Correlation Dimension

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Correlation Dimension

An estimate of the Fractal Dimension which measures the probability that two points chosen at random will be within a certain distance of each other, and examines how this probability changes as the distance is increased. White noise will fill its space since its components are uncorrelated, and its correlation dimension is equal to whatever dimension it is placed in. A dependent system will be held together by its correlations and retain its dimension whatever embedding dimension it is placed in, as long as it is greater than its fractal dimension.
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Caption: Figure 2: Box plot representing distribution of (a) correlation dimension (CD) and (b) largest Lyapunov exponent (LLE) in EEG of encephalopathic patients and EEG of normal subjects.
The correlation dimension can be expressed as follows:
When m continues to increase, the limit value of v is the correlation dimension of the system.
where C is the correlation dimension; r is the radius of the region in multidimensional space.
Chaotic features such as Lyapunov exponent, correlation integral, and correlation dimension are computed and combined with a feature vector.
We deploy two tests of chaos: (i) the Correlation Dimension of Grassberger and Procaccia (1983) and Takens (1984), (ii) and the BDS statistic of Brock, Dechert, and Scheinkman (1987) which are discussed in detail in Adrangi et al.
We estimated the correlation dimension and the largest Lyapunov exponent in order to estimate the invariants of the attractor.
Taking into account , the minimum embedding dimension is once more determined by the closest integer above the correlation dimension. In this case, this provides the proper minimum embedding dimension [m.sub.min], which in this case possesses the value [m.sub.min] = 2 for wax and lead solder, while it is [m.sub.min] = 2 for gypsum.
Correlation dimension estimation is highly dependent on the length of the time series .
By calculating the correlation dimension of DTG data to ensure that the correlation dimension will reach saturation status as shown in Figure 3, It is indicated that the nonlinear chaotic characteristics exist in system.
To determine the chaotic dynamics within the stream flows, the correlation dimension method was applied.
Distance-based fractal measure is the correlation dimension.

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