Networks analysis of Brazilian climate data based on the DCCA cross-correlation coefficient

The DCCA cross-correlation coefficient The DCCA cross-correlation coefficient proposed by Zebende (2011) [23] is centred on the ratio between the detrended covariance function F2 DCCA [32] and the variance function FDFA [22], as showed in the following five steps: 1. Consider two time series, {xi} and {yi}, i = 1, 2, . . .., N. These time series are integrated to obtain two new time series {Xk} and {Yk}: Xk ¼ X k i¼1 ðxi hxiÞ and Yk ¼ X k i¼1 ðyi hyiÞ; ð4Þ where hxi and hyi represent the average of each time series. 2. These two integrated time series, {Xk} and {Yk}, are divided into (N − n) overlapping boxes of equal length n, in which n varies between 4 and N 4 . 3. The local trend for each box is calculated by using a least-squares fit of each series, f ~Xk;ig and f ~Y k;ig, and the covariance of the residuals is calculated for each box: f 2 DCCAðn; iÞ ¼ 1 ðn þ 1Þ X iþn k¼i ðXk ~Xk;iÞðYk ~Y k;iÞ: ð5Þ The variance of the residuals is also computed by using: f 2 DFAx ðn; iÞ ¼ 1 ðn þ 1Þ X iþn k¼i ðXk ~Xk;iÞ2 ð6Þ f 2 DFAy ðn; iÞ ¼ 1 ðn þ 1Þ X iþn k¼i ðYk ~Y k;iÞ2 ð7Þ 4. The average over all (N − n) overlapping boxes is calculated to obtain the detrended covari- ance function: F2 DCCAðnÞ ¼ 1 N n X N n i¼1 f 2 DCCAðn; iÞ; ð8Þ and the detrended variance functions: FDFAx ðnÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N n X N n i¼1 f 2 DFAx ðn; iÞ s ð9Þ FDFAy ðnÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

N n X N n i¼1 f 2 DFAy ðn; iÞ s ð10Þ PLOS ONE Networks analysis of Brazilian climate data based on the DCCA cross-correlation coefficient PLOS ONE | https://doi.org/10.1371/journal.pone.0290838 September 15, 2023 5 / 18