On the Efficacy of the Combined Use of the Cross-Bicoherence with Surrogate Data Technique to Statistically Quantify the Presence of Nonlinear Interactions

Abstract

The cross-bispectrum is an approach to detect the presence of quadratic phase coupling (QPC) between different components in bivariate signals. Quantification of QPC is by means of the cross-bicoherence index (CBI). The major limitations of the CBI are that it favors only the strongly coupled signals and its accuracy becomes compromised with noise and low coupling strength. To overcome this limitation, a statistical approach which combines CBI with a surrogate data method to determine the statistical significance of the QPC derived from bivariate signals is introduced. We demonstrate the accuracy of the proposed approach using simulation examples which are designed to test its robustness against noise contamination as well as varying levels of phase coupling and data lengths. Comparisons were made to the traditional CBI and the method based on the use of cross-bispectrum followed by a surrogate data technique. Our results show that the cross-bicoherence with surrogate data technique outperforms the two other methods compared in both sensitivity and specificity, and provides an unbiased and statistical approach to determining the presence of QPC in bivariate signals. These results are in contrast to our recent study where the auto-bispectrum combined with surrogate data approach had the best performance. Application of this approach to renal hemodynamic data was applied to renal stop flow pressure data obtained in the nephrons of the normotensive (N = 18) and hypertensive (N = 15) rats. We found significant nonlinear interactions between nephrons only when they are derived from the same cortical renal artery. The accuracy was 100% and verified by comparing the results to the known vascular connectivity between nephrons.