## 4 Performance Comparison of Different Models  
### Stationary Test and Model Building  
For tropical-storm winds that often have prominent non-stationary properties [42], it is necessary to perform a stationary test for the samples before short-term forecasting, which facilitates the selection of stationary and non-stationary models. In statistics, the well-known Dickey-Fuller test is frequently utilized to test the stationarity of a sequence. Once the sequence is non-stationary, its autoregressive model would have a unit root. A general form of the conventional Dickey-Fuller test is given as [43; 44]  
\[\Delta u_{t}=\delta u_{t-1}+\alpha+\beta\lambda+\varepsilon_{t}, \tag{18}\]  
where \(\Delta u_{t}=u_{t}-u_{t-1}\) and \(u_{t}\) represents the sample; \(\alpha\), \(\beta\), and \(\delta\) are coefficients; and \(\lambda\) represents the deterministic trend. The null hypothesis \(H_{0}\): \(\delta=0\), which means the sample is non-stationary, can be tested against the stationary alternative \(H_{1}\): \(\delta<0\) based on the t-statistic [45]. A smaller t-statistic indicates a strong evidence to reject the null hypothesis, so a stationary sample has a small t-statistic. The results of the Dickey-Fuller test for the wind samples in Figure 4a are presented in Table 1 with a comparison with the critical value that the null hypothesis can be accepted with a 95% confidence level. It can be seen from Table 1 that the t-statistic of the original wind records is \(-1.067\), which is greater than the critical value, which means the null hypothesis can be accepted. Thus, the wind data are regarded as a non-stationary process. In the ARIMA model, the differenced data will be employed for forecasting. Based on this consideration, a first-order differencing step is applied to the measured wind data, and the results are depicted in Figure 5a. It is found that the differenced data show no distinct time-varying trend in the mean sense. The Dickey-Fuller test is further conducted to the differenced data, and the results are presented in Table 1 as well. It is found that the corresponding t-statistic is \(-11.894\), which is much smaller than the critical value. Hence, the null hypothesis can be rejected and the differenced data are judged as a stationary process. The PDF is also analyzed for the differenced data and then compared to the normal distribution, as shown in Figure 5b. Different from the observation in Figure 4b, the PDF of the differenced data follows the normal distribution very well. On this condition, the order of the differencing step of the ARIMA model can be taken as one. Figure 6 depicts the ACF and PACF of the differenced data. A 95%-confidence interval, in which the wind speeds are thought to be uncorrelated, is also plotted. It is observed that the ACF has a cutoff after the second lag, and the PACF cuts off after the sixth lag, which implies that the values of \(q\) and \(p\) in the time-series model can be taken as 2 and 6, respectively. Hence, an ARIMA(6,1,2) model can be utilized to characterize the selected dataset. After the orders of different terms are determined, the remaining coefficients of the ARIMA model can be estimated via the maximum likelihood method. Then, the wind speeds of the training set are reconstructed via the established ARIMA(6,1,2) model and the residual errors with respect to the measured data are calculated. Figure 7a presents the PDF of the residual component, and the ACF of residual errors is depicted in Figure 7b. A comparison between the PDF of the residual component and the normal distribution \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Items**} & \multicolumn{2}{c}{**Test Value**} \\ \cline{2-3}  & **Original Data** & **Differenced Data** \\ \hline t-statistic & \(-1.067\) & \(-11.894\) \\ Critical Value (95\%) 1 & \(-2.868\) & \(-2.868\) \\ \hline \hline \end{tabular} \({}^{\dagger}\) Note: the value in the bracket indicates the confidence to accept the null hypothesis.  
\end{table} 
Table 1: Results of the Dickey–Fuller test.