Dimension estimation with the BDS-G statistic

Abstract

The issue of time delay has been controversial among the specialized literature. In fact, there exist some contrasted methods to choose it. It is the case that even though they are investigated for chaotic series, they fail to detect which series come from a deterministic and chaotic system. In this study a new procedure for selecting the delay time which produces good results about the estimation of the correlation dimension in chaotic series is introduced. The method is based upon a statistic (BDS-G), rooted on the integral correlation function, that takes advantage of the information contained in the data in terms of dependence, and it uses it to choose proper delay times and embedding dimensions. The results for the studied series, even for small data sets, are satisfactory.

Acknowledgements

Authors are grateful for financial aid under research project PB98-0059 DGES.

Notes

 When using the terms  ‘ very small ’  and  ‘ large ’ , Buzug et al. (1990) is followed, so these expressions are understood relative to the characteristic recurrence time, which is given by the reciprocal value of the dominant frequency obtained from the power spectrum.

 The BDS-G test is a nonparametric test for independence based on BDS test, which has good power against a variety of alternatives that are common in economics (see Matilla et al., 2002).

 If an improper selection for delay time is carried out, the G-P algorithm will not collect all the available information to calculate the dimension.