# Evaluating Prediction Models for Airport Passenger Throughput Using a Hybrid Method
## 2 The Proposed Hybrid Evaluating Method
### Models
#### 2.1.2 Causal Models  
Causal models establish causal relationships between independent variables. Typical causal models include regression models and elastic coefficient models [25]. For regression models, a unary linear regression model, a multiple linear regression, a stepwise regression model, a hybrid regression model, an elastic coefficient model and a proposed elastic-like scale model are considered.  
To define the models for evaluation, consider there are \(M\) correlated variables, the variables can be collected as \(W(t)=\{W_{1}(t),W_{2}(t),\ldots,W_{m}(t),\ldots,W_{M}(t)\}\), where \(W_{m}(t)\) is the value of \(m\)th variable at time point \(t\). For APT prediction, the variables are usually the economic indicators of the cities where the airports are located in.  
The unary linear regression model can be by fitting:  
\[\hat{x}_{ultr}(t)=a_{1}W_{m}(t)+e_{1},m\in 1\cdots M \tag{17}\]  
using the least squares method. Furthermore, the multiple linear regression can be similarly described as:  
\[\hat{x}_{plr}(t)=c_{1}W_{1}(t)+c_{2}W_{2}(t)+\cdots+c_{p}W_{p}(t)+e_{1}+\cdots+ e_{p} \tag{18}\]  
By combing the regression models with the time series models, \(W_{m}(t+T)\) can be calculated first and then, \(\hat{x}_{ultr}(t+T)\) and \(\hat{x}_{plr}(t+T)\) can be obtained.  
The stepwise regression model introduces correlated variables step-by-step into the regression model until the model reflects the relationship significantly. The common methods to determine the variables include the forward method and backward method. Taking the forward method as an example, the modelling process is shown as follows.  
_Step 1_: Establishing the unary linear regression models between APT and each variable \(W_{1},W_{2},\ldots,W_{m},\ldots,W_{M}\), which can be obtained by Equation (17). Calculating the F inspection values of each regression coefficient of the models. The F inspection values can be denoted as \(F_{1}^{(  
where \(\rho(x,W_{m})\) is the correlation coefficient of the APT \(x\) and the variable \(W_{m}\), \(x(t)\) is the APT of the studied airport at time \(t\), \(\overline{x}\) is the mean value of APT, \(W_{m}(t)\) is the value of \(W_{m}\) at time \(t\) and \(\overline{W}_{m}\) is the mean value of \(W_{m}\).  
The elastic coefficient models are indirect methods to forecast the results by fitting the correlated factors. An elastic coefficient model can be defined as:  
\[E_{s}=\frac{p\prime}{q\prime} \tag{20}\]  
\[p=E_{s}*q \tag{21}\]  
\[\hat{x}_{cc}(t+T)=(1+p)^{T} \tag{22}\]  
where \(E_{s}\) is the elastic coefficient, \(T\) is the target time point, \(q^{\prime}\) is the growth rate of a correlated variable before time \(t\), \(p^{\prime}\) is the growth rate of the APT of an airport before time \(t\), \(p\) is the growth rate of a correlated variable before time \(T\), \(\hat{x}_{cc}(t+T)\) is the predicting result.  
[PERSON] et al. (2016) analyzed the causal relationship between air transport and economic growth, and the results showed the relationship was bi-directional, especially for the underdeveloped area. For the developed area, air transport only showed a positive effect on economic growth. The relationship can be reflected by the relative value of the APT and that of economic indicators, which can be represented by:  
\[R_{AI}=\frac{APT}{indicator}. \tag{23}\]  
Regarding the relative value \(R_{AI}\) as the elastic coefficient, an elastic-like scale model can be established. By fitting the indicators with trend extrapolation models, the predicted values of the indicators can be obtained and then, APT can be predicated.