## 3 Data and Methodology
For further confirmation of the possible crossover \(M_{thr}\) and for comparing our results with an independent method, we applied the Detrended Fluctuations Analysis [33] in all the 10 sub-regions. DFA is implemented under the hypothesis that if a given time series is correlated then the fluctuation function, \(F(n)\), follows an increasing power-law trend according to the factor \(n^{\alpha}\) (\(F(n)-n^{\alpha}\)), where \(n\) is the size of the window and \(\alpha\) is the scaling exponent. If the value of the exponent \(\alpha\) is larger than 0.5 (\(\alpha>0.5\)) then the data sample is positively correlated, whereas if \(\alpha\) is smaller than 0.5 (\(\alpha<0.5\)) the data are negatively correlated (anti-correlated). For the case when the exponent \(\alpha\) is equal to 0.5 (\(\alpha=0.5\)), the sample could be considered statistically independent. The exponent \(\alpha\) is calculated as the slope of the linear regression of the logarithm of the fluctuation function, \(F(n)\), against the logarithm of the window size \(n\). The next step after the detection of the \(M_{thr}\), above which the \(\Delta T\) becomes statistically uncorrelated, is the determination of the best performed statistical distribution that each \(\Delta T\) sample follows. Over the years, many statistical distributions such as the Weibull [34], the Lognormal [35; 36], the Stretched Exponential [37] and the Gamma distribution [7; 38] have been proposed. In this study, we applied the four most popular statistical distributions in the relevant studies (e.g., [39; 40]), namely the Weibull, the Gamma, the Lognormal and the Exponential, to each \(\Delta T\) data set for the 10 sub-areas. The probability density functions (_pdf_) of the four candidate distributions are given in Table 1. The parameter estimation for each distribution is achieved via the application of the Maximum Likelihood Estimation (MLE) method using the respective log-likelihood formulae [41]. In order to compare the distribution's performance and select the one that best fits the observations, we applied the Anderson-Darling non-parametric goodness of fit test (A-D test) [42]. The A-D test is implemented by calculating the distance, \(A^{2}\), between the empirical cumulative distribution function (_ecdf_) and the cumulative distribution function (_cdf_) of the distribution applied to the data. The A-D test statistic is given by  
\[A^{2}=-n-\sum_{i=1}^{n}\frac{2i-1}{n}[lnF(x_{i})+ln(1-F(x_{n+1-1}))], \tag{4}\]  
where \(\{x_{1},x_{2},\ldots,x_{n}\}\) are the ordered sample data points, \(n\) is the number of observations and \(F\) is the cdf of the distribution under study. The test compares the quantity \(A^{2}\) with a critical value, \(c\), under the null hypothesis that the data are distributed according to \(F\)If \(A^{2}\) is less than or equal to the critical value, then the null hypothesis cannot be rejected. The decision of rejecting or not the null hypothesis in the present study is based on the obtained \(p\)-values, compared with the significance level, which is equal to 0.05 (\(\alpha=0.05\)). If the \(p\)-value is greater than \(\alpha\) (\(p\)-value \(>\alpha\)) then the null hypothesis can not be rejected. On the contrary, if the \(p\)-value is lower than \(\alpha\) (\(p\)-value \(<\alpha\)) the null hypothesis can be rejected. To further compare the four distributions, we calculated the Akaike [43] and Bayesian [44]. Information Criteria (AIC and BIC, respectively). From these calculations, the distribution that displays the best performance is the one with the minimum value of the criterion in both cases. The values of AIC and BIC are given by  
\[AIC=-2 lnL+2k \tag{5}\]  
and  
\[BIC=-2 lnL+kln(n) \tag{6}\]  
where \(lnL\) stands for the value of the log-likelihood function obtained from the MLE approach, \(k\) is the number of parameters of the distribution and \(n\) is the number of the observations.