# Structural Damage Detection through Dual-Channel Pseudo-Supervised Learning
## 2 Methodology
### Data Preprocessing  
To acquire acceleration response data, it is assumed that \(n\) sensors are deployed on the \(h\)-th floor of the structure, with \(m\) sensors measuring the translational acceleration in the \(x\) direction and \(k\) sensors measuring the translational acceleration in the \(y\) direction. During response data collection, the sampling frequency of each sensor is \(f_{s}\), and the sampling time is \(t_{s}\), resulting in the number of sampling points \(S_{j}\) per sensor being \(S_{j}=f_{s}\times t_{s}\). In this paper, Equation (4) was used to fuse the acceleration response data from the co-oriented sensors on the \(h\)-th floor to obtain the fused translational accelerations in the \(x\) and \(y\) directions of this floor [40]. Equation (4) is as follows:  
\[\begin{cases}acc_{x,h}=\frac{1}{m}\times(acc_{x,1,h}+acc_{x,2,h}+\ldots+acc_{x,i,h}+\ldots+acc_{x,m,h})\\ acc_{y,h}=\frac{1}{k}\times\left(acc_{y,1,h}+acc_{y,2,h}+\ldots+acc_{y,i,h}+ \ldots+acc_{y,k,h}\right)\end{cases} \tag{4}\]  
where \(acc_{x,i,h}\) and \(acc_{y,i,h}\), respectively, denote the translational acceleration response data collected by the \(i\)-th sensor in the \(x\) and \(y\) directions of the \(h\)-th floor, and \(acc_{x,h}\) and \(acc_{y,h}\) denote the fused translational acceleration in the \(x\) and \(y\) directions of the \(h\)-th floor. The variables \(m\) and \(k\) denote the number of sensors in the \(x\) and \(y\) directions of the \(h\)-th floor, respectively. All of these data are one-dimensional time series with \(S_{j}\) data points.  
Subsequently, fused translational acceleration data are sequentially divided into \(w\) nonoverlapping samples, each consisting of 128 continuous data points [41]. Each sample is then independently normalized and marked with the corresponding damage pattern label. The processed data are defined as \(acc^{\prime}_{x,h}\) and \(acc^{\prime}_{y,h}\). Afterward, \(acc^{\prime}_{x,h}\) and \(acc^{\prime}_{y,h}\) are randomly shuffled. Assuming the damage detection task includes \(q\) damage patterns, the acceleration response data under each damage pattern are processed using the aforementioned method. After processing, the response data for the \(x\) and \(y\) directions on the \(h\)-th floor under the \(r\)-th damage pattern are denoted as \(acc^{\prime}_{x,h,r}\) and \(acc^{\prime}_{y,h,r}\), respectively. The subsets constructed from the response data of the \(q\) damage patterns in the \(x\) and \(y\) direction of the \(h\)-th floor are, respectively, expressed as follows:  
\[D_{x,h}=\begin{bmatrix}acc^{\prime}_{x,h,1}\\ acc^{\prime}_{x,h,2}\\ \ldots\\ acc^{\prime}_{x,h,q}\end{bmatrix} \tag{5}\]  
\[D_{y,h}=\begin{bmatrix}acc^{\prime}_{y,h,1}\\ acc^{\prime}_{y,h,2}\\ \ldots\\ acc^{\prime}_{y,h,q}\end{bmatrix} \tag{6}\]Assuming the structure has \(d\) floors, the intact translational acceleration dataset \(D\), composed of the subsets of the response data in the \(x\) and \(y\) directions of each floor, can be denoted as follows:  
\[D=\left[D_{x,1}\;\;D_{y,1}\;\;\cdots\;\;D_{x,h}\;\;D_{y,h}\;\;\cdots\;\;D_{x,d} \;\;D_{y,d}\right] \tag{7}\]