# Damage Identification of Stay Cables Based on a Small Amount of Deflection Monitoring Data
## 2 Theory Related to the Reduction in Measurement Points
### Damage Degree Identification  
The variation in cable damage degree is random regardless of whether it is for single-cable damage or double-cable damage. With different cable damage degrees, the corresponding girder deflection variation and distribution law are also different. In order to identify the damage degree of a cable using numerical simulation, the calculation workload is too large. By using the Kriging proxy model, an approximate function can be constructed with limited sample data to predict the girder deflection corresponding to the damage degree of the cable so as to achieve the purpose of identifying the damage degree of the cable. The Kriging proxy model is an interpolation technique for predicting unknown sample information based on the correlation of known sample information and the minimum variance criterion [22]. Let the response of an unknown vector \(\mathbf{z}=\left[\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{z}_{3},\cdots\mathbf{z}_{n}\right]^{T}\) be \(\mathbf{Y}=\left[y(\mathbf{z}_{1}),y(\mathbf{z}_{2}),y(\mathbf{z}_{3}),\cdots y(\mathbf{z}_{n}) \right]^{T}\), which can be predicted using the expression \(\hat{y}(\mathbf{z})\) obtained via the superposition of the polynomial regression model \(\mathbf{f}^{T}(\mathbf{z})\mathbf{\alpha}\) and the zero-mean normal distribution random function \(\mathbf{g}(\mathbf{z})\). \[\hat{y}(\mathbf{z})=\sum_{m=1}^{p}\alpha_{m}f_{m}(\mathbf{z})+\mathbf{g}(\mathbf{z})=\mathbf{f }^{T}(\mathbf{z})\mathbf{\alpha}+\mathbf{g}(\mathbf{z}) \tag{12}\]  
where the first part is the polynomial regression model about variable \(\mathbf{z}\), \(\mathbf{\alpha}\) is the regression coefficient, \(\mathbf{\alpha}=\left[\alpha_{1}\,\alpha_{2}\ldots\alpha_{p}\right]^{T}\), \(\mathbf{f}=\left[f_{1}(\mathbf{z})f_{2}(\mathbf{z})\,\cdots\,f_{p}(\mathbf{z})\right]^{T}\); the second part \(\mathbf{g}(\mathbf{z})\) is a random distribution with non-zero covariance and obeys the normal distribution \(N(0,\sigma^{2})\). In order to judge whether the response predicted by the Kriging proxy model can meet the accuracy requirements, the square correlation coefficient and the square error criterion is generally used to evaluate its accuracy as shown in Equations (13) and (14). \[SC=1-\frac{\sum_{i=1}^{N}\left[\hat{y}_{i}-\hat{y}_{i}^{t}\right]^{2}}{\sum_{ i=1}^{N}\left[y_{i}^{b}-\hat{y}\right]^{2}} \tag{13}\]  
\[EISE=\frac{1}{N}\sum_{i=1}^{N}\left[\hat{y}_{i}-y_{i}^{b}\right]^{2} \tag{14}\]  
In Equation (14), \(\hat{y}_{i}\) is the \(i\)-th component of the response vector predicted by the Kriging proxy model, \(\hat{y}_{i}^{b}\) is the \(i\)-th component of the actual response vector simulated by the numerical model, \(\hat{y}\) is the mean value of \(y_{i}\), and \(N\) is the length of \(\hat{y}\). In general, when \(SC>0.99\) and \(EISE<0.01\), it indicates that the response predicted by the proxy model meets the accuracy requirements; otherwise, it needs to be modified by adding criteria. In this  
Figure 12: Wavelet coefficients distribution curves of the first derivative of beam deflection difference before and after cable M16 and M7 damage. 3. Establish an objective function for the cable damage index to be identified, and then use the particle swarm optimization (PSO) algorithm [24] to find a set of data values to minimize the objective function; in addition, the corresponding data of the minimum value is the cable damage index. The deflection difference between the measured deflection difference of the girder before and after cable damage and the deflection difference predicted by the [PERSON] proxy model are compared, and the sum of the squares of the difference is taken as the objective function.