## 4 Methods: Modelling Efforts and Calibration  
When there is a need to model a variable such as air traffic, which takes integer or zero values, the use of counting data models seems appropriate [39; 40; 41]. These are specific models for situations in which the dependent variable is either integer positive or zero, i.e., it cannot, by nature, have a negative sign. In such cases, variance is a function of the expected value. In this specific case, the expected value depends on the time, day of the week, seasonality and trend. The most widely used counting regression models are the Poisson and the negative binomial. Using these models, highly accurate forecasts can be made of the expected number of events (number of flights in this case) and of volatility. In this case, volatility is caused by the uncertainty in the number of flights in each hour. For instance, some authors [42] have used the lineal Poison autoregressive (PAR) model as an alternative to the Poisson model to analyse the impacts of laws and climate on annual road traffic accidents. This alternative method may be applied in future work for comparative purposes. Note that in the Poisson model, the variance is equal to the expected value, which has its limitations. However, the negative binomial model is applied in situations in which there is overdispersion, i.e., when the variance is greater than the mean. Therefore, an initial analysis of the data is required before it can be decided which of the two models fits better for the purpose of this paper, although there are more precise statistics for making this decision. The preliminary data analysis in Section 3 suggests that a negative binomial model is more adequate for analysing air traffic. This model is an extension of Poisson regression for cases in which the variance is greater than the mean value. As stated, this is the case of the data used here. In a counting regression model such as the negative binomial, the expected value is as shown in Equation (1). \[E(Y|X)=\exp(\beta_{1}+\beta_{2}t+AC(t)+WC(t)+DC(t))=\mu \tag{1}\]  
where \(Y\) stands for the number of flights and \(X\) represents the independent variables. The calculation of expected value includes:  
1. A constant \(\beta_{1}\). 2. A trend \(\beta_{2}t\), where \(t\) is the time in years. 3. A yearly cycle \(AC(t)\) with its seasonal components: annual, semi-annual, quarterly, etc. 4. A weekly cycle \(WC(t)\) depending on the days of the week. These are dummy variables. 5. A daily cycle \(DC(t)\) based on the hours with their seasonal components in a similar way to the yearly cycle. The yearly cycle is modelled as in Equation (2). \[AC(t)=\sum_{j=1}^{j=5}\left[\beta_{1+2j}\sin(2j\pi t)+\beta_{2+2j}\cos(2j\pi t)\right] \tag{2}\]  
The weekly cycle is modelled as in Equation (3) using six dummy variables. \[WC(t)=\beta_{13}D_{1}(t)+\beta_{14}D_{2}(t)+\beta_{15}D_{3}(t)+\beta_{16}D_{4}( t)+\beta_{17}D_{5}(t)+\beta_{18}D_{6}(t) \tag{3}\]  
where the dummy variable \(D_{1}(t)=1\) if the day is Monday and \(D_{1}(t)=0\) for other days of the week. When all dummy variables are zero, it is Sunday. The hourly cycle is modelled as in Equation (4). \[DC(t)=\sum_{j=1}^{j=5}\left[\beta_{17+2j}\sin(2j\pi\tau)/24+\beta_{18+2j}\cos( 2j\pi\tau)/24\right] \tag{4}\]  
where the variable \(\tau\) indicates the hour and takes a value between 1 and 24. There are 28 parameters for calculating the expected value in the overall model, depending on trend, annual cycle, weekday and time. Table 4 presents the parameter values calculated for Germany, where the betas correspond to the constant, trend, yearly cycle parameters, weekly parameters and daily cycle parameters as specified in Equation (1) to Equation (4), also the alpha value is calculated. In this regression model, the variance is calculated using Equation (5). \[\text{Var}(Y\,|\,X)=(1+\alpha\times\mu)\times\mu \tag{5}\]  
where \(\alpha\) is a value calculated by the regression. Figure 3 shows the variance values as functions of \(\alpha\) and \(\mu\).