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Topic: Correlation dimension

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	Improved Correlation Dimension Calculation (Site not responding. Last check: 2007-10-06)
		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.
		Calculated correlation dimension for the maps in Eq.
		Convergence of the correlation dimension for the Rossler attractor resembles the logistic map.
	sprott.physics.wisc.edu /pubs/paper255.htm (245 words)

	Wikinfo \| Dimension (Site not responding. Last check: 2007-10-06)
		Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space.
		Time is frequently referred to as the "fourth dimension"; time is not the fourth dimension of space, but rather of spacetime.
		The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.wikinfo.org /wiki.php?title=Dimension (654 words)

	Correlation dimension - Wikipedia, the free encyclopedia
		In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points.
		the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, and is often in agreement with other calculations of dimension.
		Each of the sequential correlation integrals, from i = 1 to i = N m, is a function of time, because i is a function of time.
	en.wikipedia.org /wiki/Correlation_dimension (841 words)

	Reference.com/Encyclopedia/Correlation dimension
		In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points.
		the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, and is often in agreement with other calculations of dimension.
		Each of the sequential correlation integrals, from i = 1 to i = N m, is a function of time, because i is a function of time.
	www.reference.com /browse/wiki/Correlation_dimension (861 words)

	Epilepsy, a condition of chronic and recurring seizures, is a brain dysfunction that affects approximately 0
		It is vital to comprehend that the dimension represented by m is not the correlation dimension, but rather the phase space extracted dimension.
		This is done due to the fact that an optimal dimension of the phase space reconstruction of the brain (m) is not yet known and a range of dimensional values must be analyzed.
		Hence, the correlation dimension data as well as the EEG trace can systematically rule out the theory that there is a rhythm generator in one of the patient’s hemispheres.
	www.sfu.ca /~rpyke/335/projects/miron/report.html (4520 words)

	Henon Map Correlation Dimension
		The correlation dimension is given by D = d log(C) / d log(r) in the limit dr --> 0, r --> 0, and N --> infinity.
		To calculate the correlation dimension, one would normally generate a time series of some appropriate number of successive x iterates and then, from the separation of every pair of points in xy-space, construct the function C(r) whose logarithmic derivative approximates the correlation dimension.
		The accuracy with which a correlation dimension can be calculated is dictated by the number of pairs of data points that are correlated.
	sprott.physics.wisc.edu /chaos/henongp.htm (1369 words)

	TEMPORAL VARIATIONS OF THE FRACTAL PROPERTIES OF SEISMICITY (Site not responding. Last check: 2007-10-06)
		The fractal dimension of seismicity is a measure of the degree of both the heterogeneity of the process (whether fixed or self-generated) and the clustering of seismic activity.
		The fractal correlation dimension of the distribution of earthquake epicenters for the same 100-event windows was estimated using the correlation integral method (Grassberger and Procaccia, 1983).
		In this study, we found a weak negative correlation (r=-0.56) for temporal variations of the b-value and the correlation dimension Dc for earthquakes along the western part of the NAFZ and the northern Aegean sea.
	www.angelfire.com /al/geophysics/non/non.htm (3804 words)

	Capacity Dimensions and Correlation Dimensions
		As mentioned earlier, one would expect the average fractal dimension for the elliptical galaxies to be lower than the average fractal dimension for the spiral galaxies because of their less complex structure.
		Although in the example of Figure 8, correlation dimension seems to be working as a separator between the two classes, the histograms show that, in general, we cannot rely on either of the two average fractal dimensions computed for the intensity range selected here for classification.
		When we reexamine Figure 8, we notice that both capacity and correlation dimensions are higher for the spiral galaxy than the elliptical galaxy around the center of the intensity range.
	www.physics.unlv.edu /~thanki/thesis/node21.html (482 words)

	BioMedical Engineering OnLine \| Full text \| Nonlinear analysis of EEG signals at different mental states
		The dimension increases with the degree of the cognitive activity.
		In analysis of EEG data, different chaotic measures such as correlation dimension, Lyapunov exponent and entropy are used in recent literature [16-22].
		The dimension of a graph determined by experimental data is therefore very useful to the understanding of the system, particularly for deciding if the system is periodic, chaotic, or noisy.
	www.biomedical-engineering-online.com /content/3/1/7 (4662 words)

	JSpline+ API Specification: Class Deviation
		Calculates the standard deviation for vectors' variables and the correlation matrix between independent variables and variables to be predicted.
		For example, if vector dimension is 10 and 2 variables are predictable, then the first 8 variables are counted as independent variables and the last 2 variables are predictable.
		correlation values are stored in a double[] array in the following order: first (dimension-predicted) values mean the correlation between independent variables and the first predictable variables, next (dimension-predicted) values correspond to the second predictable variable, and so on.
	www.excelsior-usa.com /doc/jspline/api/ru/sscc/util/data/mining/Deviation.html (388 words)

	Fractals (Site not responding. Last check: 2007-10-06)
		The correlation dimension differs from the box dimension in the way that the box dimension measures each box equally, while the correlation dimension weights the average number of points per region.
		The correlation dimension has advantages over the box dimension in the way that it statistically accounts for the fact that scaling on a fractal surface may not be constant.
		The dimension of slightly more complex fractals can be found using either box dimension or correlation dimension methods, depending on the degree of complexity of the fractal.
	www.zpipedragon.com /Home/Science/Fractals/fractals1.htm (4715 words)

	Exercise alters fractal dimension and spatial correlation of pulmonary blood flow in the horse -- Sinclair et al. 88 ...
		dimension is a scale-independent measure of scale-dependent heterogeneity).
		The fractal dimension statistic is related to the degree of local correlation of flow among lung pieces.
		dimension from trot to canter and a lesser decline or plateau
	jap.physiology.org /cgi/content/full/88/6/2269 (4806 words)

	Comparing Correlation Dimensions and Capacity Dimensions
		If we now compare the capacity dimensions for elliptical galaxies with their correlation dimensions, for the entire range of intensities (Figure 9), not much difference is seen, but spiral galaxies show a noticeable amount of difference between both dimensions.
		Figure 11 shows histograms for the number of spiral galaxies and the number of elliptical galaxies versus the difference between the average correlation dimension and the average capacity dimension for the entire intensity range.
		It was found that the point where the correlation dimensions curve drops below the capacity dimension curve is the point where the star in the lower left corner disappears.
	www.physics.unlv.edu /~thanki/thesis/node22.html (879 words)

	Chaotic behavior of renal sympathetic nerve activity: effect of baroreceptor denervation and cardiac failure -- DiBona ...
		B: estimate of correlation dimension by the method of Grassberger and Procaccia (6, 7).
		The slope of the linear segment in a plot of log correlation sum C(r) vs. log radius r is taken as the correlation dimension d(r).
		Correlation dimension as a function of embedding dimension for 8 normal rats before and after sinoaortic and cardiac baroreceptor denervation.
	ajprenal.physiology.org /cgi/content/full/279/3/F491 (5492 words)

	A short daytime test using correlation dimension for respiratory movement in OSAHS -- Miyata et al. 23 (6): 885 -- ...
		a) Log Cm(r) versus log r of correlation integrals (m=2, 4, 6,..., 20), b) the slopes of curves for various embedding dimensions converge to a saturation value of 2.31, which is the correlation dimension of the attractor in respiration.
		Correlation dimension of the human sleep electroencephalogram: cyclic changes in the course of the night.
		Relationship between correlation dimension and indices of linear analysis in both respiratory movement and electroencephalogram.
	erj.ersjournals.com /cgi/content/full/23/6/885 (2812 words)

	More Abstracts
		We show that the correlation dimension of the attractor of a class of iterated function systems in RN is typically uniquely determined by the contraction rates of the maps which make up the system.
		When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.
		Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps.
	www.chaos.umd.edu /publications/abs2.html (7410 words)

	Comparative Fractal Dimensions
		In other words, the dimension is the number of equations (or unknown variables) necessary to describe the given system; for fractal dimensions, the number of equations is incremented to the next greater integer.
		The fractal dimension is computed for at least M embedding dimensions and D versus M is plotted as in Figure 1.
		If the length of the data series is not large enough, the estimation of the fractal dimension depends greatly on the variable chosen to model the attractor; that is, different variables may yield different fractal dimension estimates (usually underestimates).
	www.public.iastate.edu /~rwarritt/pp36.html (1672 words)

	Correlation dimension (Site not responding. Last check: 2007-10-06)
		To detect the saturation value of the correlation dimension, the function plots the computed correlation dimension as a function of the embedding dimension.
		Mathematically, the correlation dimension is a special case of the generalized dimension, and it is given by
		The correlation dimension is closely related to the correlation integral by Grassberger and Procaccia, which allows an accurate and efficient numerical computation of the attractor's dimension.
	www.ixellence.com /onlinedocu/dataplore/dp_manual62.html (281 words)

	Microcurrent Electrical Therapy (MET) with Alpha-Stim® effects on EEG spectrum and correlation dimension.
		These measures were considered to be clinically relevant since low points in the FFT have been associated with attention deficit disorder, and declining correlation dimension has been associated with onset of epilepsy.
		Correlation dimension of both trapezius and CES groups increased significantly (P<.001) as compared to placebo, with the correlation dimension measures for earlobe being 5.7, trapezius 5.6, and placebo 3.7.
		As correlation dimension increases in EEG data, the eigenvectors transverse many more paths, avoiding a collapse into fixed point attractors and resultant pathologic brain electrical patterns.
	www.alpha-stim.com /Information/Technology/Research/Index/Heffernan_96/heffernan_96.html (452 words)

	[113.15] The Correlation Dimension of Young Stars in Dwarf Galaxies (Site not responding. Last check: 2007-10-06)
		We present the integrated correlation function of resolved young stars in four dwarf galaxies, using Hubble Space Telescope images that minimize the problems of crowding, dust, stellar age identification, and foreground contamination.
		On a scale of 100 pc, cluster dimensions for stars younger than 50 Myr cover a large range, from 0.5 +/- 0.2 to 1.6 +/- 0.1; for comparison, the dimension of interstellar clouds is 1.2 â“ 1.5, typically cited as 1.35.
		The correlation dimensions for older stars are closer to two, as expected for less-clustered, two-dimensional distributions.
	www.aas.org /publications/baas/v37n4/aas207/1083.htm (191 words)

	Correlation Dimension
		This fractal dimension is a simple way to distinguish a random signal from a signal generated by a strange (possibly chaotic) set.
		In contrast, the correlation dimension for a closed curve (a periodic orbit) is 1, and for a two-dimensional surface (such as quasiperiodic motion on a torus) is 2.
		Thus, the correlation dimension is now a standard tool in a nonlinear dynamicist's toolbox that helps one distinguish between noise and low-dimensional chaos.
	cnls.lanl.gov /People/nbt/Book/node94.html (418 words)

	The Science House - Chaos & Fractals
		For example, a straight line has a dimension of 1, a sheet of paper has a dimension of 2, and a solid cube has a dimension of 3.
		That is, an object with a fractional dimension has a dimension that is not an integer but rather a fraction.
		To determine the Correlation dimension you apply the same idea as the Hausdorff dimension except use circles instead of squares.
	www.science-house.org /student/bw/chaos/cfpictures.html (521 words)

	"NG43B-1158" in fm06 (Site not responding. Last check: 2007-10-06)
		In particular, the correlation dimension of earthquake spatial distribution is discussed, techniques for its evaluation presented, and results for several earthquake catalogs are analyzed.
		The value of the correlation dimension declines to 1.8-1.9 for intermediate events (depth interval 71-280 km) and to 1.5-1.6 for deeper ones.
		For plate tectonic deformation on the time scale of millions of years, it is possible that the correlation dimension for shallow earthquakes may increase to 2.6-2.7.
	www.agu.org /cgi-bin/wais?hh=NG43B-1158 (247 words)

	Thaver Abstract (Site not responding. Last check: 2007-10-06)
		The correlation dimension and Lyapunov exponents were calculated for all the EEG data using optimal embedding dimensions and time delays.
		The correlational length (time) τ for the EEG data was more than that for the Random data but less than that for Duffing data, showing more correlation between temporal neighbor than the former but less than the latter.
		The correlation dimension Dc for EEG data was less than Random data but more Duffing data suggesting that EEG is not merely noise.
	www.wooster.edu /Physics/Lindner/SrISAbstracts/Thaver.html (319 words)

	Chaos and Fractals in Financial Markets, Part 8, by J. Orlin Grabbe
		The correlation dimension D* is related to the fractal dimension we defined previously.
		Case 1: If we increase the embedding dimension m, and the estimate of the correlation dimension D* stops increasing, then the correlation dimension is said to saturate.
		Case 3: If we increase the embedding dimension, and the correlation dimension keeps increasing in line with the embedding dimension, and stays more or less equal to the embedding dimension, then the series follows a martingale process.
	www.aci.net /Kalliste/chaos8.htm (1134 words)

	Microcurrent Electrical Therapy (MET) effects on EEG spectrum and pain control with Alpha-Stim®. (Site not responding. Last check: 2007-10-06)
		These measures were considered to be clinically relevant since low points in the FFT have been associated with attention deficit disorder, and declining correlation dimension has been associated with onset of epilepsy.
		Correlation dimension of both trapezius and CES groups increased significantly (P<.001) as compared to placebo, with the correlation dimension measures for earlobe being 5.7, trapezius 5.6, and placebo 3.7.
		As correlation dimension increases in EEG data, the eigenvectors transverse many more paths, avoiding a collapse into fixed point attractors and resultant pathologic brain electrical patterns.
	www.alpha-health.com /research/r21_heffernan_96a.html (451 words)

	EXPERIMENT AND CORRELATION DIMENSION ANALYSIS (Site not responding. Last check: 2007-10-06)
		We have estimated the correlation dimension for the sets of data: Arbitrarily Polarized Antennas (APA, 6000 data points), co-polarized (CPA, 3800 points) and Cross-polarized antennas (XPA, 3800 points) by the box-counting method of Grassberger and Procaccia [3].
		There are a number of limitations and potential pitfalls with correlation dimension estimation that have been discussed by various authors [5].
		The results we found for the various correlation dimensions agreed with all the results known in the literature to within a few percents.
	fraise.univ-brest.fr /~tannous/chaos/node3.html (652 words)

	Fractal dimension - Wikipedia, the free encyclopedia
		In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales.
		The Rényi dimension with α=0 treats all parts of the support of the attractor equally; but for larger values of α increasing weight in the calculation is given to the parts of the attractor which are visited most frequently.
		An attractor for which the Rényi dimensions are not all equal is said to be a multifractal, or to exhibit multifractal structure.
	en.wikipedia.org /wiki/Fractal_dimension (472 words)

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