2.2 Double cumulative curves
Double cumulative curves are often plotted to examine the correlations between variables and their variation. For example, they are adopted to analyze the consistency, trends, and intensity of the changes in hydrological variables. (Zhang et al., 2017; Jiang et al., 2012) The construction of the double cumulative curve for two variables, A and B, is described as follows. It is assumed that A denotes the independent variable, whereas B is the dependent variable. The time series covers N years. The values of different years are expressed as \(A_{i}\) and\(B_{i}\). The annual cumulative values were obtained for variables A and B to provide a new cumulative series \(A_{i}^{{}^{\prime}}\) and\(B_{i}^{{}^{\prime}}\), where \(i=1,2,3\cdots N\). They are expressed as follows:
\(A_{i}^{{}^{\prime}}=\sum_{i=1}^{N}A_{i}\) (8)
\(B_{i}^{{}^{\prime}}=\sum_{i=1}^{N}B_{i}\) (9)
In the double cumulative curve, the continuous cumulative value of variable A is plotted against that of B. If a straight line results, the variables vary proportionally with each other, and the slope of the line provides a constant ratio.
Nevertheless, if the slope of the curve changes at a certain point, a structural break exists. At the structural break, the relationship between the two variables is altered. The year in which the slope changes is the time of the structural break (Peng et al., 2013). Kohler (Kohler., 1949) believed that double cumulative curves give accurate results only if the dependent and independent variables exhibit strong correlations, direct proportionality, and reasonable comparability during the period of interest.