2.3.5. The Sen's Slope Estimator

In order to get the magnitude of a consistent trend (Q), the Sen's non-parametric method [19] is used. It is estimated by the slope of all data pairs (N = n(n − 1)/2) as the following formula:

$$\mathbf{Q}\_{\mathbf{i}} = \frac{\mathbf{x}\_{\mathbf{j}} - \mathbf{x}\_{\mathbf{k}}}{\mathbf{j} - \mathbf{k}} \text{ for } \mathbf{i} = \mathbf{1}, \mathbf{N} \tag{7}$$

where Q is slope between data points xj and xk, xj and xk are the data values at time j and k (j > k) respectively, j is time after time k. The Sen's is computed by the median slope as:

$$\mathbf{Q\_{med}} = \begin{cases} \begin{array}{c} \mathbf{Q\_{[\frac{N+1}{2}]}} & \text{if N is odd} \\\\ \frac{\mathbf{Q\_{[\frac{N}{2}]} + \mathbf{Q\_{[\frac{N+2}{2}]}}}}{2} & \text{if N is even} \end{array} \end{cases} \tag{8}$$

where N is the number of calculated slopes.

The confidence limits for the nonparameter slope estimator are estimated by the well-known studies [27,29,53,54].
