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 dimension d of the state space is related to the correlation integral as:

Grassberger and Procaccia (1983) and Swinney (1985) use a form of correlation integral to define the CD:

The correlation integral is the fraction of pairs ([x.

For a given embedding dimension M and a distance [member of], the correlation integral is given by

BDS (1987) employ the correlation integral to obtain a statistical test that has been shown to have strong power in detecting various types of nonlinearity as well as deterministic chaos.

C = the Grassberger and Procaccia correlation integral, and

The GP correlation dimension test utilizes the correlation integral, [C.

The correlation integral C([epsilon]) measures the fraction of the total number of pair of points of {[X.

To deal with the problems associated with using the correlation dimension, Brock, Dechert and Scheinkman (1987) devised a statistical test based on the correlation integral given in Equation 1.

One of the best DOS system programs available to calculate the

correlation integral and BDS statistics can be obtained from W.

The

correlation integral C(e) measures the fraction of the total number of pairs of points such that the distance between them is at most [Epsilon].