Estimation Of Correlation Dimension Research Articles

A Consistent Approach to Least Squares Estimation of Correlation Dimension in Weak Bernoulli Dynamical Systems

A new approach to the least squares procedure for correlation dimension estimation is suggested. Consistency of the new estimator is established for a class of dynamical systems that includes the Cantor map and the logistic map with parameter value 4. Unlike the proofs of consistency for other estimation procedures, no assumptions are made about the Grassberger-Procaccia spatial correlation integral beyond the existence of the correlation dimension.

A note on bias correction for correlation dimension

Correlation dimension estimates are often calculated by the well-known algorithm of Grassberger and Procaccia [Phys. Rev. Lett. 50 (1983) 346] for white noise data. One source of bias for such estimates is the use of log ϵ in regression calculations. It is shown by simple Taylor expansions that this bias is of order ϵ or smaller and does not need in general to be corrected. Should nevertheless a bias correction be necessary, it is suggested that extrapolation is safer than the use of a different regressand, such as log[ ϵ(2− ϵ)], which requires additional knowledge of the true white noise distribution.

Implications of recording strategy for estimates of neocortical dynamics with electroencephalography.

Neocortical dynamics evidently involves very complex, nonlinear processes including top-down and bottom-up interactions across spatial scales. The dynamics may also be strongly influenced by global (periodic) boundary conditions. The primary experimental measure of human neocortical dynamics at short time scales ( approximately few ms) is the scalp electroencephalogram (EEG). It is shown that different recording and data analysis strategies are sensitive to different parts of the spatial spectrum. Thus experimental measures of system dynamics (e.g., correlation dimension estimates) can generally be expected to depend on experimental method. These ideas are illustrated in two ways: a large scale, quasilinear theory of neocortical dynamics in which standing wave phenomenon occur with predicted frequencies in the general range of EEG, and a relatively simple nonlinear physical system consisting of a linear string with attached nonlinear springs. The string/springs system is integrated numerically to illustrate transitions from periodic to chaotic behavior as mesoscopic nonlinear influences dominate macroscopic linear effects. The implications of these results for new theories of neocortical dynamics, experimental estimates of dynamic properties, and cognitive EEG studies are considered.

Comparison of algorithms calculating optimal embedding parameters for delay time coordinates

The success of noise reduction processes and forecasting methods as well as accurate calculations of fractal dimensions and Lyapunov exponents from scalar time series depends strongly on the quality of the reconstructed phase space. We present a comparison of four methods, which simultaneously estimate the two embedding parameters (delay time τ and sufficiently large embedding dimension dim E) for Takens' delay time coordinates. Recently we introduced a global geometrical measure, the fill-factor, and a local dynamical measure, the averaged integral local deformation. We will discuss these methods briefly and compare them to the topological wavering-product. In addition a simple algorithm is presented, which observes the spreading of trajectories at the transition from dim E to dim E + 1 and gives the opportunity to estimate the correlation entropy K 2. We applied these algorithms to experimental data series measured from a rotational Taylor-Couette experiment and verified the quality of the obtained embedding parameters by the estimation of correlation dimensions for different reconstructed attractors.

Mutual information, strange attractors, and the optimal estimation of dimension

It has been shown that the appropriate setting of data windows is crucial to a successful estimation of a time-series correlation dimension using the Grassberger-Procaccia algorithm [Physica 9D, 189 (1983); Phys. Rev. Lett. 50, 346 (1983)], and it has been proposed that the first minimum of the corresponding mutual-information function may be an appropriate window value. We have tested this hypothesis against data generated by the Rossler equations, the Lorenz equations, and a three-dimensional irrational torus. We conclude that mutual information is not consistently successful inidentifying the optimal window

Improved correlation dimension estimates through change of variable

To evaluate the correlation dimension of chaotic regimes of a CO 2 laser with modulated losses, attractor reconstruction using the method of delays is performed using the logarithm of the intensity rather than the intensity itself. Improved convergence with respect to the embedding dimension and better reliability are observed.

A maximum likelihood approach to correlation dimension and entropy estimation.

To obtain the correlation dimension and entropy from an experimental time series we derive estimators for these quantities together with expressions for their variances using a maximum likelihood approach. The validity of these expressions is supported by Monte Carlo simulations. We illustrate the use of the estimators with a local recording of atrial fibrillation obtained from a conscious dog.

A maximum likelihood approach to correlation dimension and entropy estimation

A maximum likelihood approach to correlation dimension and entropy estimation

A simple noise-reduction method for real data

A method to reduce noise in experimental data with nonlinear time evolution is presented. A locally linear model is used to obtain a trajectory consistent with the dynamics as well as with the measured data. In contrast to previous methods where the fit to the dynamics and the cleaning are done in separate steps, it is an (iterated) one step procedure. This is made possible by using data both from the past and from the future in the locally linear model. The method is applied to both artificial and real data. Among others, it leads to a significant improvement of correlation dimension estimates.

Comments on ‘Deterministic chaos: the science and the fiction' by D. Ruelle

The results of several works concerning estimates of correlation dimension have recently been criticized on the basis of a new upper bound on such estimates (i. e. 2 log 10 N , where N is the number of data). It is shown here that this bound is not valid for correlation dimension estimates in general, and furthermore it is too weak to be useful for régimes where it does hold. This casts the indicated criticism into question.

ESTIMATING DIMENSIONS WITH CONFIDENCE

Estimation of correlation and other fractal dimensions is a favored approach for investigating whether time series or other dynamical data may be produced by nonlinear deterministic systems. The standard methods have disadvantages which are widely recognised; briefly put, an estimate is always produced, with no indication of its reliability or likely error. By assuming an explicit geometrical model for an attractor, we are able to produce a better estimator and to give a form of weak confidence intervals. Experience so far indicates that this approach is much less likely to give misleading answers than the conventional ones.

Measuring the Strangeness of Gold and Silver Rates of Return

The predictability of rates of return on gold and silver are examined. Econometric tests do not reject the martingale hypothesis for either asset. This failure to reject is shown to be misleading. Correlation dimension estimates indicate a structure not captured by ARCH. The correlation dimension is between 6 and 7 while the Kolmogorov entropy is about 0·2 for both assets. The evidence is consistent with a nonlinear deterministic data generating process underlying the rates of return. The evidence is certainly not sufficient to rule out the possibility of some degree of randomness being present.

Dynamics of brain electrical activity.

In addition to providing important theoretical insights into chaotic deterministic systems, dynamical systems theory has provided techniques for analyzing experimental data. These methods have been applied to a variety of physical and chemical systems. More recently, biological applications have become important. In this paper, we report applications of one of these techniques, estimation of a signal's correlation dimension, to the characterization of human electroencephalographic (EEG) signals and event-related brain potentials (ERPs). These calculations demonstrate that the magnitude of the technical difficulties encountered when attempting to estimate dimensions from noisy biological signals are substantial. However, these results also suggest that this procedure can provide a partial characterization of changes in cerebral electrical activity associated with changes in cognitive behavior that complements classical analytic procedures.