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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C (m, r, [tau]) = 1/[M.sub.2] [summation over (1[less than or equal to]j[less than or equal to]k[less than or equal to]M)] [theta](r -[parallel][X.sub.j] - [X.sub.k][parallel]) is the correlation integral of the system.
The application of the improved chaos algorithm in the football performance prediction
The correlation integral of the embedded time series is defined as follows:
Method of Quantitative Analysis for Multirobot Cooperative Hunting Behaviors
Recently, several chaotic features, including Largest Lyapunov exponent (LLE), correlation dimension and correlation integral have been used to represent time series for recognition purpose.
Exploiting chaotic feature vector for dynamic textures recognition
The BDS statistic is derived from the correlation integral and has its origins in the recent work on deterministic nonlinear dynamics and chaos theory.
Noise reduction analysis of radar rainfall using chaotic dynamics and filtering techniques
Then, if the attractor is a strange one, the correlation integral will be proportional to [r.sup.v], where v is a measure of the attractor's dimension called correlation dimension.
Analysis of rattleback chaotic oscillations
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:
Nonlinear forecasting of stream flows using a chaotic approach and artificial neural networks
The correlation integral, in fact, a distribution function of distances between all pairs of points in a set of points in a space with a distance, was introduced in [11].
Separation in data mining based on fractal nature of data
The correlation integral can then be used to calculate the fractal dimension (FD):
Clustering analysis of AE in rock
The Attractor dimension estimation was accomplished by calculating the correlation integral C(R) (Grassberger and Proccacia, 1983): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Studies on the phase space of multiscale seismicity
Grassberger and Procaccia (1983) and Swinney (1985) use a form of correlation integral to define the CD:
Nonlinearity and low deterministic chaotic behavior in insurance portfolio stock returns
de Lima, P.J.F., 1995b, "Nuisance Parameter Free Properties of Correlation Integral Based Statistics," Working Paper, John Hopkins University.
Efficiently estimated mean and volatility characteristics for the Nordic Spot electric power market
One can measure the spatial correlations among the M-histories by calculating the correlation integral. For a given embedding dimension M and a distance [member of], the correlation integral is given by
Nonlinear dependence in gold and silver sutures: Is it chaos?

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