The correlation dimension ratio: comparison with the DBS statistic

Abstract

In "An Alternative Correlation Statistic", the author compares the Correlation Dimension Ratio (CDR) with Grassberger and Procaccia's Correlation Dimension, (CD) showing it to be equally useful for distinguishing between deterministic and stochastic processes, while at the same time providing a test for the IID null. The CDR also eliminates the CD's bias towards low dimension in relatively short time series (this bias accounts for spurious low-dimensional results of 5 to 7 in the literature). Here, the CDR, and a related statistic, the Correlation Ratio Logarithm (CRL), are compared to the DBS statistic in a Monte Carlo experiment measuring the moments of their distributions. The CDR, followed by the CRL, are found to be consistently closer to the normal distribution, more sensitive to stochastic structure, and les heteroskedastic as varies.