Calculation Of Correlation Dimension Research Articles

Predictive Analysis of Economic Chaotic Time Series Based on Chaotic Genetics Combined with Fuzzy Decision Algorithm

The irreversibility in time, the multicausality on lines, and the uncertainty of feedbacks make economic systems and the predictions of economic chaotic time series possess the characteristics of high dimensionalities, multiconstraints, and complex nonlinearities. Based on genetic algorithm and fuzzy rules, the chaotic genetics combined with fuzzy decision-making can use simple, fast, and flexible means to complete the goals of automation and intelligence that are difficult to traditional predicting algorithms. Moreover, the new combined method’s ergodicity can perform nonrepetitive searches in a global scope, hence improving the algorithm’s accuracy and efficiency. On the basis of summarizing and analyzing previous research works, this paper expounded the research status and significance of the prediction of economic chaotic time series, elaborated the development background, current status, and future challenges of the combined algorithm of chaotic genetics with fuzzy decision, introduced the basic principles of chaotic genetic algorithm and fuzzy decision algorithm, constructed a prediction model for economic chaotic time series, performed parameter synchronization optimization and moderate function construction, analyzed the prediction processes of economic chaotic time series, conducted phase space reconstruction and correlation dimension calculation, and finally carried out a simulation experiment with its result analysis. The study results show that the algorithm of chaotic genetics combined with fuzzy decision-making can dynamically adjust chaotic mutation operators and summarize fussy expert experiences. The phase space of its reconstructed chaotic attractor has high-precision predictability and can find orderly processes from changeable economic results, which in turn can be used to analyze and predict the complex economic chaotic time series. The study results of this paper provide a reference for further research on predictive analysis of economic chaotic time series based on chaotic genetics combined with fuzzy decision algorithm.

The effect of sounds on nonlinear measures of heart rate variability

Purpose and methodThis purpose of the study is to determine changes in nonlinear heart rate variability (HRV) measures during the passive listening of sounds. Twenty subjects (all women) participated in the study. Subjects listened to four sounds: (1) white noise, (2) the road traffic noise, (3) excitatory music (Diamanda Galas' Women with Steak Knives, (4) lullaby (Ba Mo Leanabh (O My Baby), Arr. William Jackson and Mackenzie, Fiona McKenzie). Emotional responses were assessed with The Affect Grid. The experiment consisted of 5 min sessions (rest and listening). RR intervals were recorded during both sessions. Fractal scaling exponents of HRV were quantified by means of the detrended fluctuation analysis (DFA). Three indices of Poincaré plots were calculated: the width of the cloud (SD1), the length of the cloud (SD2) and the ratio SD2/SD1. The complexity of attractor was assessed by calculation of correlation dimension (D2) and entropy measures (ApEn, SampEn). The following measures were derived from recurrence plot (RP): mean line length (Lmean), maximum line length (Lmax), recurrence rate (REC), determinism (DET), and Shannon entropy of line length distribution (ShanEn). We chose a tolerance level to achieve the REC =5% in accordance with a suggestion of Webber (2007). Freidman ANOVA and Wilcoxon matched‐pairs signed‐ranks tests were used to determine the significance of differences between measurements. Data are mean ± SE.ResultsExcitatory music was interpreted as unpleasant and arousing (valence: −3.07 ± 0.23; arousal: 2.00 ± 0.30). Lullaby was associated with positive emotion with negative arousal (valence: 2.19 ± 0.21; arousal: −1.53 ± 0.28). White noise induced negative valence (−2.53 ± 0.2) and positive arousal (0.95 ± 0.25), road traffic noise induced negative valence (−1.19 ± 0.24) and very low arousal (0.48 ± 0.22). The levels of valence were significantly different across the 4 stimuli (Friedman ANOVA: χ2 = 131.92, p < 0.001). This difference was present in pairs of sounds (Wilcoxon: p < 0.05). The results were the same for arousal (Friedman ANOVA: χ2 = 73.14, p < 0.001). Results of calculation of complexity measures (ApEn, SampEn, D2) exhibit non‐significant differences between rest and sound listening (p > 0.05). Sound listening produces non‐significant response changes in short‐range fractal scaling exponent of DFA (p > 0.05). There was a significant increase in long‐range fractal scaling exponent during excitatory music listening (from 0.40 ± 0.06 to 0.52 ± 0.07). During excitatory music, exposure was found significantly higher values of Lmean, Lmax, and ShanEn (p < 0.05). The RP indexes when listening to the white noise, traffic noise and lullaby showed no significant changes (Wilcoxon: p > 0.05). Poincare plot analysis of RR sequences reveals a significant decrease of the standard deviation of the instantaneous beat‐to‐beat variability data (SD1) during excitatory music listening (p < 0.01). The changes in the length of the cloud (SD2) and SD2/SD1 ratio were nonsignificant (p > 0.05).Support or Funding InformationThis abstract is from the Experimental Biology 2019 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Chaos Analysis on the Acceleration Control Signals of the Piezoelectric Actuators in the Stewart Platform

In order to solve the nonlinear mechanism in the process of actual control application of six-degree-of-freedom parallel Stewart platform, the chaos theory was applied in the paper to analyze the acceleration control signal. The research included correlation dimension calculation by use of the G-P method, the maximum Lyapunov coefficient of the acceleration control signal, and Kolmogorov entropies of the acceleration control signal. The results show that the acceleration signals are of chaos characteristics, and there are lots of influencing factors to the acceleration variables.

Topographic distribution of EEG alpha attractor correlation dimension values in wake and drowsy states in humans

Organization of resting state cortical networks is of fundamental importance for the phenomenon of awareness, which is altered in the first part of hypnagogic period (Hori stages 1–4). Our aim was to investigate the change in brain topography pattern of EEG alpha attractor correlation dimension (CD) in the period of transition from Hori stage 1 to 4. EEG of ten healthy adult individuals was recorded in the wake and drowsy states, using a 14 channel average reference montage, from which 91 bipolar channels were derived and filtered in the wider alpha (6–14Hz) range. Sixty 1s long epochs of each state and individual were subjected to CD calculation according to the Grassberger–Procaccia method. For such a collection of signals, two embedding dimensions, d={5, 10}, and 22 time delays τ=2–23 samples were explored. Optimal values were d=10 and τ=18, where both saturation and second zero crossing of the autocorrelation function occurred. Bipolar channel CD underwent a significant decrease during the transition and showed a positive linear correlation with electrode distance, stronger in the wake individuals. Topographic distribution of bipolar channels with above median CD changed from longitudinal anterior–posterior pattern (awake) to a more diagonal pattern, with localization in posterior regions (drowsiness). Our data are in line with the literature reporting functional segregation of neuronal assemblies in anterior and posterior regions during this transition. Our results should contribute to understanding of complex reorganization of the cortical part of alpha generators during the wake/drowsy transition.

Nonlinear dynamic analysis of voice: A normative study in the Indian population

Background: The aim of this study was to establish normative data for the Indian population using Nonlinear dynamic analysis. In this study, correlation dimension, a measure of nonlinear dynamic analysis was performed for normophonic young, middle aged and elderly voices. Materials and Methods: For this purpose, normophonic young, middle aged and elderly individuals were selected without a history of voice/respiratory problems and vocal abuse/ misuse. 60 participants were selected in each group. All of these individuals had a normal voice as evaluated through GRBAS scale. Sound Recorder, on a computer desktop was used for voice recording and “convert” code in MATLAB as well as D2.ini.writer software based on TISEAN package (Hegger, Kantz & Schreiber, 1999) was used for the calculation of Correlation dimension (D2). Correlation dimension measures were obtained for each participant, for both steady vowel phonations (/a/, /i/, /u/) as well as narration samples. Results: The correlation dimension measures across the group revealed a significant main effect of the groups indicating correlation dimension increases with increase in age. Conclusions: The application of nonlinear dynamic measures in the assessment of voice is a novel venture and thus this study provides normative data for correlation dimensions in the Indian population for future comparisons against the disordered voice samples. Further studies are warranted to investigate the same in the clinical population. Also other nonlinear dynamic analysis methods need to be investigated to obtain the normative data in the Indian population.

Chaotic air pressure fluctuations during departure of air bubbles from two neighbouring nozzles

Chaotic air pressure fluctuations during departure of air bubbles from two neighbouring nozzlesIn the experiment, bubbles were generated from two brass nozzles with inner diameters of 1.1 mm. They were submerged in the glass tank filled with distilled water. There have been measured the air pressure fluctuations and the signal from the laser-phototransistor sensor. For analysis of the pressure signal the correlation (the normalized cross - correlation exponent) and non-linear analyses have been used. It has been shown that hydrodynamic interactions between bubbles can lead to bubble departure synchronization. In this case the bubble departures become periodic. The results of calculation of correlation dimension and the largest Lyapunov exponent confirm that hydrodynamic bubble interactions observed for 4 mm spacing between nozzels cause the periodic bubble departures from two neighbouring nozzles.

A novel method to identify the scaling region for chaotic time series correlation dimension calculation

To obtain more accurate correlation dimension estimations for chaotic time series, a novel scaling region identification method is developed. First, points that obviously do not belong to the scaling region associated with the whole double logarithm correlation integral curve are removed using the K-means algorithm. Second, a point-slope-error algorithm is developed to recognize a possible scaling region. Third, the K-means algorithm is used again to further remove a small interval of interfering points in the possible scaling region to obtain a more precise scaling region. The correlation dimension of four typical chaotic attractors and five curves generated by the Weierstrass-Mandelbrot fractal function were calculated using the proposed method. These calculated values were very close to the respective theoretical fractal dimensions. Moreover, the effectiveness of our method in identifying the scaling region was compared with existing methods. Results show that our method can distinguish the scaling region objectively, accurately, automatically and quickly, making estimations of the correlation dimension more precise and affording significant improvements in nonlinear analysis.

S1601-2-1 非線形時系列解析手法による生体状態の過渡的変動の評価(生物医学工学における計測と制御(2),社会変革を技術で廻す機械工学)

We propose a new correlation dimension calculation method for the transient response analysis of human condition. To achieve the monitoring of nonstational human condition, we apply a moving window technique and an overlapping reconstructed orbit technique to the point correlation dimension estimation method. This method is applied to transient electrocardiogram (ECG) time series on standing up. We find a significant difference between a response pattern of fatigued state and unfatigued state.

Correlation Dimension Analysis of Doppler Signals in Children with Aortic Valve Disorders

In this study, the correlation dimension analysis has been applied to the aortic valve Doppler signals to investigate the complexity of the Doppler signals which belong to aortic stenosis (AS) and aortic insufficiency (AI) diseases and healthy case. The Doppler signals of 20 healthy subjects, ten AS and ten AI patients were acquired via the Doppler echocardiography system that is a noninvasive and reliable technique for assessment of AS and AI diseases. The correlation dimension estimations have been performed for different time delay values to investigate the influence of time delay on the correlation dimension calculation. The correlation dimension of healthy group has been found lower those found in AI and AS disorder groups and the correlation dimension of AS group has also been found higher than those found in AI group, significantly. The results of this study have indicated that the aortic valve Doppler signals exhibit high level chaotic behaviour in AI and AS diseases than healthy case. Additionally, the correlation dimension analysis is sensitive to the time delay and has successfully characterized the blood flow dynamics for proper time delay value. As a result, the correlation dimension can be used as an efficient method to determine the healthy or pathological cases of aortic valve.

3707 移動ポイント相関次元によるリアルタイム人間状態評価の試み(S55-1 生物医学工学における計測と制御(1),21世紀地球環境革命の機械工学:人・マイクロナノ・エネルギー・環境)

We propose a new correlation dimension calculation method for the real-time monitoring of human condition. To achieve the real-time monitoring of long and nonstational time series, we apply a moving window technique to the point correlation dimension estimation. This moving point correlation method is applied to electrocardiogram (ECG) time series during calculation task. The results show the positive correlation between calculation performance and point correlation dimension values.

The nonlinear current behaviour of a driven R–L-Varactor in the low frequency range

The nonlinear behaviour of an R–L-Varactor circuit, simulated by Multisim 7.0 at a driving frequency that is below the circuit resonance frequency, is reported and evaluated. A new high amplitude oscillation is observed and attributed to the emergence of a large diode junction capacitance. Increasing the driving signal amplitude, the circuit is led to a non-periodic mode of operation, producing trajectories of increasing chaotic content in a four-dimensional phase space. At specific amplitudes a two-dimensional tori was monitored, where trajectories are periodic, like in a two oscillator system with commensurate frequencies. Poincaré cross-sections, FFT spectra and correlation dimension calculations, suggest the quasi-periodicity route to chaos.

Dynamical Model Reconstruction and Accurate Prediction of Power-Pool Time Series

The emergence of the power pool as a popular institution for trading of power in different countries has led to increased interest in the prediction of power demand and price. In this paper, the authors investigate whether the time series of power-pool demand and price can be modeled as the output of a low-dimensional chaotic dynamical system by using delay embedding and estimation of the embedding dimension, attractor-dimension or correlation-dimension calculation, Lyapunov-spectrum and Lyapunov-dimension calculation, stationarity and nonlinearity tests, as well as prediction analysis. Different dimension estimates are consistent and show close similarity, thus increasing the credibility of the fractal-dimension estimates. The Lyapunov spectrum consistently shows one positive Lyapunov exponent and one zero exponent with the rest being negative, pointing to the existence of chaos. The authors then propose a least squares genetic programming (LS-GP) to reconstruct the nonlinear dynamics from the power-pool time series. Compared to some standard predictors including the radial basis function (RBF) neural network and the local state-space predictor, the proposed method does not only achieve good prediction of the power-pool time series but also accurately predicts the peaks in the power price and demand based on the data sets used in the present study.

Correlation dimension analysis of heart rate variability in patients with dilated cardiomyopathy

A correlation dimension analysis of heart rate variability (HRV) was applied to a group of 55 patients with dilated cardiomyopathy (DCM) and 55 healthy subjects as controls. The 24-h RR time series for each subject was divided into segments of 10,000 beats to determine the correlation dimension (CD) per segment. A study of the influence of the time delay (lag) in the calculation of CD was performed. Good discrimination between both groups (p<0.005) was obtained with lag values of 5 or greater. CD values of DCM patients (8.4+/-1.9) were significantly lower than CD values for controls (9.5+/-1.9). An analysis of CD values of HRV showed that for healthy people, CD night values (10.6+/-1.8) were significant greater than CD day values (9.2+/-1.9), revealing a circadian rhythm. In DCM patients, this circadian rhythm was lost and there were no differences between CD values in day (8.8+/-2.4) and night (8.9+/-2.1).

On reconstruction of strange attractors using their noise related directional properties

We study directional properties of strange attractor trajectory vectors projected on selected subset of principal component eigenvectors in m-dimensional embedding spaces. We introduce a simple measure of anisotropicity for chosen principal component subspace, based on transformed projected trajectory matrix, which changes smoothly as a function of window width of the original trajectory matrix. The value of such defined measure is dependent on amount of noise in the data. For isotropically distributed noise (or close to isotropic), that fact allows us to set up the criterion for minimal window width for satisfactory reconstruction, as a function of amount of noise in the data. For data with low noise content we introduce a simple quantity that is closely related to “potential energy” of attractor's trajectory. It turns out that the minimum of such defined quantity determines the adequate window width for reconstruction of the attractor. The method is assessed through correlation dimension calculations.

Analyses of transient chaotic time series.

We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits, from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

IMPROVED CORRELATION DIMENSION CALCULATION

For many chaotic systems, accurate calculation of the correlation dimension from measured data is difficult because of very slow convergence as the scale size is reduced. This problem is often caused by the highly nonuniform measure on the attractor. This paper proposes a method for collecting data at large scales and extrapolating to the limit of zero scale. The result is a vastly reduced required number of data points for a given accuracy in the measured dimension. The method is illustrated in detail for one-dimensional maps and then applied to more complicated maps and flows. Values are given for the correlation dimension of many standard chaotic systems.

A Method for Detecting Chaos in Canine Myocardial Microcirculatory Red Cell Flux

To determine whether red cell movement, as measured by laser Doppler velocimetry, in the capillary net of the beating heart is chaotic. Using two dog hearts, in situ red blood cell flux was measured at many sites. Simultaneously, epicardial arterial flow and left ventricular pressure were recorded via transit-time flowmeter and catheter manometer, respectively. The presence or absence of chaos was tested by two methods: Lyapunov exponents and correlation dimension. For capillary red cell flux, the Lyapunov was strongly positive at most sites. It was less so for coronary arterial flow and least for left ventricular pressure. Correlation dimension calculation was less able to distinguish the presence or absence of chaos in capillary red cell tissue flux, coronary arterial flow, and left ventricular pressure. Capillary red cell flux (movement of red cells in capillaries) is nonlinear, (i.e., chaotic). This complexity suggests that the primary control for oxygen delivery to cardiac myocytes by red blood cells resides in the microcirculation. Also, capillary red cell flux is bifractal, suggesting an ordering of control.

A Method for Detecting Chaos in Canine Myocardial Microcirculatory Red Cell Flux

Objective: To determine whether red cell movement, as measured by laser Doppler velocimetry, in the capillary net of the beating heart is chaotic. Methods: Using two dog hearts, in situ red blood cell flux was measured at many sites. Simultaneously, epicardial arterial flow and left ventricular pressure were recorded via transit-time flowmeter and catheter manometer, respectively. The presence or absence of chaos was tested by two methods: Lyapunov exponents and correlation dimension. Results: For capillary red cell flux, the Lyapunov was strongly positive at most sites. It was less so for coronary arterial flow and least for left ventricular pressure. Correlation dimension calculation was less able to distinguish the presence or absence of chaos in capillary red cell tissue flux, coronary arterial flow, and left ventricular pressure. Conclusions: Capillary red cell flux (movement of red cells in capillaries) is nonlinear, (i.e., chaotic). This complexity suggests that the primary control for oxygen delivery to cardiac myocytes by red blood cells resides in the microcirculation. Also, capillary red cell flux is bifractal, suggesting an ordering of control.

Chaos in the Portevin–Le Châtelier Effect

We report the verification of the prediction of chaos in the Portevin–Le Châtelier effect or the jerky flow by analyzing the stress signals obtained from samples of polycrystalline Al–Mg alloys subjected to a constant strain rate test. Particular care is taken to obtain reasonably long and accurate stress signals. The analysis of these signals is carried out by using several complementary methods such as calculation of correlation dimension, singular value decomposition and the spectrum of Lyapunov exponents. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent. Using the existence of a positive Lyapunov exponent and finite correlation dimension as a discriminator, we also carry out a surrogate analysis of the time series to ascertain that the signals are not from a power law stochastic process. The analysis provides an unambiguous support for the existence of chaos in Portevin–Le Châtelier effect thus verifying the prediction of the model. Further, from the analysis we find that the minimum number of variables required for a dynamical description of the jerky flow appears to be four or five consistent with the model.

Analysis of Class of Nonlinear System under Deterministic and Stochastic Excitations

In this paper, some analysis techniques of nonlinear dynamics are applied to physical systems which may be modeled by the Duffing nonlinear differential equation. The response of the Duffing oscillator to both deterministic sinusoidal and stochastic loadings is investigated and distinct regimes of the response motion are discerned and discussed. The stochastic input to the system is low-pass Gaussian white noise. The efficacy of studying the variation in time of the probability density of one or more of the system output states to determine the type of motion of the system is examined. Attractors in phase space are defined via Poincaré mapping and bounds on motion which serve as signatures for particular types of motion (e.g., chaotic, periodic) are identified by a hypervolume measurement technique. An accepted method for adapting one measured output state into a higher dimensional space by using time-delayed coordinates is used in conjunction with correlation dimension calculation to supply a lower-bound estimate of the fractal dimension and insight into the character of the motion of a nonlinear dynamic system.