## 3 Performance Evaluation  
### Simulation Study  
The performance of the proposed damage identification methodology was first illustrated through numerical simulations carried out on a ten-story shear building model with sensor locations along with a non-uniform distribution of floor masses (\(m\)) and story stiffnesses (\(k\)), as shown in Figure 3. Damping ratio was assumed to be proportional and assigned as 2% for all modes. Damage scenarios included 10% loss of floor stiffness in each floor, one at a time. The analytically computed natural frequencies for the undamaged as well as the damaged cases are listed in Table 1.  
Data for the baseline state, necessary for initial training and subsequent testing, were generated using the finite element model of the frame by assigning an initial velocity at each sensor location, one at a time. Acceleration measurements were simulated at these positions with a sampling time of 0.01 s. over a 60 s period and these data were then contaminated with varying levels of noise, randomly distributed within the range of 2% to 5%. The resulting signal pool included 60 sets of acceleration measurements utilized to compute the TFs. From this baseline data pool, the first 40 were used to train the classification model, while the remaining 20 were reserved for validation. Figure 3: Simulated system with sensor locations and system properties. Figure 5 compares the MSE of the training and the validation (unseen) data for \(n=3\) neurons. This value is found to be optimal with the number of epochs set to 20, since it ensures that the model captures the underlying structure of the transmissibility data without compromising performance on unseen cases. With the selected architecture, the 512 dimensional input data were compressed into a three-dimensional latent space while preserving the most significant features of the input transmissibility data. Since transmissibility, defined as the ratio of amplitudes across different frequencies, is a positive-valued function, a log-sigmoid activation function was employed for the encoder for an effective mapping of the input to a constrained range. For the decoder, however, a linear transfer function was employed for the accurate reconstruction of the input data. The training of the AE is based on minimizing the reconstruction error, calculated as the MSE between the input and output vectors and the network weights are adjusted through backpropagation, with gradients computed using stochastic gradient descent. In this process, each TF was treated as an independent input and trained separately. To assess the AE's effectiveness in capturing the data structure and mitigating measurement noise, its performance was compared to PCA, a linear dimensionality reduction technique that projects the data onto a subspace defined by the principal components. In PCA-based data compression, the projection \(P\), of the input transmissibility \(T\) is given as follows:  
\[P=UT \tag{12}\]  
where \(U\) represents the principal components obtained from the eigenvectors of the covariance matrix of the transmissibility data. For consistency with the AE's latent space, the number of principal components was set to \(n=3\). Figure 6 illustrates this comparison for the distribution of damage sensitive features (DSF) for the training and validation data as extracted by the AE and PCA. A key observation is that the validation data align more closely with the training data in the AE's latent space, whereas in PCA, a significant portion of the validation data lies outside the scatter of the training data. This separation in PCA arises because variations in the validation data that do not align with the major principal components are projected into less significant components, resulting in a more dispersed representation. In contrast, the AE's latent space is smoother, minimizing the influence of outliers and minor variations in the validation data. Figure 5: MSE comparison for training versus validation data for \(n=3\) neurons.