# Evaluating Prediction Models for Airport Passenger Throughput Using a Hybrid Method
## 2 The Proposed Hybrid Evaluating Method
### Models  
#### 2.1.1 Time Series Models  
Time series models regard the systems for prediction as black boxes, without considering the factors which influence the results. Instead, the models are established by fitting several historical data. Trend extrapolation models, exponential smoothing models, grey models, autoregressive models and moving average models are typical time series models, which have been widely used in nearly every prediction scene. Considering the impact of specific events, complex methods may perform better. For example, [PERSON] et al. [23] predicted the passenger demand of Djalaluddin Gorontalo Airport using a multiplicative of Holt-Winters exponential smoothing. [PERSON] et al. [24] compared the performance of various models including linear trend, quadratic trend, exponential trend, linear exponential smoothing (Holt's Model), and autoregressive integrated moving average models on APT prediction, and the results showed that linear exponential smoothing model performed best facing the impact of COVID-19, with a level of reliability of 95%. Concerning the large number of research samples, linear trend, quadratic trend, cubic trend, power trend, exponential trend, logarithmic trend, quadratic exponential smoothing, cubic exponential smoothing and grey GM (1,1) models are considered in this paper. To define the models for evaluation, an APT observation sequence \(x_{0k}=\{x_{0},x_{1},\ldots,x_{k}\}\) is proposed, where \(k\) denotes a time metric. The parameters of the linear trend model, the quadratic trend model, the cubic trend model, the power trend model, the exponential trend model and the logarithmic trend model by using the least squares method and the models can be described as:  
\[ \begin{split}\text{Linear trend model : }\hat{x}_{lf}=a_{lf}t+b_{lf}\\ \text{Quadratic trend model : }\hat{x}_{qf}=a_{qf}t^{2}+b_{qf}t+c_{ qf}{}^{2}\\ \text{Cubic trend model : }\hat{x}_{cf}=a_{cf}t^{3}+b_{cf}t^{2}+c_{ cf}t+d_{cf}\\ \text{Power trend model : }\hat{x}_{ef}=a_{ef}e^{b_{f}x}\\ \text{Exponential trend model : }\hat{x}_{pf}=a_{pf}x^{b_{pf}}\\ \text{Logarithmic trend model : }\hat{x}_{\log f}=a_{\log f}\log t+b_{ \log f}.\end{split}  \tag{1}\]  
The quadratic exponential smoothing model is based on a single exponential smoothing model, and the single exponential smoothing model can be obtained by:  
\[S_{t+1}^{\text{es1}}=\alpha_{1}x_{t}+(1-\alpha_{1})S_{t}^{\text{es1}} \tag{2}\]  
where \(\alpha_{1}\) is the smoothing coefficient of the model, \(S_{t}^{\text{es1}}\) is the single smoothing result at time \(t\) and \(S_{t+1}^{\text{es1}}\) is the single smoothing result at time \(t+1\). When \(t=1\), \(S_{0}^{\text{es1}}\) is set to be the value of \(x_{0}\). Then, the quadratic exponential smoothing model can be deduced as:  
\[S_{t+1}^{\text{es2}}=\alpha_{2}S_{t}^{\text{es1}}+(1-\alpha_{2})S_{t}^{\text {es2}} \tag{3}\]where \(\alpha_{2}\) is the smoothing coefficient of the model, \(S_{t}^{\rm es1}\) is the single smoothing result at time \(t\), \(S_{t}^{\rm es2}\) is the quadratic smoothing result at time \(t\) and \(S_{t+1}^{\rm es2}\) is quadratic the smoothing result at time \(t+1\). When \(t=1\), \(S_{0}^{\rm es2}\) is set to be the value of \(S_{0}^{\rm es1}\). The predicted result of APT \(\hat{x}_{t+T}^{\rm es2}\) at time \(\hat{x}_{t+T}^{\rm es2}=a_{2t}+b_{2t}*Tt+T\) can be calculated by:  
\[\hat{x}_{t+T}^{\rm es2}=a_{2t}+b_{2t}*T \tag{4}\]  
\[a_{t}^{\rm es2}=2S_{t}^{\rm es1}-S_{t}^{\rm es2} \tag{5}\]  
\[b_{t}^{\rm es2}=(\alpha_{2}/(1-\alpha_{2}))*(S_{t}^{\rm es1}-S_{t}^{\rm es2}) \tag{6}\]  
where \(a_{t}^{\rm es2}\) and \(b_{t}^{\rm es2}\) are the parameters which can be calculated by Equations (5) and (6).