Dimension correlation

Once the reconstructed portrait is found, the same type of analyses as were described for the simulated data—that is, construction of Poincare sections and Poincare maps, calculation of the fractal dimension, etc.—can be carried out. Of these, the calculation of the fractal dimension is often of interest, although, as cautioned earlier, the knowledge of this number cannot, alone, distinguish chaotic data from nonchaotic data. An understanding of the route by which the suspected chaotic state arises is also necessary before a definitive statement can be made. 

Nevertheless, the determination of the fractal dimension from a data set thought to be chaotic is often of interest. A number of different dimensions exist in the literature, including the Hausdorff dimension, the information dimension, the correlation dimension, and the Lyapunov dimension. Which of these is the true fractal dimension Of the ones in this list, the information dimension, Di, has the most basic and fundamental definition, so we often think of it as the true fractal dimension. Because the information dimension is impractical to calculate directly, however, most investigators have taken to finding the correlation dimension, Dq, as an estimate of the fractal dimension. Grassberger and Procaccia published a straightforward and widely used algorithm for the calculation of the correlation dimension. On the other hand, the Lyapunov 

Reference 93 can be consulted for a full explanation of the differences between the dimensions used to estimate the fractal dimension of an attractor. 

the function H is the Heaviside step function which is zero if e u — Uy and one if e u, - u. The correlation sum is found to follow a simple power law form which can be used to determine the correlation dimension, D  

Taking the natural logarithm of both sides of this equation yields 

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Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

Correlation dimension. The correlation dimension is calculated by measuring the Hausdorff dimension according to the method of Grassberger [36,39]. The dimension of the system relates to the fewest number of independent variables necessary to specify a point in the state space [40]. With random data, the dimension increases with increase of the embedding space. In deterministic data sets, the dimension levels off, even though the presence of noise may yield a slow rise. 

In that spirit, Widman et al. [576] adapted a prescription for an overall index of nonlinear coherence that has been found powerful for anticipating epileptic seizures from implanted electrode recordings. This index based on phase space reconstruction and correlation sums was called d, and it contains many ingredients familiar from the Grassberger-Procaccia algorithm for the correlation dimension [577]. 

Nonlinear analysis requires the use of new techniques such as embedding of data, calculating correlation dimensions, Lyapunov exponents, eigenvalues of singular-valued matrices, and drawing trajectories in phase space. There are many excellent reviews and books that introduce the subject matter of nonlinear dynamics and chaos [515,596-599]. 

Albano, A., Mess, A., DeGuzman, G., and Rapp, P., Data requirements for reliable estimation of correlation dimensions, Chaos in Biological Systems, edited by H. Demi, A. Holden, and L. Olsen, Plenum Press, New York, 1987, pp. 207-220. 

Silva, C., Pimentel, I., Andrade, A., Foreid, J., and Ducla-Soares, E., Correlation dimension maps of EEG from epileptic absences, Brain Topography, Vol. 11, No. 3, 1999, pp. 201-209. 

Fig. 16. A chaotic front moving through a fixed-bed reactor during CO oxidation over Pt/ AljOj. Ta, Tk, Tc, and Tj are pellet temperatures in the entrance cross section T) and T2 are thermocouple temperatures 24 and 46 mm downstream, respectively. Chaos was characterized by correlation dimension and Liapunov exponent. (From Ref. 700.)...

Effective Temperatures, Energies, and Correlation Dimensions for the Argon Trimera... 

P", y" Empirical constants in gas-liquid mass transfer correlations (dimensioned appropriately)... 

The correlation dimension takes account of the density of points on the attractor, and thus differs from the box dimension, which weights all occupied boxes equally, no matter how many points they contain, (Mathematically speaking, the correlation dimension involves an invariant measure supported on a fractal, not just the fractal itself.) In general, c/corretaUon - > although they are usually very... 

Estimate the correlation dimension of the Lorenz attractor, for the standard pa-... 

Soluiion Figure 11.5.3 shows the results of Grassberger and Procaccia (1983). (Note that in their notation, the radius of the balls is ( and the correlation dimension is V.) A line of slope =2.05 0.01 gives an excellent fit to the data, except for large e, where the expected saturation occurs. 

These results were obtained by numerically integrating the system with a Runge-Kutta method. The time step was 0.25, and 15,000 points were computed. Grassberger and Procaccia also report that the convergence was rapid the correlation dimension could be estimated to within 5 percent using only a few thousand points. ... 

The correlation dimension ofthe limiting set has been estimated by Grassberger and Procaccia (1983). They generated a single trajectory of 30,000 points, starting from. ro = i. Their plot of logC( ) vs. log is well fit by a straight line of slope... 

Project) Write a program to compute the correlation dimension of the Lorenz attractor. Reproduce the results in Figure 11.5.3. Then try other values of r. How does the dimension depend on r ... 

In principle, attractor reconstruction can distinguish low-dimensional chaos from noise as we increase the embedding dimension, the computed correlation dimension levels off for chaos, but keeps increasing for noise (see Eckmann and Ruelle (1985) for examples). Armed with this technique, many optimists have asked questions like. Is there any evidence for deterministic chaos in stock market prices, brain waves, heart rhythms, or sunspots If so, there may be simple laws waiting to be discovered (and in the case of the stock market, fortunes to be made). Beware Much of this research is dubious. For a sensible discussion, along with a state-of-the-art method for distinguishing chaos from noise, see Kaplan and Glass (1993). 

Wongchaowart NT, Rinehart DG, Hsi ED. The correlation dimension identifles B-cell immunoglobulin hght chain restriction in peripheral blood flow cytometry data. Am J Chn Pathol 2006 125(4) 600-607. 

Figure 38 Definition of the distance U -Uy used in determining the correlation sum, Eq. [118], for the correlation dimension.

Because the Grassberger-Procaccia algorithm is the most widely used for the calculation of the correlation dimension, we briefly review it here. As discussed above, a phase space portrait must first be reconstructed from the measured data. The resulting attractor consists of a set of ordered points, uj, U2,.. ., U ,.. . , where u,- = x(tj), y(tj), z(tj)) is the fth point in the trajectory from which the attractor has been reconstructed. The correlation dimension is calculated by comparing the distance between any two points on the attractor (see Figure 38) with a given small distance, e, which will be varied. The number of pairs of points, (u u,), that both fall inside a ball of diameter e are counted. The correlation sum C(e) is the average number of point pairs inside balls of diameter e spread over the attractor... 

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are ... 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

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