Generalized theory for detrending moving-average cross-correlation analysis: A practical guide

To evaluate the long-range cross-correlation in non-stationary bi-variate time-series, detrending-operation-based analysis methods such as the detrending moving-average cross-correlation analysis (DMCA), are widely used. However, its mathematical foundation has not been well established. In this paper, we propose a generalized theory to form the foundation of DMCA-type methods and introduce the higher-order DMCA in which Savitzky-Golay filters are employed as the detrending operator. Using this theory, we can understand the mathematical basis of DMCA-type methods. Our theory establishes a rigorous relationship between the DMCA-type analysis, the cross-correlation function analysis, and the cross-power spectral analysis. Based on the mathematical validity, we provide a practical guide for the use of higher-order DMCA. Additionally, we present illustrative results of a numerical and real-world analysis. To achieve reliable and accurate detection of the long-range cross-correlation, we emphasize the importance of time-lag estimation and time scale correction in DMCA, which has not been pointed out in the previous studies.