Correlation Integral


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Correlation Integral

The probability that two points are within a certain distance from one another. Used in the calculation of the correlation dimension.
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The correlation integral of the embedded time series is defined as follows:
With a properly selected time delay, the considered time series can be reconstructed in the 7M-dimensional phase space by calculating the correlation exponent from the correlation integral (C(r)) as follows:
The correlation integral can be rewritten in the form
The process used to obtain the fractal dimension is the correlation integral, which compares the proximity of events to one another:
The Attractor dimension estimation was accomplished by calculating the correlation integral C(R) (Grassberger and Proccacia, 1983):
Grassberger and Procaccia (1983) and Swinney (1985) use a form of correlation integral to define the CD:
For a given embedding dimension M and a distance [member of], the correlation integral is given by
9 shows log-log plots of the correlation integral against length scale for different embedding dimensions.
C = the Grassberger and Procaccia correlation integral, and
m, T]([Epsilon]) = the correlation integral of the finite (T-length) series
The correlation integral for a data series {[[alpha].
One of the best DOS system programs available to calculate the correlation integral and BDS statistics can be obtained from W.

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