## 2 Method
### Fractional-Order Polynomial Prediction Model and Genetic Algorithm-Based Optimization
#### 2.2.1 Fractional-Order Polynomial Prediction Model  
A basis carries the crucial information required for training, testing, validation and future predictions of the model. Henceforth, this basis must be formed to reflect the most informative features of the input data. The fractional-order polynomial basis covering a bias, fractional orders, and linear and quadratic data is constructed to create a smooth  
Figure 3: Percentages of the departing foreign visitors by reason.  
parameter space with a second-order derivative. For example, the batch type basis \(\mathbf{b}^{\mathbf{T}_{\epsilon}}\) of the departing foreign visitors for traveling reason \(\mathbf{T}_{\epsilon}\) is built as follows:  
\[\mathbf{b}^{\mathbf{T}_{\epsilon}}=\begin{bmatrix}1^{N\times 1}&\mathbf{T}_{ \epsilon}^{0.5}\bullet\mathbf{T}_{\epsilon}^{0.5}&\mathbf{T}_{\epsilon}^{0.8} \bullet\mathbf{T}_{\epsilon}^{0.8}&\mathbf{T}_{\epsilon}^{1}\bullet\mathbf{T}_ {\epsilon}^{1}&\mathbf{T}_{\epsilon}^{2}\bullet\mathbf{T}_{\epsilon}^{2} \end{bmatrix} \tag{1}\]  
where \(\mathbf{T}_{\epsilon}\in\mathbb{R}^{N}\) has dimension of \(N\) and \(\bullet\) is the vector dot product. The basis in Equation (1) is a batch type that can average the effects of the unstructured uncertainties and biases in the data. If a limited amount of data is biased, then their overall impact on the model will be ignorable. However, if the whole data is biased, the algorithm will learn the mean of the biased data in this case. To ensure that the training data is unbiased, it is collected from the official governmental TUIK office and also confirmed with open sources. The corresponding unknown parameter vector \(\mathbf{w}^{\mathbf{T}_{\epsilon}}\) is:  
\[\mathbf{w}^{\mathbf{T}_{\epsilon}}=\begin{bmatrix}w_{1}^{\mathbf{T}_{\epsilon }}&w_{2}^{\mathbf{T}_{\epsilon}}&w_{3}^{\mathbf{T}_{\epsilon}}&w_{4}^{ \mathbf{T}_{\epsilon}}&w_{5}^{\mathbf{T}_{\epsilon}}\end{bmatrix} \tag{2}\]  
The estimated model output \(\mathbf{\hat{y}}^{\mathbf{T}_{\epsilon}}\) is:  
\[\mathbf{\hat{y}}^{\mathbf{T}_{\epsilon}}=\mathbf{w}^{\mathbf{T}_{\epsilon}} \ast\mathbf{b}^{\mathbf{T}_{\epsilon}^{T}} \tag{3}\]  
The training error \(\mathbf{e}^{\mathbf{T}_{\epsilon}}\) is:  
\[\mathbf{e}^{\mathbf{T}}=\mathbf{y}^{\mathbf{T}}-\mathbf{\hat{y}}^{\mathbf{T}} \tag{4}\]  
The constrained cost function to be minimized for the optimal \(\mathbf{w}^{\mathbf{T}_{\epsilon}}\), which is the unknown parameter vector in Equation (2), is:  
\[\begin{array}{rl}J^{\mathbf{T}_{\epsilon}}=&\text{min}&\mathbf{e}^{\mathbf{ T}_{\epsilon}}\ast\mathbf{e}^{\mathbf{T}_{\epsilon}^{T}}\\ &\text{subject to}&\\ &\mathbf{w}^{\mathbf{T}_{\epsilon}}_{\text{min}}\leq\mathbf{w}^{\mathbf{T}_{ \epsilon}}\leq\mathbf{w}^{\mathbf{T}_{\epsilon}}_{\text{max}}\end{array} \tag{5}\]  
where \(\mathbf{w}^{\mathbf{T}_{\epsilon}}_{\text{min}}\) and \(\mathbf{w}^{\mathbf{T}_{\epsilon}}_{\text{max}}\) are the minimum and maximum boundaries of the unknown parameter vector \(\mathbf{w}^{\mathbf{T}_{\epsilon}}\).  
The next section presents the genetic algorithm-based optimization of the fractional-order polynomial prediction model in Equation (3).