The problem with 'fractal dimensio' is that the differnt quantities are called so. One of those is 'correlation dimension'.

The correlation dimension is one of many ways to calculate the dimension of a fractal. Thus it is a specific method usually applied when one has time-series data, whereas fractal dimension is a general term. Note that the different ways of calculating a fractal dimension do not generally give the same result, especially if the object is multifractal.

The state of chaos exhibited in nonlinear systems may appear through various

roots. As a measure of chaos, the most suitable tools to be considered are: Lyapunov

characteristic exponents (LCE) and topological entropies.. A chaotic

system exhibits a chaotic set, strange attractor, that may have fractal property.

Calculation of correlation dimension is required to obtain the dimension of such

attractor. The correlation dimension actually gives a measure of complexity for the

underlying attractor of the system. Strange attractors are conceptual objects

that exist in the state space of chaotic dynamical systems. The Grassberger-Procaccia Algorithm is used for estimating the correlation dimension of some fractal measure from a given set of points. A fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. Among the techniques discussed by Mandelbrot the box counting method is also found like the most adapted for the estimation of fractal dimension

You may want to have a look at Hentschel and Proccaccia's paper defining an infinite number of generalized fractal dimensions D_q (Physica 8D (1983) 435---444). In general D_q>=D_q', if q

Dimension is a property of a mathematical object that refers to the extent it occupies the space in which it is embedded. There are many formal definitions of dimension, if the definition allows non-integer values (a fraction), it is a fractal dimension. The most rigorous definition is the Hausdorff-Besicovitch dimension, because it is defined over a measure space. The box-counting dimension is defined over a vector space, but there is also the packing dimension, compass dimension, etc. etc. See the recent research topic in FrontiersIn Physiology (open access): http://www.frontiersin.org/Fractal_Physiology/researchtopics/Fractal_Analyses_Statistical_a/555 which contains many tutorials and suggestions for reliable dimension estimation.

Two good answers already - correlation dimension is one type of fractal dimension, and there are many different fractal dimensions. I would that the correlation dimension of random points on a fractal (random with respect to its Hasudorff measure) agrees with the Hausdorff dimension (cf Hastings and Sugihara, Fractals ..., Oxford Press, 1993) and also cite Ruelle's caution (1990 I think) about computing dimensions of attractors from too small a set of points.

see : Wireless Sensor Network, 2010, 2, 67-73

doi:10.4236/wsn.2010.21010 anuary 2010 (http://www.SciRP.org/journal/wsn/).

Copyright © 2010 SciRes. WSN

Published Online J:

> by Behzad AHMADI and Rassoul AMIRFATTAHI

Yes it is important in analysis of experimental time series to explore by using different kinds of measurement. I also suggest in D2 measurement to control by using surrogate data.

For a person whose main iterested lies in quantifying chaos from experimental data, but is not a specialist, I think it could be useful the book of Neil Geshenfeld "The nature of mathematical modelling" (Cambridge University Press, 1999) and the references given there. There is an older chapter of Gershenfeld, "An experimentalist's introduction to the observation of dynamic systems" in a book of Hao (1990) that introduces the reader to many techniques that could be of interest for an experimentalist.

As said before by other people, in spite of Hausdorff-Besicovitch rigorous definition, when landing in practical matters there is no single fractal dimension: correlation dimension is a possible and often convenient one.

In some cases, please, also remember to evaluate the possible trend of the data.It is neccessary in some cases.

I agree Harlan, just Hurst exponent estimation often requires to detrend first the time series data.

Amit, I just remember that we have a Research Center specialized in non linear methodologies in analyisis of time series and we have still courses at distances in case this could be of some interest.

The correlation dimension is one member of the big family of fractal dimensions.

1) The most popular, probably because this is the first one introduce by B.Mandelbrot, is the capacity dimension (also known as box counting dimension). It consists in computing the number of small cubes needed to cover the whole fractal object and to compare the result with the boxes size on a log scale. The limit of the ration when cubes size tends toward 0 defines the box counting dimension (denoted D0)

* The drawback of the capacity dimension is that it is a purely geometric notion and that is not sensitive to the density of points. If the object is a dynamical trajectory (as it is probable for your EGG) or a sampled data series issued from a dynamical process, this remark means that Boxe counting dimension do not take into account the dynamic behavior. This fractal is not sensitive to time spent in a cube.

2) Since points are not spread out with a uniform density, some regions are more often visited than others. This fact is taken into account by the entropy notion which evolves the probability of occupancy of each cover cube. From the entropy, the information dimension is defined. This second fractal dimension is denoted (D1)

** If you use some toolboxes to perform estimations of fractals, be very aware with the consistency of results because they are very sensitive to the algorithm parameters. One way to check the consistency is to verify D0 > D1 (D0 = D1 means a uniform density of points).

3) Now, take a look to another fractal dimension: the correlation dimension (denoted D2). If you are interesting in fractal matter, I suppose that you EGG is dealing with chaotic dynamic. Then, through a proximity property of two random points on the chaotic attractor, the correlation dimension takes into account the fact that the attractor is a dense set of points. The correlation integral is first computed versus the cube size and then the correlation dimension by a limit when the cube size tends toward 0.

*** Because your data set is not infinite, you will not be able to compute the limit when the cube size tends towards 0. Also, the statistical properties of noise become prevailing for very small boxes. Then the slope of the correlation integral versus the cube size is a sufficient estimator of the correlation dimension.

**** An other important remark is the necessity to not perform fractal estimations on original attractor but only on an embedded attractor using the embedding theory but it's an other topic (see at the end of my answer).

***** You have to compute several estimations of the correlation dimension by increasing the embedding dimension. When it is high enough, the correlation dimension converges to a constant value.

****** An other checkpoint for the consistency is D0 > D1 > D2

4) Reny's spectrum

The previous fractal dimensions can be considered as the first elements of the Rényi dimensions spectrum.

The Rényi’s dimensions are defined from the notion of generalized entropy. The principle is to weight the probability of the most often visited cubes according to the order of the dimension.

******* A relationship has to to be check by Renyi's dimensions: D_q+1 < D_q.

EMBEDDING THEORY

Embedding process have to precede any estimations of fractals from a data series. First of all, the question is "why do we embed"?

Generally speaking, one is not able to sense all state variables. Theoretically, according to the Takens theorem, any state variable can be used to calculate the invariants of the dynamics. But practically, it is not a calculation that is made, but only an estimation performed by an algorithm. This raises the problem of convergence and it is directly related to the more or less good conditioning of the useful information into the variable.

The main practical task are:

A) The choice of the embedding variable.

B) The choice of the embedding delay.

After having chosen the embedding variable, the next step is the determination of the right embedding delay. It can be estimated by the first zero crossing of the autocorrelation function or, better, by the first local minimum of the mutual information.

A way is to program the Fraser and Swinney algorithm.

C) After that the determination of the embedding dimension (it's not a fractal but an integer usually denoted De).

The embedding dimension is the minimal dimension of the state space that allows the attractor reconstruction without topological ambiguity and able to represents all dynamic behaviors.

In order to find the right embedding dimension, you can use the False Nearest Neighbors algorithm. The good dimensions are the values with the number of false neighbors near to zero. You will see that the result depend on the size of the data set (the biggest is the best) but also of your initial choice for the embedding variable... So be very careful...

Once again, remember that algorithms are useful but the expertise is essential for interpreting results and doing the good choices. It can be seen the importance to have a critical glance in order to align the results of various estimations with the other invariants like fractal dimensions.

Fore example, base on a mathematical conjecture, it is expected that De >= 2Df, where Df denotes the fractal dimension.

Pr. B.G.M. ROBERT - University of Reims Champagne-Ardenne, France.

Hello Bruno Robert, you write: " The drawback of the capacity dimension is that it is a purely geometric notion and that is not sensitive to the density of points." That may be true (about the density), but in a setting of of time series analysis, an advantage of that the geometric notion is that it does not require any assumptions about the physical change process underlying the measurements. e.g. many analyses estimate a self-affinity parameter (H) and then convert to D via some rule (2-H). The conversion rule however is based on an assumption about the measured process (compare fGn vs. fBm). Here http://www.frontiersin.org/Fractal_Physiology/10.3389/fphys.2013.00075/full

I make an argument in favour of this view (includes scaling exponent conversion schemes).

Also you write: "An other important remark is the necessity to not perform fractal estimations on original attractor but only on an embedded attractor" I am very interested finding about about the arguments that support this, can you point me to some papers? Reason is, in my field nobody is embedding!

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Hi Amit here is another thought for EEG analysis: Recurrence Quantification Analysis

Schinkel, S., Marwan, N., & Kurths, J. (2009). Brain signal analysis based on recurrences. Journal of physiology, Paris, 103(6), 315–23. doi:10.1016/j.jphysparis.2009.05.007

Schinkel, S., Marwan, N., & Kurths, J. (2007). Order patterns recurrence plots in the analysis of ERP data. Cognitive neurodynamics, 1(4), 317–25. doi:10.1007/s11571-007-9023-z

Given two EEG signals (left and right hemisphere) is there a way to use fractal analysis such as Highuchi's fractal dimension, in a way that takes both channels into account? Has anyone ever explored whether this yields any useful information?

Dear Amit, you can calculate fractal dimension both through box counting (Df) and correlation integral (Dc) methods. The difference between the two results is that Df is slightly higher than Dc (about 5%), but Dc presents a higher correlation coefficient (R^2) than Df. Usually, I apply the correlation integral of Grassberger & Procaccia (1983) as a tool to analyse natural morphologies (rivers, coasts, etc.). The reference is: GRASSBERGER, P., and PROCACCIA, I. (1983), Characterization of

strange attractors, Phys. Rev. Lett. 50, 346–349.

Best, Carlo