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

5. Finally, the cross-correlation coefficient is computed as: rðXi;XjÞðnÞ ¼ F2 DCCAðnÞ FDFAx ðnÞ FDFAy ðnÞ ; ð11Þ where rðXi;XjÞðnÞ ranges from 1 � rðXi;XjÞðnÞ � 1, in which: i. rðXi;XjÞðnÞ ¼ 1 means a perfect cross-correlation; ii. rðXi;Xj ÞðnÞ ¼ 0 there is no cross-correlation; and iii. rðXi;Xj ÞðnÞ ¼ 1 means a perfectly anti cross-correlation. It is important to notice that rðXi;XjÞ depends on n (time scale). So, one of the advantages of this detrended cross-correlation coefficient is to measure cross-correlations between two non- stationary time series at different time scales [24, 33–37]. Network analysis A graph is a mathematical structure that models the pairwise relations between objects. It includes vertices/nodes/points connected by edges/links/lines. There are two main types of graphs: (i) undirected graphs, where edges link two vertices without a specific direction, and (ii) directed graphs, where edges link two vertices in a specific direction. Formally, a graph can be defined as an ordered pair G = (V, E), being V a set of vertices and E � {{x, y}|x, y 2 V, x 6 ¼ y} a set of edges that connects two distinct edges x and y. Network theory is related to the study of graphs in terms of symmetric or asymmetric relations between objects. In this study, due to the nature of the variables, we consider undirected graphs and network analysis with symmetric relations between objects (in this case, the locations of the environ- mental variables). To conduct the network representation and analysis, we consider the

DCCA cross-correlation coefficient between pairs of the Brazilian state capitals for each of the four environmental variables, temperature, humidity, radiation, and wind speed, for short median and long-distance correlations (box values of 13–one day, 1261–97 days, and 3328– 256 days). The graph, based on the DCCA cross-correlation coefficients, was created using the function graph_from_adjacency_matrix of the R package igraph, and the network plot was obtained using the function ggnet2 of the R package GGally. To avoid overcrowded networks, we only depicted edges associated with the top 10% DCCA cross-correlation coefficients between objects. Results In this section, we will include some of the main results and their discussion of the analysis of Brazilian climate data per state capital, for the DFA—detrended fluctuation analysis, DCCA cross-correlation analysis, and network analysis. Further tables and figures can be found in the supplementary material. General descriptive statistics for global radiation, temperature, air humidity, and wind speed, can be found in S1-S4 Tables in S1 File. Detrended fluctuation analysis (DFA) We considered time series with 52.234 data points (hours), generated by the observation of 4.018 days (1/01/2010—31/12/2020), with 13 observations per day (daily local time between 6 am to 6 pm; UTC 9 am to 9 pm), for the variables radiation in KJ/m2, maximum air tempera- ture in ˚C (dry bulb), the air humidity in %, and wind speed in m/s, and for the 26 Brazilian 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 6 / 18