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.