## 3 Materials and Methods
### Methodology  
The study's stages are presented below. 1. Compute the data series' basic statistics to assess the series' characteristics. 2. Perform the following statistical tests to determine the series properties and assess their similarity/dissimilarity, as presented below. * Test the series normality using the Shapiro-Wilk [31] and the Q-Q plot to cross-validate the results [32]. The Q-Q plot is obtained by plotting the estimated quantiles of a distribution against those of another distribution (generally considered as a reference, in our case, Gaussian). * Test the series homoskedasticity by the Levene test [33]. * Drawing the boxplots to assess the outliers' existence. An outlier is a value of the data series outside the interval [\(Q_{1}-1.5\times IQR\), \(Q_{3}+1.5\times IQR\)], where \(Q_{1}\) (\(Q_{3}\)) is the lower (upper) quartile is, and IQR is the interquartile range = \(Q_{3}-Q_{1}\). * In this study, we did not eliminate the outliers. Excluding outliers would not be beneficial for detecting the areas with more accentuated pollution. As presented in the analysis, the outliers' existence is one of the features used in the second phase of clustering. * Compute the autocorrelation function (ACF) and draw the correlogram to see the variation in the correlations between couples of the series values at different lags. * For a time series (\(X_{t}\)), ACF is defined by: \[\mathbf{\rho_{k}}=\frac{\mathbf{Cov}(\mathbf{X_{t}},\mathbf{X_{t+k}})}{\sqrt{\mathbf{Var}(\mathbf{X_{t} })\mathbf{Var}(\mathbf{X_{t+k}})}}\] (1) where \(Cov(X_{t},X_{t+k})\) is the covariance of \(X_{t}\) and \(X_{t+k}\), and \(Var(X_{t})(Var(X_{t+k}))\) is the variance of \(X_{t}\) (\(X_{t+k}\)). * Compute the partial autocorrelation function (PACF), which indicates the correlation between \(X_{t}\) and \(X_{t+k}\) considers intermediary lagged values, in contrast to the ACF, which takes into account all the intermediary values. * Study the relationships between series (and their characteristics) as a group. Evaluate the correlations between the series in each zone. The first approach involved building the correlation matrices, whose elements are the Pearson correlation coefficients of couples of series. The second one was checking the multivariate independence of the series in each group (not pairwise, as in the correlation matrix), using the distance correlation \(t\)-test of independence [34]. The advantage of the last procedure is that it uses a modified distance that makes it robust compared to other tests based on second-order moments. 2. Test the hypothesis that all series in a zone come from the same continuous distribution against the hypothesis that at least one is issued from another distribution using the Anderson-Darling test implemented in the 'kSamples' package in R [35]. 3. Use [PERSON]'s median test [36] to check the hypothesis that the medians of the series in each group are equal against the alternative that at least two differ. 4. Compare the variances of the collected samples utilizing the Levene test (with medians instead of means) and the Fligner-Killeen [37] nonparametric test (based on ranks). The null hypothesis is that the variances of all samples are equal, and the alternative is that at least two variances differ. The above tests were utilized due to their robustness to the normality hypothesis violation. 4. Group the data series in clusters using the k-means algorithm [38; 39]. The algorithm was employed to cluster the raw data series, on the one hand, and the series of features determined from statistical analysis, on the other hand, after normalizing the datasets. Finding the optimal number of clusters (\(k\)) is essential for robust clustering. When selecting \(k\), different methods can be used, like those implemented in the NBC Clust package in R 4.4.2 [40]. They are based on various mathematical approaches, which are not equivalent to each other, so the provided optimal number of clusters differs. This is why the maximum criterion is utilized to select the best \(k\).