Estimation Of Correlation Dimension Research Articles

Correlation dimension in empirical networks.

Network correlation dimension governs the distribution of network distance in terms of a power-law model and profoundly impacts both structural properties and dynamical processes. We develop new maximum likelihood methods which allow us robustly and objectively to identify network correlation dimension and a bounded interval of distances over which the model faithfully represents structure. We also compare the traditional practice of estimating correlation dimension by modeling as a power law the fraction of nodes within a distance to a proposed alternative of modeling as a power law the fraction of nodes at a distance. In addition, we illustrate a likelihood ratio technique for comparing the correlation dimension and small-world descriptions of network structure. Improvements from our innovations are demonstrated on a diverse selection of synthetic and empirical networks. We show that the network correlation dimension model accurately captures empirical network structure over neighborhoods of substantial size and span and outperforms the alternative small-world network scaling model. Our improved methods tend to lead to higher estimates of network correlation dimension, implying that prior studies could have produced or utilized systematic underestimates of dimension.

Prediction of environmental missing data time series by Support Vector Machine Regression and Correlation Dimension estimation

Environmental time series are often affected by missing data, namely data unavailability at certain time points. This paper presents the Iterated Imputation and Prediction algorithm, that allows the prediction of time series with missing data. The algorithm uses iteratively the Correlation Dimension Estimation of the underlying dynamic system generating the time series to fix the model order (i.e., how many past samples are required to model the time series accurately), and the Support Vector Machine Regression to estimate the skeleton of time series. Experimental validation of the algorithm on three environmental time series with missing data, expressing the concentration of Ozone in three European sites, shows a small average percentage prediction error for all time series on the test set.

Coexistence of chaotic and complexity dynamics of fluctuations with long-range temporal correlations under typical condition for formation of multiple anodic double layers in DC glow discharge plasma

Analysis of electrostatic floating potential fluctuations associated with multiple anodic double layer revealed a complexity dynamic coexisting with chaotic behavior. These externally controllable nonequilibrium quasi-stationary multiple anodic double layers were created in a cold cathode dc discharge setup in front of an extra anode which was used to supplement additional ionization. Despite the fact that chaos and complexity dynamics are often defined with entirely different properties, this study provides scenarios under which both can exist simultaneously. A stable multiple double-layer structure consisting of four successive double layers, each with positive and negative charged particles arranged in opposite sheets, was created when the anode potential exceeds a certain threshold value for a minimum gas breakdown bias between cathode and ground. After the stable multiple double layers was created, it was further controlled externally by varying the cathode bias between minimum gas breakdown bias of − 335 V and − 610 V at a pressure of 0.3 mbar for studying its advanced stages. With increase in cathode bias, multiple anodic double-layer structure advances towards the anode surface with the collapse of innermost layer due to the arrival of more energetic electrons in the anode zone. This process results from the self-organization and re-organization of charged particles in each double layer mediated by energetic electrons. The process continued until all the double layers disappeared and only an intense anode glow remained at anode for higher values of cathode bias. The chaotic dynamics of the system was studied at every stage by analyzing the corresponding floating potential fluctuations using FFT, phase space trajectories and nonlinear technique such as Lyapunov exponent, time-delay reconstruction etc. The analysis revealed the onset of chaos beyond − 410 V was triggered by the collapse of double layers. Estimation of correlation dimension, autocorrelation function and Hurst exponent unfolded the complexity features such as self-similarity and longtime dependence in the fluctuations. The value of correlation dimension reaches the maximum with an increase in cathode bias. Estimation of Hurst exponent using rescaled range analysis technique with values of H between 0.5 and 1 and algebraic decay of autocorrelation function provide signatures of long-range temporal correlations in the chaotic fluctuations and underlines the coexistence of complexity behavior.

Asymptotic behavior of the time-dependent divergence exponent.

The divergence rate method, which is used to determine the maximum Lyapunov exponent out of time series, is based on the evaluation of the time-dependent divergence exponent. For chaotic systems and in the small time regime, this exponent grows linearly in time. The asymptotic regime is instead characterized by a time-independent behavior due to the system eventually losing its memory of the starting conditions. The amplitude of this "plateau"-like divergence exponent depends both on the choice of the embedding dimension and lag and on the maximum distance of nearby starting trajectories in a way that is characteristic of the underlying dynamical system. In this paper, upon introducing the basic mathematical tools, we address the plateau evaluation for two classes of time series, those generated by a white noise source and those generated by a finite-dimensional chaotic system. The different behavior provides a novel tool to distinguish purely stochastic sources from deterministic ones, as well as to provide a precise estimate of the correlation dimension in the latter case. The method is also sensitive to correlated noise sources.

Estimating correlation dimension of high-dimensional signals - quick algorithm

The paper describes the improvements and corrections to the method of estimation of correlation dimension of d for high-dimensional signals (HDS). The following problems are described: (a) presentation of the quick version of the algorithm that reduces computation time, (b) improvement of the precision of the elementary Takens-Ellner formula estimating d, (c) the problem of strongly non-gaussian signals and its possible normalisation. Appendix discusses the use of author’s Matlab function estimating d which is available for public use.

Correlation dimension and phase space contraction via extreme value theory.

We show how to obtain theoretical and numerical estimates of correlation dimension and phase space contraction by using the extreme value theory. The maxima of suitable observables sampled along the trajectory of a chaotic dynamical system converge asymptotically to classical extreme value laws where: (i) the inverse of the scale parameter gives the correlation dimension and (ii) the extremal index is associated with the rate of phase space contraction for backward iteration, which in dimension 1 and 2, is closely related to the positive Lyapunov exponent and in higher dimensions is related to the metric entropy. We call it the Dynamical Extremal Index. Numerical estimates are straightforward to obtain as they imply just a simple fit to a univariate distribution. Numerical tests range from low dimensional maps, to generalized Henon maps and climate data. The estimates of the indicators are particularly robust even with relatively short time series.

Statistical analysis of Geopotential Height (GH) timeseries based on Tsallis non-extensive statistical mechanics

In this paper, we perform statistical analysis of time series deriving from Earth’s climate. The time series are concerned with Geopotential Height (GH) and correspond to temporal and spatial components of the global distribution of month average values, during the period (1948–2012). The analysis is based on Tsallis non-extensive statistical mechanics and in particular on the estimation of Tsallis’ q-triplet, namely {qstat, qsens, qrel}, the reconstructed phase space and the estimation of correlation dimension and the Hurst exponent of rescaled range analysis (R/S). The deviation of Tsallis q-triplet from unity indicates non-Gaussian (Tsallis q-Gaussian) non-extensive character with heavy tails probability density functions (PDFs), multifractal behavior and long range dependences for all timeseries considered. Also noticeable differences of the q-triplet estimation found in the timeseries at distinct local or temporal regions. Moreover, in the reconstructive phase space revealed a lower-dimensional fractal set in the GH dynamical phase space (strong self-organization) and the estimation of Hurst exponent indicated multifractality, non-Gaussianity and persistence. The analysis is giving significant information identifying and characterizing the dynamical characteristics of the earth’s climate.

Regenerating time series from ordinal networks.

Recently proposed ordinal networks not only afford novel methods of nonlinear time series analysis but also constitute stochastic approximations of the deterministic flow time series from which the network models are constructed. In this paper, we construct ordinal networks from discrete sampled continuous chaotic time series and then regenerate new time series by taking random walks on the ordinal network. We then investigate the extent to which the dynamics of the original time series are encoded in the ordinal networks and retained through the process of regenerating new time series by using several distinct quantitative approaches. First, we use recurrence quantification analysis on traditional recurrence plots and order recurrence plots to compare the temporal structure of the original time series with random walk surrogate time series. Second, we estimate the largest Lyapunov exponent from the original time series and investigate the extent to which this invariant measure can be estimated from the surrogate time series. Finally, estimates of correlation dimension are computed to compare the topological properties of the original and surrogate time series dynamics. Our findings show that ordinal networks constructed from univariate time series data constitute stochastic models which approximate important dynamical properties of the original systems.

A real-time detection method of abnormal building energy consumption data coupled POD-LSE and FCD

To understand the energy consumption characteristics of buildings, numerous building energy consumption monitoring platforms and internet of building energy systems have been established in the large public buildings, government office buildings, as well as colleges and universities. Although the platforms provide us with a large amount of energy consumption in-formation, there are serious problems. However, the validity of the energy consumption data is one of the most common problems. In this paper, a real-time detection method, which coupled fractal correlation dimension and proper orthogonal decomposition linear stochastic estimation, is presented to identify abnormal energy consumption data. The proper threshold is selected with varying operational conditions. The result shows that using this real-time meth-od yields a higher correctness rate than using traditional method in fault detection of data loss and outlier. It’s worth pointing out that a hybrid method combining with other intelligent algorithms should be promising to identify and classify small bias abnormal data fault in further investigation.

Robust to noise and outliers estimator of correlation dimension

The estimation of correlation dimension of continuous and discreet deterministic chaotic processes corrupted by an additive noise and outliers observations is investigated. In this paper we propose a new estimator of correlation dimension based on similarity between the evolution of Gaussian kernel correlation sum (Gkcs) and that of modified Boltzmann sigmoidal function (mBsf), this estimator is given by the maximum value of the first derivative of logarithmic transform of Gkcs against logarithmic transform of bandwidth, so the proposed estimator is independent of the choice of regression region like other regression estimators of correlation dimension. Simulation study indicates the robustness of proposed estimator to the presence of different types of noise such us independent Gaussian noise, non independent Gaussian noise and uniform noise for high noise level, moreover, this estimator is also robust to presence of 60% of outliers observations. Application of this new estimator with determination of their confidence interval using the moving block bootstrap method to adjusted closed price of S&P500 index daily time series revels the stochastic behavior of such financial time series.

Irregular-regular-irregular mixed mode oscillations in a glow discharge plasma

Floating potential fluctuations of a glow discharge plasma are found to exhibit different kinds of mixed mode oscillations. Power spectrum analysis reveals that with change in the nature of the mixed mode oscillation (MMO), there occurs a transfer of power between the different harmonics and subharmonics. The variation in the chaoticity of different types of mmo was observed with the study of Lyapunov exponents. Estimates of correlation dimension and the Hurst exponent suggest that these MMOs are of low dimensional nature with an anti persistent character. Numerical modeling also reflects the experimentally found transitions between the different MMOs.

Time series analysis on marine wind-wave characteristics using chaos theory

This paper discusses an attempt to identify the chaotic behavior of the dynamic system of wind-wave characteristics including significant wave height, wave period and wave direction in southern, central and northern regions of the Caspian Sea. Surrogate data method and spectral analysis are utilized to detect the nonlinearity and aperiodicity of the time series. The delay embedding theorem of reconstructing the phase space is applied to identify characteristics of the time series dynamic system. The false nearest neighbor analysis along with the correlation dimension estimation indicate the presence of high dimensional chaotic behavior with dimensions of the chaotic attractors at the range of 5.91, 6.30 and 7.55 for the significant wave height series and 6.59, 7.64 and 7.87 for the wave period for the southern, central and northern parts of the Sea. Higher correlation dimension of the northern station could be related to the physical characteristics of the sea bathymetry. However, the estimated correlation dimension indicates the low choaticity of the wave direction set (3.65, 3.70 and 3.77). Positive Lyapunov exponents obtained for all 9 data series prove the exponential divergence of the trajectories which support the presence of chaos in the wind-wave characteristics.

Correlation Dimension of Fractional Gaussian Noise: New Evidence from Wavelets

In this paper, we study the behavior of the correlation dimension estimated using the Grassberger–Procaccia (GP) algorithm [Grassberger & Procaccia, 1983] in the wavelet domain for functions belonging to Hölder space. We prove that, as the wavelet scale level tends to infinity, the GP correlation dimension estimate tends to zero. Applying this result to the trajectories of the fractional Brownian motion process and using basic properties of the wavelet transform, we show that, for the fractional Gaussian noise process (fGn), the correlation dimension estimated by the GP procedure converges to the zero value. As the fractional Gaussian noise is a stochastic process with 1/fα spectrum, -1 < α < 1, our results confirm Osborne and Provenzale's assertion that colored random noise leads to the convergence of the GP-based correlation dimension estimator. However, our result holds for a different range of the spectrum exponent values. Moreover, for the fGn class of random processes, we found no correspondence between the value of the scaling exponent H and the value of the correlation dimension estimated by the GP algorithm as the latter is simply zero.

Methodological Framework for Estimating the Correlation Dimension in HRV Signals

This paper presents a methodological framework for robust estimation of the correlation dimension in HRV signals. It includes (i) a fast algorithm for on-line computation of correlation sums; (ii) log-log curves fitting to a sigmoidal function for robust maximum slope estimation discarding the estimation according to fitting requirements; (iii) three different approaches for linear region slope estimation based on latter point; and (iv) exponential fitting for robust estimation of saturation level of slope series with increasing embedded dimension to finally obtain the correlation dimension estimate. Each approach for slope estimation leads to a correlation dimension estimate, called , , and . and estimate the theoretical value of correlation dimension for the Lorenz attractor with relative error of 4%, and with 1%. The three approaches are applied to HRV signals of pregnant women before spinal anesthesia for cesarean delivery in order to identify patients at risk for hypotension. keeps the 81% of accuracy previously described in the literature while and approaches reach 91% of accuracy in the same database.

ANÁLISE DA COMPLEXIDADE DO SALDO DE RADIAÇÃO EM UMA FLORESTA SAZONALMENTE INUNDADA NO NORTE DO PANTANAL

The ecosystem net radiation has a complex variation, and represents the energy available for physical and biological processes that occurbetween the vegetated surface and atmosphere. Therefore, the objective of this study was to analyze the complexity of net radiation measuredabove the canopy of a seasonal flooded forest in the northern Pantanal from the perspective of complexity theory. The attractors wereseasonal reconstructed by the estimation of the exponents correlation dimension (D2(wet) = 1,73±0,05 and D2(dry) = 1,36±0,18), embeddeddimension (m = 7), rate of recurrence (RR(wet) = 0,144±0,004 and RR(dry) = 0,169±0,002) and determinism (DET(wet) = 0,896±0,013 andDET(dry) = 0,973±0,015) data sets analyzed. The results showed the existence of strange attractors of low fractal dimension, and sensitiveto the seasonality of radiation balance.

Nonlinear analysis of rainfall variability in Australia

This study examines the utility of nonlinear dynamic concepts for analysis of rainfall variability across Western Australia. The correlation dimension method, which involves data reconstruction and nearest neighbor concepts, is employed to estimate the dimensionality of rainfall time series and assess the degree of variability (or complexity) of rainfall dynamics. Monthly rainfall data observed over a period of 67 years (January 1937–December 2003) from each of 62 raingage stations across Western Australia are analyzed. The results indicate the utility of the dimensionality concept for identification of rainfall variability at point locations and rainfall patterns across space. The dimension estimates (ranging from 4.63 to 8.29) suggest a medium-level complexity in rainfall dynamics at each of the 62 stations and, hence, across Western Australia. Rainfall dynamics are generally more complex in the southwest part of Western Australia than in other parts of the state, with the far south exhibiting a greater complexity. Also, the mid-western region exhibits slightly more complex rainfall dynamics when compared to the northeast. The reliability of the correlation dimension estimates with respect to data size (i.e. number of data points in time series) is also discussed. The outcomes of the present study, the first ever based on nonlinear dynamic and chaos theories for rainfall in Australia and also across a far more extensive network of raingages, can provide new avenues for rainfall (and other hydrologic) studies in Australia and for spatial/spatio-temporal analysis.

Correlation dimension estimates of human postural sway

Human postural sway during quiet standing demonstrates a complex structured dynamics, which has been studied by applying numerous methods, such as linear system identification methods, stochastic analysis, and nonlinear system dynamics tools. Although each of the methods applied revealed some particular features of the sway data none of them have succeeded to present a global picture of the quiet stance dynamics, which probably has both stochastic and deterministic properties. In this study we have started applying ergodic theory of dynamical systems to explore statistical characteristic of the sway dynamics observed in successive trials of a subject, different subjects in an age group, and finally different age groups constituted by children, adults, and elderly subjects. Five successive 180-s long trials were performed by each of 28 subjects in four age groups at quiet stance with eyes open. Stationary and ergodic signal characteristics of five successive center of pressure time series collected from a subject in antero-posterior direction (CoPx) were examined. 97% of the trials were found to be stationary by applying Run Test while children and elderly groups demonstrated significant nonstationary behavior. On the other hand 13 out of 24 subjects were found to be nonergodic. We expected to observe differences in complexity of CoPx dynamics due to aging (Farmer, Ott, & Yorke, 1983). However linear metrics such as standard deviation and Fourier spectra of CoPx signals did not show differences due to the age groups. Correlation dimension (Dk) estimates of stationary CoPx signals being an invariant measure of nonlinear system dynamics were computed by using the average displacement method (Eckmann & Ruelle, 1985). Postural dynamics was expanded in m-dimensional space through CoPx signal by introducing optimum time delays, τcritical. 112 out of 136 stationary CoPx signals for 24 stationary subjects converged to Dk estimates. Average of Dk estimates for children and adult groups was 3.67±0.28, whereas mean of Dk estimates for elderly subjects was 4.12±0.59. Nonlinear metrics of postural sway (τcritical, msaturated, and Dk estimates) showed significant differences with respect to the age groups. Dk estimates computed from ergodic subjects’ CoPx sway trajectories revealed that human quiet standing demonstrates multiple degree of freedom dynamics having a fractal structure with a considerable level of noise embedded in the signal whose characteristics is determined individually for each subject. Furthermore by using ergodic theory of complex systems, we have been able to show that the ability to independently control multiple degrees of freedom has been affected by aging.

Investigations of a thermal plasma jet structure by generalized correlation dimension

Dynamics of thermal plasma jets is characterized by significant turbulent phenomena influencing various technological applications. In two types of experiments, we have analysed (i) tomographical reconstructions of plasma jet radiation captured from four sides by photodiode arrays in two planes perpendicular to the jet axis and (ii) recordings of the jet radiation acquired by a high-speed CCD camera. The analysis was based on the method of generalized correlation dimension stemming from the correlation dimension estimates by the Grassberger–Procaccia algorithm common in the domain of nonlinear dynamic systems. Our results map spatial distributions of complexity within recorded data including their temporal evolution and provide more information about the jet dynamics than, e.g., calculations of standard deviations. The most stable region of the plasma jet was identified at a small distance from the jet axis and the most turbulent region about 4 mm from the plasma torch nozzle. Signs of short-time collective rotational movement of the jet core were also identified.

CHAOS, SELF ORGANIZED CRITICALITY, INTERMITTENT TURBULENCE AND NONEXTENSIVITY REVEALED FROM SEISMOGENESIS IN NORTH AEGEAN AREA

Strong evidence is provided for significant far from equilibrium complex processes in the seismogenic layer of the North Aegean region (Greece), after applying modern nonlinear methods to various seismicity time series. The data used are subsets of the regional catalogue compiled in the central Seismological Station of Geophysics Department, Aristotle University of Thessaloniki and concern 4367 earthquakes of magnitude greater than 3.8, which took place during the period of 1968–2008. We present results, derived from the application of nonlinear algorithms, concerning the estimation of correlation dimension, mutual information, largest Lyapunov exponent, flatness coefficient and q-value which correspond to Tsallis nonextensive statistics. These quantities are estimated for two seismic time series corresponding to the basic focal parameters of earthquakes, namely origin time and magnitude. The obtained results can be associated with novel far from equilibrium complex dynamics such as low dimensional chaos, Self Organized Criticality (SOC) and intermittent turbulence. Furthermore, in this study, new information is provided about the nonlinear turbulent character of the Hellenic lithospheric dynamics related to the Tsallis nonextensive statistical theory. Our analysis indicates the coexistence of two different lithospheric processes, one low dimensional (chaotic) and the other high dimensional (SOC), revealing the strongly turbulent character of the Greek lithospheric system. In particular, the low dimensional chaotic process corresponds to the temporal manifestation of earthquakes, whereas the high dimensional nonlinear (SOC) process corresponds to the burst energy releases, a result that has significant implications concerning the ability of earthquake prediction.

Mathematical analysis of the navigational process in homing pigeons

In a novel approach based on the principles of dynamic systems theory, we analyzed the tracks of pigeons recorded with the help of miniaturized GPS recorders. Using the method of time lag embedding, we calculated the largest Lyapunov exponent to determine the system’s predictability and the correlation dimension to estimate the number of factors involved. A low Lyapunov exponent around 0.02, which proved to be rather constant over all calculations, indicates that the navigational process is almost deterministic. In the distribution of the correlation dimension estimates we found three distinctive peaks, at 3.3, 3.7 and 4.2, indicating that avian navigation is a complex multi-dimensional process, involving at least four or five independent factors. Additional factors, as indicated by an increase in the correlation dimension, seem to be included as the pigeons approach their home loft. This increase in correlation dimension and its fractal nature suggest that the various navigational factors can be included as required and weighted independently. Neither the correlation dimension nor the Lyapunov exponent is affected by increasing familiarity of the pigeons with the terrain. This suggests that the navigational strategy is stable with the same process controlling the flight across familiar as well as unfamiliar terrain.