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

correlation integral of the embedded time series is defined as follows:

Recently, several chaotic features, including Largest Lyapunov exponent (LLE), correlation dimension and

correlation integral have been used to represent time series for recognition purpose.

The BDS statistic is derived from the

correlation integral and has its origins in the recent work on deterministic nonlinear dynamics and chaos theory.

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.

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, 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].

The

correlation integral can then be used to calculate the fractal dimension (FD):

The Attractor dimension estimation was accomplished by calculating the

correlation integral C(R) (Grassberger and Proccacia, 1983): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Grassberger and Procaccia (1983) and Swinney (1985) use a form of

correlation integral to define the CD:

de Lima, P.J.F., 1995b, "Nuisance Parameter Free Properties of

Correlation Integral Based Statistics," Working Paper, John Hopkins University.

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