## 4 Methods and Preliminary Analysis
Figure 12: Clustering of the time series of detrended groundwater levels corresponding to the HYs by applying the AHP method. The years presented in the legend correspond to the years in the dendrogram (Figure 11). Figure 6:0 Clustering of the time series of detrended groundwater levels corresponding to the winter seasons by applying the AHP method. The years in the legend are corresponding to the years in the dendrogram (Figure 13). Figure 6:1 Dendrogram for time series of detrended groundwater level corresponding to winter seasons from applying the AHP clustering. Figure 6:2 The clustering of the time series of detrended groundwater levels corresponding to the summer seasons by applying the AHP method. The years in legend are corresponding to the years in the dendrogram (Figure 15). Figure 6:3 Dendrogram for time series of detrended groundwater levels corresponding to the summer seasons from applying the AHP clustering. Intuitively, the groundwater level depends on precipitation. Consequently, in the next step, the dependence between the groundwater level and precipitation will be examined to identify a relationship and if it evolves in time. Except for the basic measure of a linear dependence, we also examined the two rank correlation coefficient measures: [PERSON] tau [57] and Spearman rho [58]. In contrast to the Pearson correlation coefficient, they show the monotonic dependence, not only the linear one. Moreover, due to the specific nature of data, namely the presence of outliers, the quality of dependence measures from several approaches was validated. The Kendall's tau correlation coefficient of two time series \(\{z_{t}^{i}\}_{t=1}^{n}\) and \(\{z_{t}^{j}\}_{t=1}^{n}\) is defined as follows:  
Figure 6:4  
where \(n\) is the sample size, \(d_{ki}^{z^{i}}=sign(z_{k}^{i}-z_{l}^{i})\) and \(d_{kl}^{z^{j}}=sign(z_{k}^{j}-z_{l}^{j})\). The second considered rank measure is the Spearman rho correlation coefficient. It can be defined based on the Pearson correlation coefficient (\(\rho(\cdot,\cdot)\)) described in (2) between the rank variables:  
Figure 6:5  
where \(rg_{z^{i}}\) and \(rg_{z^{j}}\) are the ranks obtained from two time series \(z_{tt=1}^{i^{n}}\) and \(\{z_{t}^{j}\}_{t=1}^{n}\) respectively. The results obtained for the winter and summer seasons are presented alternately (Figure 17). Based on the results (Figure 17), one might conclude that a relationship between the time series of the groundwater level and precipitation does not exist which seems to contradict the mentioned intuitive expectations. What might be the reason for the small correlation? It seems as if the explanation is hidden in the two data sets (Figure 18). Based on the data, the groundwater level is low at low precipitation periods (up to day 150), and it increases thereafter. This, intuitively, might result in a strong cross-correlation, yet, the statistical analysis does not confirm this observation. Therefore, the relationship between the cumulative sums of the precipitation (CSP) and the groundwater level (Figure 19) was analyzed, where the cumulative sum of the precipitation vector \(P(1),...P(N)\) is given by:  
Figure 6:6  
where \(N\) is the vector size. The CSP considers also the \"history\" of precipitation and indirectly the specificity of the location and the permeability of the ground. Due to different types of pavement, varied geological conditions, specific permeability and actual weather conditions, not all the rain can  
Figure 6:7 The Pearson (\(\rho\)), Kendall (\(\tau\)) tau and Spearman rho (\(\tau\)) correlation coefficient between time series of detrended groundwater level and precipitation. W: winter season, S: summer season. easily and immediately reach the groundwater table. This happens with a delay and only when there is enough intensive or long-lasting precipitation. Considering a situation, where past precipitation causes an elevated groundwater level in the present, using the cumulative precipitation sum, this dependency can be identified as well (Figure 20).