## 3 Laboratory Experiment
### Damage Detection  
Feedforward ANNs were used for anomaly detection. All patterns were divided (evenly for each case of the considered scenario) into learning and testing sets in the ratio of 2:1. The division remained constant throughout the experiment. Every set of patterns \(P\) was described according to the following formula:  
\[P=\{\mathbf{x}^{i},\mathbf{t}^{i}\}^{n} \tag{3}\]  
where  
\(\mathbf{x}^{i}_{(Nx1)}\) -- the input vector for the \(i\)-th pattern, the number of inputs \(N\in\{6,8,9\}\);  
\(\mathbf{t}^{i}_{(Kx1)}\) -- the target vector of the \(i\)-th pattern, the number of outputs \(K\in\{1,4\}\);  
\(n\) -- the number of patterns in the considered set; in the learning set \(n=1826\) and in the testing one \(n=913\). A number of variants of input and output parameters were considered, see Table 1:  
\[\mathbf{y}^{i,j}_{(Kx1)}=\text{ANN}^{j}(\mathbf{x}^{i},\mathbf{w}^{j}) \tag{4}\]  
where  
\(\mathbf{y}^{i,j}_{(Kx1)}\) -- the output vector of the \(i\)-th pattern and the \(j\)-th neural network of the specified architecture; ANN-- the ANN of the \(N-Hi-K\) architecture, with \(Hi\) hidden neurons;  
\(\mathbf{w}^{i}\) -- the free parameters vector for the \(j\)-th neural network of the analyzed architecture  
\((N,Hi,K=const)\). The output vectors took one of the following forms (see Figure 2d and Table 1):  
* \(\mathbf{y}_{(1x1)}=\{C\}\), where \(C\) was the class of anomaly; adopted classes:  
Figure 3: (**a**) Horizontal displacement of point OO (middle point in the observed area, shown in Figure 2e) in one selected set of 20 consecutive pictures, (**b**) the selected triangular area in two opposite states, called the reference picture and Picture No. 2, and (**c**) changes in its shape. concerning the determination of anomaly location:  
\(C=0\) for Class S00, \(C=1\) for S33, \(C=2\) for S22, \(C=3\) for S23, \(C=4\) for S44, \(C=5\) for S34, \(C=6\) for S14, \(C=7\) for S11, and \(C=8\) for S12,  
concerning the determination of size of anomaly:  
\(C=0\) for Class S00, \(C=1\) for S11, S22, S33, and S44, and \(C=2\) for S12, S14, S23, and S34;  
\(\mathbf{\gamma}_{(4x1)}=\{R1,R2,R3,R4\}\), where \(Rr\) is the described condition of the \(r\)-th row of bolts, starting from the top and ending with the bottom one; \(Rr=0\) means that the \(r\)-th row is properly fastened, \(Rr=1\) means that the \(r\)-th row is loosened;  
\(\mathbf{\gamma}_{(1x1)}=\{D\}\), where \(D\) determines the occurrence of the anomaly of the selected type, \(D=0\) means that the anomaly in the selected type does not exist, \(D=1\) means that the anomaly in the selected type exists. It should be noted that the elements of output vectors were continuous values in specified ranges, while the elements of target vectors were discrete values (see Table 1). The number of input and output parameters determined the network architecture. The learning process was repeated 30 times for each architecture, because of the random nature of weight adoption in the first step of iteration. All networks of the analyzed architecture were used for the purpose of anomaly appearance detection and the output value \(\hat{\mathbf{y}}^{i}\) for a single pattern was given after the statistical calculation, conducted using Equations (5) though (8).