# Midspan Deflection Prediction of Long-Span Cable-Stayed Bridge Based on DIWPSO-SVM Algorithm
## 2 DIWPSO-SVM Algorithm
### Wavelet Transform-Based Separation of Load Effects  
Wavelet transform is an effective analytical method for signal processing with better localization and multiresolution characteristics, which can simultaneously analyze signals in the time and frequency domains, overcoming the shortcomings of the traditional Fourier analysis [31]. Wavelet transform was initially limited due to drawbacks such as excessive arithmetic. However, the signal was decomposed into detailed and approximate components with the introduction of excellent decomposition algorithms such as Mallat multiresolution analysis. The wavelet transform harmonic extraction method based on the fast algorithm has become a widely used harmonic detection method [32]. Therefore, the Mallat wavelet transform algorithm was used in this study to deconstruct the mid-span deflection signal.  
Figure 2: Parameter optimization process of SVM algorithm by DIWPSO.  
The vertical deflection of a bridge is mainly affected by the ambient temperature and vehicle load, in which the total response of bridges is obtained under the combined effect of temperature load and vehicle load in usual deflection monitoring. However, the deflection curves of bridges under two kinds of load effects have completely different characteristics. The data under different load effects can be learned and analyzed separately, and then re-stacked, which can improve the prediction accuracy of the SVM algorithm. Therefore, it is necessary to separate the sample data from the load effects.  
The effect of temperature load on bridge deflection is mainly classified into annual and daily temperature difference effects, which are measured on time scales of years and days, respectively. The vehicle load is a moving load, and the corresponding vertical deflection is characterized by high-frequency signals measured in seconds. The annual temperature difference effect has the characteristics of a low-frequency signal, which has the most negligible effect on the short-term deflection of the bridge. Thus, the effect of long-term temperature differences on the short-term deflection of bridges was ignored in this study; only the short-term temperature deflection curve and the deflection curve could be extracted for analysis.  
In order to accurately analyze the characteristics of the mid-span deflection curves of the main bridge girder under two kinds of load effects, the Mallat wavelet transform algorithm was adopted to deconstruct the mid-span deflection signals [11]. The basic algorithm of the Mallat wavelet transform is shown in Equation (16).  
\[\begin{cases}\begin{array}{c}A_{o}[f(t)]=f(t)\\ A_{j}[f(t)]=\sum\limits_{k}H(2t-k)A_{j-1}[f(t)]\\ D_{j}[f(t)]=\sum\limits_{k}G(2t-k)A_{j-1}[f(t)]\end{array}\end{cases} \tag{16}\]  
where \(A_{j}\) and \(D_{j}\) are the wavelet coefficients of the signal _f(t)_ in the low-frequency and high-frequency parts, respectively. \(H\) and \(G\) are the low-pass and high-pass filters of Daubechies six-order (Db6) wavelet basis functions, respectively. \(t\) is the discrete time series, and \(j\) is the number of decomposition layers, \(j=5\).  
The reconstruction algorithm is shown in Equation (17).  
\[A_{j}[f(t)]=\sum\limits_{k}h(2t-k)A_{j+1}[f(t)]+\sum\limits_{k}g(2t-k)D_{j+1} [f(t)] \tag{17}\]  
where \(h\) and \(g\) are the reconstructed wavelet low-pass and high-pass filters, respectively.