## 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): (1) where Q and −1 if not. If _{j}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:(2) |
If there were repeated values in the time series, the variance of the time series, (3) where ti represents the number of same values in the time series. If there were two equal values in the time series, then 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 |

(4)

where r are the values of the autocorrelation function at the _{τ}^{R}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 CF._{2}The Mann-Kendall test statistic for the time series was computed as: (5) adhering to standard normal distribution (Kendall, 1962). If α=0.05, then there was a trend in the time series and the hypothesis H was adopted. If not, the hypothesis was rejected._{0} |
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 ( |

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