# On Wind-Induced Fatigue of Curtain Wall Supporting Structure of a High-Rise Building
## 3 Analysis of Results
### Hypothesis Test for Fatigue Life Distribution  
In view of the obtained RBF neural network and joint distribution model parameters of wind velocity and direction, this study predicted the life of each node of the curtain wall supporting structure, as shown in Figure 8.  
Figure 7: Training and test of RBF regression. The degree of fitting is roughly judged by correlation coefficient (\(R^{2}\)) and mean square error (_MSE_), as shown in Table 5. It indicates the kernel distribution has the best fitting effect on nodes among all distributions, whose correlation coefficient is closer to 1 than other distributions, and the mean square error is more closer to 0. Normal distribution and GEV (generalized extreme value) distribution have a slightly poor-fitting effect on node 1, but the fitting effect on other nodes is not significantly different from the kernel distribution. The chi-square test is an important method for goodness-of-fit and it determines whether a data sample comes from specified probability distributions and parameters are estimated from the data. In the test, data were grouped into different intervals whose actual and theoretical frequencies were calculated, and chi-square test statistic was measured [15]. Figure 8: Probability distribution statistics of lifespan. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{2}{c}{**Normal**} & \multicolumn{2}{c}{**GEV**} & \multicolumn{2}{c}{**Kernel**} \\ \cline{2-7}  & \(R^{2}\) & _MSE_ & \(R^{2}\) & _MSE_ & \(R^{2}\) & _MSE_ \\ \hline
1 & 0.8204 & \(3.17\times 10^{-4}\) & 0.8660 & \(2.38\times 10^{-4}\) & 0.9807 & \(3.51\times 10^{-5}\) \\ \hline
2 & 0.9843 & \(2.31\times 10^{-4}\) & 0.9824 & \(2.52\times 10^{-4}\) & 0.9924 & \(1.54\times 10^{-4}\) \\ \hline
3 & 0.9628 & \(5.41\times 10^{-4}\) & 0.9655 & \(5.26\times 10^{-4}\) & 0.9824 & \(3.51\times 10^{-4}\) \\ \hline
4 & 0.9538 & \(4.92\times 10^{-4}\) & 0.9713 & \(3.36\times 10^{-4}\) & 0.9818 & \(2.33\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Correlation coefficient and mean square error of fitting life of several distributions. The chi-square test aims to check the difference level between actual frequency and theoretical frequency; therefore, the core content is to calculate the statistic of the overall difference between actual frequency and theoretical frequency, which is the chi-square distance. This distance is equal to cumulative sum of ratio of square of difference between actual frequency and theoretical frequency to expected frequency [16]:  
\[\chi^{2}=\sum_{k=1}^{r}\frac{\left(n_{k}-np_{k}\right)^{2}}{np_{k}} \tag{16}\]  
where \(n_{k}\) stands for actual frequency and \(np_{k}\) is the theoretical frequency based on a hypothesis distribution. When the frequency is large, the statistic is in an approximate chi-square distribution. The bigger chi-square value means longer distance and stronger difference. The table below explains the actual frequency and theoretical frequency calculated by each distribution. For easy comparison, the actual frequency is a positive integer, and the theoretical frequency is accurate to one decimal place. Table 6 lists test frequencies under various hypothetical distributions, including actual and theoretical frequencies. The difference in actual and theoretical frequencies of the three distributions is far from obvious. Provided that \(k\) unknown parameters in theoretical distribution need to be replaced by a corresponding estimator, then \(n\rightarrow\infty\), statistic \(\chi^{2}\) distribution is asymptotical to \(\chi^{2}\) distribution with \(l-k-1\) degrees of freedom, and \(l\) is the number of intervals of test data.