# Quantifying the Occurrence of Multi-Hazards Due to Climate Change
## 2 Materials and Methods
### Statistical Tools and Data Processing  
All model-data processing and figure drawing were executed with R software and the ARC.GIS (MAP) 10.0 environment. Figure 1 depicts the basic steps of the process followed for the assessment of the extreme values of the variables, namely those of temperature  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Dataset** & **Period** \\ \hline EC–EARTH–WRF & 1980–2004 (historical) \\ EC–EARTH–WRF RCP4.5 & 2025–2049 (near future) \\ EC–EARTH–WRF RCP8.5 & 2025–2049 (near future) \\ EC–EARTH–WRF RCP4.5 & 2075–2099 (far future) \\ EC–EARTH–WRF RCP8.5 & 2075–2099 (far future) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The global (EC–Earth) model datasets downscaled by WRF model used in the study and corresponding periods.  
(maximum and minimum), precipitation rate, snowfall and wind speed. Firstly, the maximum and minimum temperature values were retrieved for the summer and winter seasons, respectively, while data were extracted throughout the year from the gridded datasets to determine the extremes of the precipitation rate, snowfall and wind speed. In this manner, at each grid cell and for each period (see Table 1), the 25 maximum values of each variable were obtained. The process was applied to all three time periods of interest and both RCP scenarios (see Table 1).  
Then, the modeled values of the variables of interest to this study were categorized in terms of likelihood, i.e., probability of occurrence, according to Table 2. The likelihood categories, six in total, and the threshold values of Table 2 were retrieved from the EU-CIRCLE project report [51] for the characterization of the hazard (maximum (summer) temperature, minimum (winter) temperature, precipitation rate, snow rate and wind speed). The probability of occurrence of each variable was calculated for all six categories. As the study was aimed at determining the likelihood of extremes, the focus was placed only on the specific categories characterizing the highest threshold values and hence, the probabilities of occurrence were calculated as totals of the three classes \"High\", \"Very High\" and \"Exceptional\".  
In this work, the method for calculating the probability of exceeding a value was based on the Extreme Value Theory (EVT). The determination of the extreme values of the studied variables required the selection and definition of the underlying distribution functions. The estimations were made using the R package \"Extremes\" [52], fitting a Generalized Extreme Value distribution (GEV) to block maxima data (annual maxima) under the assumption of non-stationarity [53]. The GEV distribution has three parameters: the shape factor \(\xi\), the scale or dispersion parameter \(\sigma\) and the location or mode parameter \(\mu\). The GEV-distribution function, \(G(y)\), is given by:  
\[\text{For }\xi\ \
eq\ 0,\ G(y)=\exp\left(-\left[1+\xi\left(\frac{y-\mu}{ \sigma}\right)\right]^{-1/\xi}\right) \tag{1}\]  
\[\text{For }\xi\ =\ 0,\ G(y)=\exp\left(-\exp\left(-\frac{y-\mu}{\sigma}\right)\right) \tag{2}\]  
The GEV has three types depending on shape parameter \(\xi\) as follows:  
1. When \(\xi=0\), GEV is known also as Type I Extreme Value Distribution (or Gumbel Distribution, light tail)
2. When \(\xi>0\), GEV is known also as Type II Extreme Value Distribution (or Frechet Distribution, heavy tail)  
Figure 1: Schematic representation of the main methodological steps for determining a quantified climatic multi-hazard assessment.  
When \(\xi<0\), GEV is known also as Type III Extreme Value Distribution (or Weibull Distribution, upper finite end point).  
The final step of the applied methodology included the calculation of the most frequently appearing hazard at each grid cell, time period and RCP scenario. In this manner, we performed the spatial assessment of the occurrence of multi-hazards in the historical period and in the two studied future periods according to the two RCP projections in order to illustrate the areas susceptible to hazards over long time scales.