## 3 Results
### Development and Evaluation of Wind Speed Models
Therefore, we consider first introducing a time-varying function for the location parameter and, on this basis, introducing a time-varying function for the location parameter. Then, we compare and evaluate the stationary GEV model with these two non-stationary models. Regarding the location parameter, combining the results of the long-term trend analysis from the previous section, we first introduce a linear function, as shown in Equation (14):  
\[\mu(t)=\beta_{0}+\beta_{1}\cdot T \tag{14}\]  
where \(\mu(t)\) is the location parameter at time \(t\), \(T\) is the time variable, \(\beta_{0}\) is the base location parameter, and \(\beta_{1}\) is the coefficient related to the time variable. Meanwhile, due to the appar \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{M_{n}}\) & **NLL** & **AIC** & **BIC** & **K-S(D-\(p\))** \\ \hline Gumbel & 160.596 & 325.191 & 330.053 & 0.081–0.606 \\ GEV & 160.038 & 326.077 & 333.369 & 0.063–0.869 \\ Weibull & 164.394 & 334.787 & 342.080 & 0.073–0.730 \\ Gamma & 159.686 & 325.371 & 332.664 & 0.070–0.769 \\ \hline \hline \end{tabular} \(K\)-S(D-\(p\))—Kolmogorov–Smirnov test results, where D is the test statistic and \(p\) is the \(p\)-value.  
\end{table} 
Table 5: Results of the test on four maximum wind speed distribution models. ent seasonal characteristics of the wind speed throughout the entire time series (as shown in Figure 9), to capture this periodic feature, we decompose the potentially existing periodic functions into a sum of sine and cosine functions using Fourier series. We set the month as the periodic variable, \(M\), and combine it with Equation (14) to obtain Equation (15):  
\[\mu(t)=\beta_{0}+\beta_{1}\cdot T+\beta_{2}\cdot cos\left(\frac{2\pi\cdot M}{12} \right)+\beta_{3}\cdot sin\left(\frac{2\pi\cdot M}{12}\right) \tag{15}\]  
where \(\beta_{2}\) and \(\beta_{3}\) are the coefficients related to the cosine and sine terms, and \(M\) is the value of the month. For the scale parameter, we introduce a function identical to the function for the location parameter. Research indicates that, since the scale parameter must be positive, the scale parameter is set using the logarithmic form [48]. However, setting parameters in logarithmic form makes the interpretation of parameters in the results more complex and requires the re-transformation to calculate the corresponding estimated values. We use the \"extRemes\" package in R to perform a simulation of non-stationary GEV, where the scale parameter is set to positive values by default [49], so the form of the function does not change, as shown in Equation (16):  
\[\sigma(t)=\gamma_{0}+\gamma_{1}\cdot T+\gamma_{2}\cdot cos\left(\frac{2\pi \cdot M}{12}\right)+\gamma_{3}\cdot sin\left(\frac{2\pi\cdot M}{12}\right) \tag{16}\]  
where the meaning of the parameters is consistent with Equation (15). After obtaining the best fit parameters through Maximum Likelihood Estimation (MLE), we compared the stationary GEV distribution with two types of non-stationary GEV distributions (considering only the variation in the location parameter and considering the variations in both the location and scale parameters). First, we compared the quantiles between the actual data and the model-predicted data, as shown in Figure 11.  
Figure 11: _Cont._  
In Figure 11, the horizontal axis represents quantiles from actual samples, while the vertical axis represents quantiles predicted by the theoretical distribution of the model. In theory, if the samples perfectly match the theoretical distribution, data points should fall on the reference line labeled \"1-1 line\". The solid line represents the fit between sample quantiles and theoretical quantiles; if actual data closely match the model data, this line will be very close to the \"1-1 line\". The dashed line represents the 95% confidence interval of the regression line. If this interval includes the \"1-1 line\", it typically indicates acceptable model-data matching.