# 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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When m continues to increase, the limit value of v is the correlation dimension of the system.
2) By calculating the correlation dimension d, the embedding dimension m can be determined according to the formula m [greater than or equal to] 2d + 1.
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.
In the latter situation, the correlation exponents reach a saturation value, which provides the correlation dimension of the system.
Distance-based fractal measure is the correlation dimension.
By analysing the correlation dimension (method of Grassberger & Procaccia, 1983) the author found that in five of the six series analysed (the exception being one of the pairs in the experimental study) the decision-making processes showed chaotic dynamics.
2] ([micro]) is the correlation dimension of the measure [micro] (see Sect.
Also it is possible to obtain the correlation dimension because the dimension of each mode converges to a constant value.
For each of these four variables the correlation dimension [D.
Tests based on the correlation dimension partially confirm the presence of nonlinearity.

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