# On Wind-Induced Fatigue of Curtain Wall Supporting Structure of a High-Rise Building
## 3 Analysis of Results  
### Regression Analysis  
Since countless working conditions are required to analyze wind-induced fatigue of the curtain wall, regression methods can be considered to reduce working conditions to be calculated. Traditional regression methods must be based on mathematical models, but it is difficult to find the relationship between wind velocity and direction with fatigue life. There is no mature mathematical model as reference. Even if there is, it cannot serve as a general model for different scenarios. In summary, the RBF neural network was adopted for regression analysis because it could not only approximate functions but also make predictions. The learning steps of the RBF neural network are:  
1 Initialization. Determine input vector \(\mathbf{X}\), output vector \(\mathbf{Y}\), expected output vector \(\mathbf{O}\), link weight from hidden layer to output layer, central parameter \(\mathbf{C}_{j}\) of each neuron in hidden layer, width vector \(\mathbf{D}_{j}\) and other neural network parameters:  
\[\mathbf{X}=\left[ \begin{array}{cccccc}x_{1}&x_{2}&\cdots&x_{n}\end{array}  \right]^{\mathrm{T}},\ \mathbf{Y}=\left[ \begin{array}{cccccc}y_{1}&y_{2}&\cdots&y_{q}\end{array}  \right]^{\mathrm{T}},\ \mathbf{O}=\left[ \begin{array}{cccccc}o_{1}&o_{2}&\cdots&o_{q}\end{array}  \right]^{\mathrm{T}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(a\) & \(b\) & \(c\) & \(d\) \\ \hline
0.0099 & \(-\)0.0016 & \(-\)0.3167 & \(-\)0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Probability density model parameters of azimuth. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wind Direction** & \(\mu\) **(Location Parameter)** & \(\sigma\) **(Scale Parameter)** & \(k\) **(Shape Parameter)** \\ \hline
13 & \(-\)0.065 & 1.207 & 2.222 \\ \hline
14 & \(-\)0.069 & 1.317 & 2.085 \\ \hline
15 & 0.062 & 1.012 & 1.672 \\ \hline
16 & 0.178 & 0.735 & 1.387 \\ \hline
17 & \(-\)0.005 & 0.545 & 1.079 \\ \hline
18 & \(-\)0.207 & 0.448 & 1.200\[\mathbf{W}_{k}=\left[ \begin{array}{cccccc}w_{k1}&w_{k2}&\cdots&w_{kp}\end{array}  \right]^{\mathrm{T}}\left(k=1,2,3,\cdots,q\right) \tag{6}\]  
\[\mathbf{C}_{j}=\left[ \begin{array}{cccccc}c_{j1}&c_{j2}&\cdots&c_{jn}\end{array}  \right]^{\mathrm{T}},\ \mathbf{D}_{j}=\left[ \begin{array}{cccccc}d_{j1}&d_{j2}&\cdots&d_{jn}\end{array}  \right]^{\mathrm{T}} \tag{7}\]  
where \(n\) is the number of input layer units; \(q\) means the number of output layer units. The initial value of the neural network center parameter and width vector can be output by the following formula:  
\[c_{ri}=\min i+\frac{\max i-\min i}{2p}+(j-1)\frac{\max i-\min i}{p}\ (j=1,2,3, \cdots,p) \tag{8}\]  
\[d_{y}=d_{f}\sqrt{\frac{1}{n}\sum_{k=1}^{5}\left(x_{i}^{t}-c_{j}\right)} \tag{9}\]  
In the formula, \(p\) represents the total number of hidden layers; \(d_{f}\) is the width adjustment coefficient, and the value should be less than 1, in order to allow each hidden layer neuron to easily perceive local information. However, the nonlinear mapping ability of the RBF network is reflected in the hidden layer function, so the adjustment of the hidden layer is conducive to improving the local response ability of the RBF neural network. 2. Calculate output \(z_{j}\) of the \(j\)-th neuron in the hidden layer:  
\[z_{j}=\exp\left(-\left|\frac{\mathbf{X}-\mathbf{C}_{j}}{\mathbf{D}}\right|^{2}\right) \tag{10}\]  
3. Calculate output of neurons in the output layer. \[y_{k}=\sum_{j=1}^{\infty}w_{kj}z_{j}\ (k=1,2,3,\cdots,q) \tag{11}\]  
In the formula, \(w_{kj}\) refers to adjustment weight between \(k\)-th neuron in the output layer and \(j\)-th neuron in the hidden layer. 4. Iteration of weight parameters.