# Demand Forecasting Model for Airline Flights Based on Historical Passenger Flow Data
## 2 Materials and Methods
### Algorithm for Medium-Term Flight Demand Forecasting
The multiple regression equation is as follows [42] (see Equation (9)):  
\[f_{1}(t)\ =\ 1+k_{m}, \tag{2}\]0  
where \(Y_{t}\) is the value of the total passenger demand for year \(t\), \(\beta_{0}\) is the intercept, \(\beta_{1t},\beta_{2t},\ldots,\beta_{6t}\) are the regression coefficients determining the level of influence of independent variables on the value of the total passenger demand for year \(t\), \(x_{1t},x_{2t},\ldots,x_{gt}\) are the values of selected macroeconomic indicators corresponding to the region of the departure and arrival points of the considered destination by time for year \(t\): \(x_{1t},x_{2t}\) correspond to the values of the GRP indicators, \(x_{3t},x_{4t}\) correspond to the values of the median per capita income, \(x_{5t},x_{6t}\) correspond to the values of the population of departure and arrival points, \(\zeta_{t}\) is the model error, and \(t\) is the time index. Additionally, in order to facilitate the incorporation of the independent values \(x_{1t},x_{2t},\ldots,x_{6t}\) into the multiple regression model, it is essential to employ time series modeling of macroeconomic indicators for the independent variable X prior to the forecast period conclusion. It is then proposed that an average linear model be constructed, equal to the average of the linear annual models over the historical annual periods. In order to achieve this, it is necessary to calculate the averaged regression coefficient for each historical year (see Equation (10)). \[f_{1}(t)\ =\ 1+k_{m}, \tag{2}\]1  
where \(\overline{\beta}_{j}\) is the average coefficient for the \(j\)th independent variable, \(\beta_{ji}\) is the coefficient for \(j\)-th independent variable in a year \(i\), \(n\) is the number of years in the forecast period for which data needs to be averaged. The averaged model \(\overline{Y}_{t}\) has coefficients averaged over all years (see Equation (11)):  
\[f_{1}(t)\ =\ 1+k_{m}, \tag{2}\]2  
where \(\overline{Y}_{t}\) is the predicted value of demand according to the averaged model for the year \(t\), \(\beta_{0}\) is the intercept, \(\beta_{1},\ \beta_{2},\ \ldots,\ \beta_{n}\) are the averaged regression coefficients (the average values of the coefficients for all periods), \(x_{1t},\ x_{2t},\ \ldots,\ x_{nt}\) are independent variables (the same as in Equation (9)), \(\zeta_{t}\) is a model error. Furthermore, the model data must be normalized in accordance with the historical data from Equation (9), relative to the averaged model. Therefore, the parameter \(f_{external}(t)\) from Equation (1) will be expressed by the following dependence (see Equation (12)):  
\[f_{1}(t)\ =\ 1+k_{m}, \tag{2}\]3  
where \(\overline{Y}_{t}\) is the predicted value of demand according to the averaged for the year \(t\) (see Equation (11)), \(Y_{t}\) is the actual value of the final passenger demand for the year \(t\) (see Equation (10)). The result is a series of data reflecting the functional dependence of flight demand on fluctuations in macroeconomic indicators. The third stage of development proposes the utilization of a system dynamics approach to describe the influence of external factors on demand [43]. System dynamics models provide more reliable forecasts of short- and medium-term trends than statistical models. Additionally, they allow for the calibration of forecast demand distributions by incorporating external factors, thereby enhancing the forecast's informational value in the medium term [43]. The conceptual framework illustrated in Figure 2 depicts the interrelationship between the historical distribution of flight demand, forecast demand, and the impact of macroeconomic variables on the forecast.