Effects of Match Location,Quality of Opposition and Match Outcome on Match Running Performance in a Portuguese Professional Football Team

The aim of this study was to analyze the effects of match location, quality of opposition and match outcome on match running performance according to playing position in a Portuguese professional football team. Twenty-three male professional football players were monitored from eighteen Portuguese Football League matches during the 2019–2020 season. Global positioning system technology (GPS) was used to collect time-motion data. The match running performance was obtained from five playing positions: central defenders (CD), fullbacks (FB), central midfielders (CM), wide midfielders (WM) and forwards (FW). Match running performance was analyzed within specific position and contextual factors using one-way analysis of variance (ANOVA) for repeated measures, standardized (Cohen) differences and smallest worthwhile change. CM and WM players covered significantly greater total distance (F = 15.45, p = 0.000, η² = 0.334) and average speed (F = 12.79, p < 0.001, η² = 0.294). WM and FB players covered higher distances at high-speed running (F = 16.93, p = 0.000, η² = 0.355) and sprinting (F = 13.49; p < 0.001, η² = 0.305). WM players covered the highest number of accelerations (F = 4.69, p < 0.001, η² = 0.132) and decelerations (F = 12.21, p < 0.001, η² = 0.284). The match running performance was influenced by match location (d = 0.06–2.04; CI: −0.42–2.31; SWC = 0.01–1.10), quality of opposition (d = 0.13–2.14; CI: –0.02–2.60; SWC = 0.01–1.55) and match outcome (d = 0.01–2.49; CI: −0.01–2.31; SWC = 0.01–0.35). Contextual factors influenced the match running performance with differential effects between playing positions. This study provides the first report about the contextual influence on match running performance in a Portuguese professional football team. Future research should also integrate tactical and technical key indicators when analyzing the match-related contextual influence on match running performance.     

The “Hockey” Assist Makes the Difference—Validation of a Defensive Disruptiveness Model to Evaluate Passing Sequences in Elite Soccer

With the growing availability of position data in sports, spatiotemporal analysis in soccer is a topic of rising interest. The aim of this study is to validate a performance indicator, namely D-Def, measuring passing effectiveness. D-Def calculates the change of the teams’ centroid, centroids of formation lines (e.g., defensive line), teams’ surface area, and teams’ spread in the following three seconds after a pass and therefore results in a measure of disruption of the opponents’ defense following a pass. While this measure was introduced earlier, in this study we aim to prove the usefulness to evaluate attacking sequences. In this study, 258 games of Dutch Eredivisie season 2018/19 were included, resulting in 13,094 attacks. D-Def, pass length, pass velocity, and pass angle of the last four passes of each attack were calculated and compared between successful and unsuccessful attacks. D-Def showed higher values for passes of successful compared to unsuccessful attacks (0.001 < p ≤ 0.029, 0.06 ≤ d ≤ 0.23). This difference showed the highest effects sizes in the penultimate pass (d = 0.23) and the maximal D-Def value of an attack (d = 0.23). Passing length (0.001 < p ≤ 0.236, 0.08 ≤ d ≤ 0.17) and passing velocity (0.001 < p ≤ 0.690, −0.09 ≤ d ≤ 0.12) showed inconsistent results in discriminating between successful and unsuccessful attacks. The results indicate that D-Def is a useful indicator for the measurement of pass effectiveness in attacking sequences, highlighting that successful attacks are connected to disruptive passing. Within successful attacks, at least one high disruptive action (pass with D-Def > 28) needs to be present. In addition, the penultimate pass (“hockey assist”) of an attack seems crucial in characterizing successful attacks.     

Reduced System Complexity of Heart Rate Dynamics in Patients with Hyperthyroidism: A Multiscale Entropy Analysis

Studying heart rate dynamics would help understand the effects caused by a hyperkinetic heart in patients with hyperthyroidism. By using a multiscale entropy (MSE) analysis of heart rate dynamics derived from one-channel electrocardiogram recording, we aimed to compare the system complexity of heart rate dynamics between hyperthyroid patients and control subjects. A decreased MSE complexity index (CI) computed from MSE analysis reflects reduced system complexity. Compared with the control subjects (n = 37), the hyperthyroid patients (n = 37) revealed a significant decrease (p < 0.001) in MSE CI (hyperthyroid patients 10.21 ± 0.37 versus control subjects 14.08 ± 0.21), sample entropy for each scale factor (from 1 to 9), and high frequency power (HF) as well as a significant increase (p < 0.001) in low frequency power (LF) in normalized units (LF%) and ratio of LF to HF (LF/HF). In conclusion, besides cardiac autonomic dysfunction, the system complexity of heart rate dynamics is reduced in hyperthyroidism. This finding implies that the adaptability of the heart rate regulating system is impaired in hyperthyroid patients. Additionally, it might explain the exercise intolerance experienced by hyperthyroid patients. In addition, hyperthyroid patients and control subjects could be distinguished by the MSE CI computed from MSE analysis of heart rate dynamics.     

Cardio-Hypothalamic-Pituitary Coupling during Rest and in Response to Exercise

The objective of this study was to examine cardio hypothalamic-pituitary coupling and to better understand how the temporal relations between these systems are altered during rest and exercise conditions. An intensive within subjects study design was used. Seven adult males completed two visits, each consisting of either a 24 h period of complete rest or a 24 h period containing a high-intensity exercise bout. An intravenous catheter was used to collect serum samples every 10 min throughout the 24 h period (i.e., 145 samples/person/condition) to assess growth hormone (GH) dynamics throughout the 24 h period. Cardiac dynamics were also collected throughout the 24 h period and epoched into 3 min windows every 10 min, providing serial short-time measurements of heart rate variability (HRV) concurrent to the GH sampling. The standard deviation of the normal RR interval (SDNN), the root mean square of successive differences (rMSSD), and sample entropy (SampEn) was calculated for each epoch and used to create new profiles. The dynamics of these profiles were individually quantified using SampEn and recurrence quantification analysis (RQA). To address our central question, the coupling between these profiles with GH was assessed using cross-SampEn and cross-RQA (cRQA). A comparison between the epoched HRV profiles indicated a main effect between profiles for sample entropy (p < 0.001) and several measures from RQA. An interaction between profile and condition was observed for cross-SampEn (p = 0.04) and several measures from cRQA. These findings highlight the potential application of epoched HRV to assess changes in cardiac dynamics, with specific applications to assessing cardio hypothalamic-pituitary coupling.     

Effects of Therapeutic Ultrasound on Intramuscular Blood Circulation and Oxygen Dynamics

Purpose: This study aimed to clarify the effects of therapeutic ultrasound on intramuscular local blood circulation (and oxygen dynamics) using near-infrared spectroscopy (NIRS). Participants: The participants were 11 healthy males. Methods: All participants performed all three trials; (1) the ultrasound (US group), (2) without powered ultrasound (placebo group), and (3) rest (control group). Ultrasound was applied at 3 MHz, 1.0 W/cm², and 100% duty cycle for 10 minutes. Evaluation index were oxygenated, deoxygenated, and total hemoglobin (Hb) concentrations in the intramuscular and skin surface temperature (SST). The experimental protocol was a total of 40 minutes, that is, 10 minutes before trial (rest), 10 minutes during the trial (ultrasound, placebo, and control), and 20 minutes after trial (rest). The NIRS and SST data collected before and after the trial were divided into 5 minutes intervals for further analysis. Results: Oxygenated and total hemoglobin levels were significantly higher in the US group than in the placebo and control groups for the 20 minutes after ultrasound (p < 0.01). The SST was significantly higher in the US group than in the control for 15 minutes after ultrasound (p < 0.05), while it was significantly lower in the placebo group than in the US and control groups for 20 minutes after the trials (p < 0.01). Conclusion: The effects of ultrasound were maintained for 20 minutes after the trial on intramuscular blood circulation and oxygen dynamics. These effects were caused by a combination of thermal and mechanical effects of the ultrasound.     

Microwave spectrum and structure of ethylene ozonide: Effects of large axes rotations in structure calculations

Rotational spectra for 14 isotopic species of ethylene ozonide have now been assigned. The consistency of the Kraitchman substitution structure was checked by calculating the O_p---O_p bond distance six ways; the values ranged from 1.458 to 1.502 Å. This variation was attributed to an amplification of residual vibrational effects by large axes rotations upon isotopic substitution. Estimates of errors produced from this effect were made and a procedure was developed for choosing r_s parameters in which the effect is minimized. This gave the following ring parameters: d(CO_e) = 1.416 Å, d(CO_p) = 1.412 Å, d(OO) = 1.461 Å, <CO_eC = 104.8°, <O_eCO_p = 105.5°, <CO_pO_p = 99.3°.     

On the Variability of Heart Rate Variability—Evidence from Prospective Study of Healthy Young College Students

Heart rate variability (HRV) has been widely used as indices for autonomic regulation, including linear analyses, entropy and multi-scale entropy based nonlinear analyses, and however, it is strongly influenced by the conditions under which the signal is being recorded. To investigate the variability of healthy HRV under different settings, we recorded electrocardiograph (ECG) signals from 56 healthy young college students (20 h for each participant) at campus using wearable single-lead ECG device. Accurate R peak to R peak (RR) intervals were extracted by combing the advantages of five commonly used R-peak detection algorithms to eliminate data quality influence. Thorough and detailed linear and nonlinear HRV analyses were performed. Variability of HRV metrics were evaluated from five categories: (1) different states of daily activities; (2) different recording time period in the same day during free-running daily activities; (3) body postures of sitting and lying; (4) lying on the left, right and back; and (5) gender influence. For most of the analyzed HRV metrics, significant differences (p < 0.05) were found among different recording conditions within the five categories except lying on different positions. Results suggested that the standardization of ECG data collection and HRV analysis should be implemented in HRV related studies, especially for entropy and multi-scale entropy based analyses. Furthermore, this preliminary study provides reference values of HRV indices under various recording conditions of healthy young subjects that could be useful information for different applications (e.g., health monitoring and management).     

The XY model and Kadanoff's variational real space renormalization group

The XY model in d dimensions is studied by means of a variational real space renormalization group transformation. Contrary to an earlier computation in the same framework for the d = 2 case, we find that a low order operator basis truncation is highly unstable. For certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime. The behavior in the d = 3 case is very similar.     

Statistical Inference for Ergodic Algorithmic Model (EAM), Applied to Hydrophobic Hydration Processes

The thermodynamic properties of hydrophobic hydration processes can be represented in probability space by a Dual-Structure Partition Function {DS-PF} = {M-PF} · {T-PF}, which is the product of a Motive Partition Function {M-PF} multiplied by a Thermal Partition Function {T-PF}. By development of {DS-PF}, parabolic binding potential functions α) RlnK_dual = (−ΔG°_dual/T) ={f(1/T)*g(T)} and β) RTlnK_dual = (−ΔG°_dual) = {f(T)*g(lnT)} have been calculated. The resulting binding functions are “convoluted” functions dependent on the reciprocal interactions between the primary function f(1/T) or f(T) with the secondary function g(T) or g(lnT), respectively. The binding potential functions carry the essential thermodynamic information elements of each system. The analysis of the binding potential functions experimentally determined at different temperatures by means of the Thermal Equivalent Dilution (TED) principle has made possible the evaluation, for each compound, of the pseudo-stoichiometric coefficient ±ξ_w, from the curvature of the binding potential functions. The positive value indicates convex binding functions (Class A), whereas the negative value indicates concave binding function (Class B). All the information elements concern sets of compounds that are very different from one set to another, in molecular dimension, in chemical function, and in aggregation state. Notwithstanding the differences between, surprising equal unitary values of niche (cavity) formation in Class A <Δh_for>_A = −22.7 ± 0.7 kJ·mol⁻¹ ·ξ_w⁻¹ sets with standard deviation σ = ±3.1% and <Δs_for>_A = −445 ± 3J·K⁻¹·mol⁻¹·ξ_w⁻¹J·K⁻¹·mol⁻¹·ξ_w⁻¹ with standard deviation σ = ±0.7%. Other surprising similarities have been found, demonstrating that all the data analyzed belong to the same normal statistical population. The Ergodic Algorithmic Model (EAM) has been applied to the analysis of important classes of reactions, such as thermal and chemical denaturation, denaturation of proteins, iceberg formation or reduction, hydrophobic bonding, and null thermal free energy. The statistical analysis of errors has shown that EAM has a general validity, well beyond the limits of our experiments. Specifically, the properties of hydrophobic hydration processes as biphasic systems generating convoluted binding potential functions, with water as the implicit solvent, hold for all biochemical and biological solutions, on the ground that they also are necessarily diluted solutions, statistically validated.     

Using Bidimensional Multiscale Entropy Analysis of Ultrasound Images to Assess the Effect of Various Walking Intensities on Plantar Soft Tissues

Walking performance is usually assessed by linear analysis of walking outcome measures. However, human movements consist of both linear and nonlinear complexity components. The purpose of this study was to use bidimensional multiscale entropy analysis of ultrasound images to evaluate the effects of various walking intensities on plantar soft tissues. Twelve participants were recruited to perform six walking protocols, consisting of three speeds (slow at 1.8 mph, moderate at 3.6 mph, and fast at 5.4 mph) for two durations (10 and 20 min). A B-mode ultrasound was used to assess plantar soft tissues before and after six walking protocols. Bidimensional multiscale entropy (MSE_2D) and the Complexity Index (CI) were used to quantify the changes in irregularity of the ultrasound images of the plantar soft tissues. The results showed that the CI of ultrasound images after 20 min walking increased when compared to before walking (CI₄: 0.39 vs. 0.35; CI₅: 0.48 vs. 0.43, p < 0.05). When comparing 20 and 10 min walking protocols at 3.6 mph, the CI was higher after 20 min walking than after 10 min walking (CI₄: 0.39 vs. 0.36, p < 0.05; and CI₅: 0.48 vs. 0.44, p < 0.05). This is the first study to use bidimensional multiscale entropy analysis of ultrasound images to assess plantar soft tissues after various walking intensities.     

The Value of Heart Rhythm Complexity in Identifying High-Risk Pulmonary Hypertension Patients

Pulmonary hypertension (PH) is a fatal disease—even with state-of-the-art medical treatment. Non-invasive clinical tools for risk stratification are still lacking. The aim of this study was to investigate the clinical utility of heart rhythm complexity in risk stratification for PH patients. We prospectively enrolled 54 PH patients, including 20 high-risk patients (group A; defined as WHO functional class IV or class III with severely compromised hemodynamics), and 34 low-risk patients (group B). Both linear and non-linear heart rate variability (HRV) variables, including detrended fluctuation analysis (DFA) and multiscale entropy (MSE), were analyzed. In linear and non-linear HRV analysis, low frequency and high frequency ratio, DFAα1, MSE slope 5, scale 5, and area 6–20 were significantly lower in group A. Among all HRV variables, MSE scale 5 (AUC: 0.758) had the best predictive power to discriminate the two groups. In multivariable analysis, MSE scale 5 (p = 0.010) was the only significantly predictor of severe PH in all HRV variables. In conclusion, the patients with severe PH had worse heart rhythm complexity. MSE parameters, especially scale 5, can help to identify high-risk PH patients.     

Sublimation Study of Six 5-Substituted-1,10-Phenanthrolines by Knudsen Effusion Mass Loss and Solution Calorimetry

The vapor pressures of six solid 5-X-1,10-phenanthrolines (where X = Cl, CH₃, CN, OCH₃, NH₂, NO₂) were determined in suitable temperature ranges by Knudsen Effusion Mass Loss (KEML). From the temperature dependencies of vapor pressure, the molar sublimation enthalpies, Δ_cr^gH_m⁰(⟨T⟩), were calculated at the corresponding average ⟨T⟩ of the explored temperature ranges. Since to the best of our knowledge no thermochemical data seem to be available in the literature regarding these compounds, the Δ_cr^gH_m⁰(⟨T⟩) values obtained by KEML experiments were adjusted to 298.15 K using a well known empirical procedure reported in the literature. The standard (p⁰ = 0.1 MPa) molar sublimation enthalpies, Δ_cr^gH_m⁰(298.15 K), were compared with those determined using a recently proposed solution calorimetry approach, which was validated using a remarkable amount of thermochemical data of molecular compounds. For this purpose, solution enthalpies at infinite dilution of the studied 5-chloro and 5-methylphenantrolines in benzene were measured at 298.15 K. Good agreement was found between the values derived by the two different approaches, and final mean values of Δ_cr^gH_m⁰(298.15 K) were recommended. Finally, the standard molar entropies and Gibbs energies of sublimation were also derived at T = 298.15 K. The volatilities of the six compounds were found to vary over a range of three orders of magnitude in the explored temperature range. The large difference in volatility was analyzed in the light of enthalpies and entropies of sublimation. The latter was tentatively put in relation to the rotational contribution of the substituent group on the phenanthroline unit.     

Isotope and Hyperfine Structure in the "Orange" System of FeO: Evidence for Two Δi Excited States

The orange system of FeO has been reinvestigated using low-temperature molecular beam laser-induced fluorescence spectra, obtained by supersonic jet cooling. Two new weak bands have been found, and analyses of some of the previously known bands extended. Measurements of the ⁵⁴Fe-⁵⁶Fe isotope shifts have been made for most of the bands, and the hyperfine structure of the low-J lines has been recorded for two of the strongest bands of ⁵⁷FeO. The isotope shifts are consistent with the presence of two ⁵Δ_i-⁵Δ_i transitions lying within 1000 cm⁻¹; the origins of the Ω = 4 spin components lie at 5583 and 6110 Å, respectively. The hyperfine patterns and the spin-orbit structure indicate that the upper state electron configurations are (3dδ)³ (3dπ)² (3dσ)¹, (D⁵Δ_i, 5583 Å) and O(2pπ)³ (4sσ)¹ (3dδ)³(3dπ)³, (D′⁵Δ_i, 6110 Å). The bond length in the D′ state (r₀ = 1.654 Å) has been obtained from a deperturbation of the 6110 Å band; it is only 0.035 Å longer than in the ground state, which indicates that electron promotion between the two π orbitals, nominally O(2pπ) and Fe(3dπ), has only a small effect on the strength of the bonding. The new isotope data still do not clarify the vibrational assignments of the higher levels, which are disorganized by extensive electronic perturbations.     

Effects of Tau and Sampling Frequency on the Regularity Analysis of ECG and EEG Signals Using ApEn and SampEn Entropy Estimators

Electrocardiography (ECG) and electroencephalography (EEG) signals provide clinical information relevant to determine a patient’s health status. The nonlinear analysis of ECG and EEG signals allows for discovering characteristics that could not be found with traditional methods based on amplitude and frequency. Approximate entropy (ApEn) and sampling entropy (SampEn) are nonlinear data analysis algorithms that measure the data’s regularity, and these are used to classify different electrophysiological signals as normal or pathological. Entropy calculation requires setting the parameters r (tolerance threshold), m (immersion dimension), and τ (time delay), with the last one being related to how the time series is downsampled. In this study, we showed the dependence of ApEn and SampEn on different values of τ, for ECG and EEG signals with different sampling frequencies (F_s), extracted from a digital repository. We considered four values of F_s (128, 256, 384, and 512 Hz for the ECG signals, and 160, 320, 480, and 640 Hz for the EEG signals) and five values of τ (from 1 to 5). We performed parametric and nonparametric statistical tests to confirm that the groups of normal and pathological ECG and EEG signals were significantly different (p < 0.05) for each F and τ value. The separation between the entropy values of regular and irregular signals was variable, demonstrating the dependence of ApEn and SampEn with F_s and τ. For ECG signals, the separation between the conditions was more robust when using SampEn, the lowest value of F_s, and τ larger than 1. For EEG signals, the separation between the conditions was more robust when using SampEn with large values of F_s and τ larger than 1. Therefore, adjusting τ may be convenient for signals that were acquired with different F_s to ensure a reliable clinical classification. Furthermore, it is useful to set τ to values larger than 1 to reduce the computational cost.     

Magnetische untersuchungen an mischkristallen der quasibinären legierungssysteme (Cd Te)1-x (Mn Te)x und (Sn Te)1-x (Mn Te)x

The magnetic susceptibility and magnetization are presented for polycrystalline samples of the alloy systems Cd_1-xMn_xTe 0 < x 0.1 and Sn_1-xMn_xTe 0 < 4 0.4. The magnetic measurements were performed between 2.3 K and 300 K in external magnetic fields up to 11 kOe. At sufficiently high temperatures the susceptibility can be described by a Curie-Weiss law. In the system Sn_1-xMn_xTe θ_p is positive. A linear dependence θ_p ∞ x was found with θ_p(0.4) = 49 K. In the series Cd_1-xMn_xTe θ_p changes sign. For θ < x < 0.04 θ_p is positive with a maximum θ_p ≈ 10 K at x = 0.02. In the region x #62; 0.04 θ_p becomes negative with θ_p = -35 K at x = 0.1. The effective spin value of manganese is S_eff #62; 5/2 for all the samples. The investigation was done to check the assumption that ferromagnetic coupling may exist in tellurides of manganese if the shortest distance d_MnMn is greater than 3.4 Å. This hypothesis has been stated. In the case θ_p #62; 0 the results are partly explained by the RKKY exchange coupling.     

Information Dark Energy Can Resolve the Hubble Tension and Is Falsifiable by Experiment

We consider the role information energy can play as a source of dark energy. Firstly, we note that if stars and structure had not formed in the universe, elemental bits of information describing the attributes of particles would have exhibited properties similar to the cosmological constant. The Landauer equivalent energy of such elemental bits would be defined in form and value identical to the characteristic energy of the cosmological constant. However, with the formation of stars and structure, stellar heated gas and dust now provide the dominant contribution to information energy with the characteristics of a dynamic, transitional, dark energy. At low redshift, z < ~1.35, this dark energy emulates the cosmological constant with a near-constant energy density, w = −1.03 ± 0.05, and an energy total similar to the mc² of the universe’s ∼10⁵³ kg of baryons. At earlier times, z > ~1.35, information energy was phantom, differing from the cosmological constant, Λ, with a CPL parameter difference of ∆w_o = −0.03 ± 0.05 and ∆w_a = −0.79 ± 0.08, values sufficient to account for the H₀ tension. Information dark energy agrees with most phenomena as well as Λ, while exhibiting characteristics that resolve many tensions and problems of ΛCDM: the cosmological constant problem; the cosmological coincidence problem; the H₀ tension, and the σ₈ tension. As this proposed dark energy source is not usually considered, we identify the expected signature in H(a) that will enable the role of information dark energy to be falsified by experimental observation.     

Cause and risk of catastrophic eruptions in the Japanese Archipelago

The Japanese Archipelago is characterized by active volcanism with variable eruption styles. The magnitude (M)-frequency relationships of catastrophic caldera-forming eruptions (M ≥ 7) are statistically different from those of smaller eruptions (M ≤ 5.7), suggesting that different mechanisms control these eruptions. We also find that volcanoes prone to catastrophic eruptions are located in regions of low crustal strain rate (<0.5 × 10⁸/y) and propose, as one possible mechanism, that the viscous silicic melts that cause such eruptions can be readily segregated from the partially molten lower crust and form a large magma reservoir in such a tectonic regime. Finally we show that there is a ∼1% probability of a catastrophic eruption in the next 100 years based on the eruption records for the last 120 ky. More than 110 million people live in an area at risk of being covered by tephra >20 cm thick, which would severely disrupt every day life, from such an eruption on Kyushu Island, SW Japan.     

Linear scaling of the interlayer relaxations of the vicinal Cu(p,p,p − 2) surfaces with the number of atom-rows in the terraces

We investigate the multilayer relaxation trends in the vicinal Cu(p,p,p − 2) surfaces employing the all-electron full-potential linearized augmented plane-wave method. Calculations are performed for the (3 3 1), (2 2 1), (5 5 3), (3 3 2), (7 7 5), and (4 4 3) surfaces, which have 3, 4, 5, 6, 7, and 8 atom-rows in the terrace, respectively. The following trends are identified: (i) The interlayer relaxations perpendicular to the surface scale almost linearly with the number of atom-rows in the terraces. (ii) The nearest-neighbor distances do not depend on the surface termination, but only on the local coordination. (iii) For Cu(p,p,p − 2) in which the topmost n surface layers have nearest-neighbor coordination smaller than the bulk Cu (calculated for the unrelaxed surfaces), the topmost (n − 1) interlayer spacings (d₁₂, … , d_n−1,n) contract compared with the unrelaxed spacing, while the nth interlayer spacing (d_n,n+1) expands. The next (n − 2) interlayer spacings (d_n+1,n+2, … , d_{2n−2,2n−1}) contract, while the interlayer spacing indicated by d_2n−1,2n expands. A similar rule was found for the relaxations parallel to the surfaces. These trends provide a better understanding of the atomic structure of vicinal Cu surfaces.     

Selection of Embedding Dimension and Delay Time in Phase Space Reconstruction via Symbolic Dynamics

The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay τ∗ and embedding dimension p for phase space reconstruction. The value of τ∗ can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that τ∗ should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay τw=(p−1)τ∗, which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally τw and τ∗. In this paper, we suggest a simple method for estimating τ∗ and τw based on symbolic analysis and symbolic entropy. As in the C-C method, τ∗ is estimated as the first local optimal time delay and τw as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes