*2.3. Statistical Analysis*

In order to investigate the effects of anthropogenic land cover changes to the annual actual evapotranspiration and river discharge, the following statistical tests were applied to the time-series of each land cover case study examined.

The sequential version of the Mann-Kendall (SMK) test [58] and the non-parametric rank-based distribution-free cumulative sum CUSUM test [59] were applied, so as to detect the approximate change of trend with time. The sequential version of the Mann-Kendall test (SMK) is calculated so that rank (*xi*) > rank (*xj*) (*i* > *j*). The number of cases *xi* > *xj* is counted and denoted by *ni*. The *t* statistic is calculated as Equation (1):

$$t = \sum\_{i=1}^{n} n\_i \tag{1}$$

The distribution of *t* is assumed to be asymptotically normal with the following expectations (Equations (2) and (3)):

$$E(t) = \mu = \frac{n(n-1)}{4} \tag{2}$$

and

$$Var(t) = \sigma^2 = \frac{n(n-1)(2n+5)}{72}.\tag{3}$$

The null hypothesis that there is no trend is rejected for high values of the reduced variable |*u*(*t*)|, which is calculated as Equation (4):

$$u(t) = \frac{t - E(t)}{\sqrt{Var(t)}}.\tag{4}$$

Similar to the calculation of the sequential progressive series *u*(*t*), the retrograde series *u*(*t*) is computed backwards starting from the end of the time-series [58]. The intersection of the curves *u*(*t*) and *u*(*t*) indicates the approximate turning point of the trend of the original time-series. For the trend to be

significant, the point of intersection must exceed the critical values of the confidence level. The sign of the curve *u*(*t*) indicates whether the trend is increasing or decreasing.

In CUSUM test, the test statistic *Vk* is defined as Equation (5):

$$V\_k = \sum\_{i=1}^k \text{sgn}(\mathbf{x}\_i - \mathbf{x}\_{\text{median}}), \text{ k } = 1, \ 2, \ \dots, n \tag{5}$$

Where *xmedian* the median value of the *xi* data set and *sgn*(*x*). CUSUM test allows the detection of changes in mean value of a sequence of observations ordered in time, by comparing successive observations with the median of the series. If a significant trend develops in the plotted points either upward or downward, it is evidence that the process mean has shifted and further investigation is required [59–61].

After estimating the approximate change of trend with time, the rank-based non-parametric Mann-Kendall test [62,63] was applied to each sub-period, so as to identify the trend significance. The Mann-Kendall statistic *S* compares each value of the series (*xt*) with all subsequent values (*xt*+<sup>1</sup>) and is defined as Equation (6):

$$S = \sum\_{t'=1}^{n-1} \sum\_{t=t'+1}^{n} \text{sgn}(\mathbf{x}\_t - \mathbf{x}\_{t'}) \tag{6}$$

where sgn is the sign function (Equation (7)).

$$\text{sgn}(\mathbf{x}\_{l} - \mathbf{x}\_{l'}) = \begin{cases} 1, \; \text{if } \mathbf{x}\_{l} > \mathbf{x}\_{l'} \\ 0, \; \text{if } \mathbf{x}\_{l} = \mathbf{x}\_{l'} \\ -1, \; \text{if } \mathbf{x}\_{l} < \mathbf{x}\_{l'} \end{cases} \tag{7}$$

If *n* < 10, the absolute value of *S* is compared directly to the theoretical distribution of *S* derived by Mann and Kendall [64]. When *n* ≥ 10, the statistic *S* is approximately normally distributed with the mean *m* and the variance *V* as follows [62,63] (Equation (8)).

$$ES = 0, \; V(S) = \frac{1}{18} [n(n-1)(2n+5) - \sum\_{i=1}^{\mathcal{S}} \varepsilon\_i (\varepsilon\_i - 1)(2\varepsilon\_i + 5)] \tag{8}$$

*g* is the number of tied groups, and *ei* is the number of data in the *i*th tied group. The values of *S* and *VAR*(*S*) are used to compute the test statistic *Z*. The standardized test statistic *Z* is defined as follows (Equation (9)).

$$Z = \frac{\mathbb{S} + m}{\sqrt{V(\mathbb{S})}},\tag{9}$$

Finally, the trend magnitude for each trend period identified from the abovementioned statistical test was calculated based on Sen's estimator of slope. This non-parametric statistic can be applied in cases of linear trend and determines the magnitude of change per unit time [65]. The Sen's slope estimation test is defined for a season *g* as follows (Equation (10)):

$$\beta = \operatorname{Median} \left( \frac{\mathbf{x}\_i - \mathbf{x}\_j}{i - j} \right), \text{ i } < \mathbf{j} \tag{10}$$

where β; the slope between points *xi* and *xj*, *xi* data measurement at time *i*, and *xj* data measurement at time *j*. The positive value of the β; implies the slope of the upward trend and negative value for the downward trend [66].
