Internal Stochastic Structure of Annual Discharge Time Series of Serbia’s Large Rivers - page 02

Time Series Trend

Trend assessment was based on the well-known Mann-Kendall trend test (Douglas et al., 2000). Kendall's statistic was computed as follows (Kendall, 1962):

For1     (1)

where Q is the mean annual discharge at time steps i and j, sign( ) is equal to +1 if Qi is greater than Qj and −1 if not. If S>0, the time series exhibits a downward trend and if S<0, then there is an upward trend. For an independent random variable without repeated values, the mathematical expectation and variance are:

For2     (2)


If there were repeated values in the time series, the variance of the time series, Var(S), was corrected as follows:

For3     (3)

where ti represents the number of same values in the time series. If there were two equal values in the time series, then i=2 and ti=1.

In order to account for the correlation between the members of the time series, Hamed and Rao (1998) and Yue and Wang (2004) proposed that the variance of Mann-Kendall's statistic S be corrected using the effective lengths of the non-correlated members of the time series N*. The length of the non-correlated time series N* was determined indirectly, modifying the variance of statistic S as follows:


For4     (4)


where rτ and rτR are the values of the autocorrelation function at the τ-th step of time series Q and ranked time series S, respectively. As recommended by Khaliq et al., (2009), the trend test assumed that S represented an AR(1) process, where only the first step was taken and the variance reduced using the correction parameter CF2.

The Mann-Kendall test statistic for the time series was computed as:

For5     (5)

adhering to standard normal distribution (Kendall, 1962). If Zs was greater than 1.96, corresponding to a confidence threshold of α=0.05, then there was a trend in the time series and the hypothesis H0 was adopted. If not, the hypothesis was rejected.


Compared to the conventional approach, the multi-temporal approach is an alternative trend assessment methods that involves a combination of subseries, from the first to the last member of the time series. A subseries is a set of time series members in a continuous multiple-year period. Relying on the experience of Hannaford et al., (2013), the river discharge trend was assessed using subseries segments and combining different segments. In the first step (i=1), the time series was divided into ten equal subseries. The discharge trend was calculated according to Mann-Kendall's test, taking the segment from member 1 to member N/10 (Fig. 1). The subseries were then consecutively combined in the following row: from 1 to 2N/10, from 1 to 3N/10,..., from 1 to N. In the second step (i=2), the first segment up to N/10 was discarded and subseries trends calculated from N/10 to 2N/10, from N/10 to 3N/10,..., from N/10 to N. The procedure was repeated up to the 10th step (i=10), where the remaining subseries was from 9N/10 to N.



Figure 1: Model for calculating time series trends following the multi-temporal approach.