## 3 Approaches Compared for Multi-Parametric Data Analysis
### Statistics-Based Approach  
The SBA adopts, in our case, cross-correlation functions among different parameters and their rate variation to point out statistical relationships between two independent time series by testing the reciprocal dependence. More in particular, the cross-correlation measures the similarity of two series as a function of the time lag (LAG), which indicates the time delay between two signals and outputs a statistical coefficient (i.e., correlation function; CF) whose value increases at increasing coincidence between the time series.  
The analysis presented herein was performed on the different time series using R open-source software [73], which returns a correlogram graph (i.e., CF vs. LAG). The LAG value indicates the time delay between two signals and is expressed as a multiple of the sampling step. After calculating the cross-correlation between the two signals, the maximum CF (a-dimensional) corresponds to the time at which the signals are best correlated. Two dotted lines are also depicted and represent the statistical significance threshold of the level of correlation (Figure 4).  
The SBA was applied to the data series, expressed as the rate of monitored values: temperature, rainfall and wind with related strain series. The rate can: (i) increase, when transgressive processes can evolve towards a paroxysmal episodes; (ii) decrease, when regressive processes can reduce the failure occurrence taking into account a short and/or medium time window; (iii) remain constant, when stationary processes cannot evolve towards critical phases close in time, but could do it considering a long-time windows [74; 75]. The rate of discontinuous data, as well as rainfall, can be analysed by accumulating them to derive typical ramp-and-flat variation curves. The ramps correspond to event occurrences; the flats indicate events did not occur. This analysis returns zero to non-null values in cases of occurrence or non-occurrence of triggering episodes. The rate of continuous data, as well as temperature, can be obtained by carrying out a linear regression analysis to detect a suite of events, as well as their variations over time.  
In outputting possible response delays of the system with respect to different kinds of forcing actions, a correlation coefficient (R\({}^{2}\)) can be attributed at each value of the rate, evaluating the effectiveness of a trend within a selected time window that must consist of at least 3 points. If the datasets are strongly scattered, the analysis return very low values of R\({}^{2}\), indicating an aleatory  
Figure 4: Sketch explaining how to read the correlogram and the interpretation of the peak (a) and an example of a correlogram obtained with R software (b). On the y-axis is shown the cross-correlation coefficient value and on the x-axis is shown the LAG value: y(t) anticipates the x(t) of a LAG time equal to 1 (that corresponds to the sampling step of the series) (b). The dashed lines indicate the values of statistical significance of the results.  
variation of the considered variable. At the contrary, if the data are aligned the analysis return R\({}^{2}\) values up to 1, indicating a well-delineated trend of the considered variable. In our specific case, R\({}^{2}\geq 0.75\) was considered as a reliable correlation coefficient value of both transgressive and regressive trend [76].  
The rates of different datasets were here computed using a customised seismic analysis code (SAC) and FORTRAN UNIX scripts that firstly perform some pre-processing operations applied to each time histories of continuous (like as smoothing tool) and discontinuous (like as cumulative tool) monitored data. Successively, the linear regression, the R\({}^{2}\) factors and the rates of several time series within a specific time window were calculated (Figure 5). The time series of rates were cross correlated to determine which series influence the others and over what time range.