## 3 Data Used  
The data used in this study were obtained from a number of European institutions involved in monitoring the performance of ATM systems, including the Performance Review Body (PRB), the Performance Review Unit (PRU), and EASA. The PRB is an advisory body of the European Commission that assists the Commission and national authorities in the supervision of the performance of ATM systems. The PRU is part of the EUROCONTROL Single Sky Directorate. It provides information and data analysis on the performance of European ATM. EASA collaborates with the two aforementioned institutions by collecting the necessary safety data in each state. The study covers a total of 29 European states. The value to be estimated and predicted \(y_{i,s}\) is the number of annual SMIs that have taken place in each of the 29 ATM systems included in the study. The variable \(y\) is to the number of annual SMIs. Subscript \(i\) relates to the year, and subscript \(s\) relates to the specific ATM system. Specialist literature on the topic suggests that the two factors that best explain the occurrence of SMIs are the volume of air traffic and the complexity of this traffic. To verify these dependencies, the following explanatory variables are analyzed in the study. \(x1_{i,\;s}\) is the volume of air traffic, i.e., number of annual flight hours, defined as the number of total flight hours per year of aircraft operating under Instruments Flight Rules (IFR) in the airspace of each of the countries considered in the study. The subscripts \(i\) and \(s\) refer to the year and ATM system, respectively. To assess the complexity of air traffic, Eurocontrol has defined a set of indicators that can be used for ANSP benchmarking [28]. The complexity indicators are based on the \"interactions\" that arise when there are two aircraft in the same \"place\" at the same \"time\". The variable \"complexity score\" \(x2_{i,\;s}\) is defined as the total duration in minutes per flight hour of all interactions between controlled aircraft in a given volume of airspace. The indicator \"complexity score\" is the product of two components: the adjusted density \(x3_{i,\;s}\) and the structural index \(x4_{i,\;s}\). The variable \(x3_{i,\;s}\) measures the relative concentration of aircraft in the airspace. The airspace is divided into a discrete grid of cells measuring 20 \(\times\) 20 nautical miles in the horizontal and 3000 feet high. An interaction is defined as the simultaneous presence of two planes in one of these cells. The variable \(x4_{i,\;s}\) is the sum of horizontal, vertical, and velocity interactions. In this study, the values for complexity score, adjusted density, and structural index are aggregated on an annual basis. Therefore, the variables \(x2,\;x3,\;x4\) refer to the annual mean value (calculated as the sum of the daily values of a year divided by the total flight hours in the year). The subscript \(i\) indicates the year in question, and the subscript \(s\) indicates the ATM system analyzed. The above data was obtained for 29 countries and their ATM systems over seven consecutive years. The identity of the countries is not given in the study. Each has been assigned a number from 1 to 29. In total, the data sample comprises five variables with 203 records per variable. Figure 2 shows the relationship between variables \(y,\ x1,\ x2,\ x3,\ x4\) by means of scatterplots and the calculation of the correlations between them, as well as the density functions of each variable. The value of the variable \(x1\) \"number of flight hours\" is divided by 100,000 so that it is more easily readable in the figure. Figure 3 shows a boxplot for the number of SMIs per year. Figure 4 gives a boxplot of the number of SMIs for each ANSP. Figure 2: Relationship between the variables. Scatterplots, correlations, and density functions of each variable. The number of * in the right part of the figure indicates the relevance of the correlation. Figure 5 shows the evolution of the number of SMIs versus the number of flight hours.