the latter gauged by the correlation dimensions of RR intervals; and (2) that this chronome is altered in patients with coronary artery disease (CAD).

METHODS

From 24-h Holter records of 11 healthy controls and 10 patients with CAD, 500-s sections around 02:00, 06:00, 10:00, 14:00, 18:00 and 22:00 hours were analyzed for smoothed RR intervals sampled at 4 Hz. Correlation integrals were estimated for embedding dimensions from 1 to 20 with a 1.0-s time lag, using an algorithm modified from Grassberger and Procaccia. The Wilcoxon signed-rank test compares circadian end points assessed by cosinor between the CAD patients and age-matched controls.

RESULTS

A circadian rhythm characterizes the correlation dimension of healthy subjects peaking during the night (p < 0.005). Patients with CAD have a lowered correlation dimension (p < 0.05) and an altered circadian variation which requires the consideration of an approximately 12-h (circasemidian) component.

CONCLUSION

The results demonstrate the sensitivity of circadian rhythms for the detection of disease. A partial 24- to 12-h (circadian-to-circasemidian) frequency multiplication (or partial variance transposition) in CAD of the correlation dimension, apart from being a potential clue to the etiology of the disease, adds a new feature to a chronocardiology combining, with the fractal scaling, an assessment of circadian and circasemidian components as measures of predictable variability to be tested for use in diagnosis, prognosis, and as putative guides to treatment timing.