Long-Range Cross-Correlation Analysis Between Local Variance Modulations in Bivariate Time Series

We propose a method to assess the long-range cross-correlations observed in local variance modulations in bivariate time series. Our method transforms the observed time series into the logarithm of the absolute difference and characterizes the cross-correlation between them using detrending moving-average cross-correlation analysis (DMCA). DMCA employs a Savitzky-Golay detrending filter to remove non-stationary trends embedded in the observed time series, and its mathematical foundation is well established. We demonstrate the validity of our method using numerical experiments assuming a multiplicative lognormal process with long-range-correlated local variances in which a long-range correlated lognormal process modulates the variance of the local Gaussian process. In addition, we applied this methodological framework to cardiorespiratory time series analysis. Although no interaction between interbeat intervals and inter-breath intervals (IBI) has been observed, our study shows the existence of long-range cross-correlation between local variances of those time series.