## 4 Experimental Validation
### The Efficacy of an SAE-DNN-Based HI Constructor  
This stage starts with SAE-DNN-based HI constructor training. Initially, fast Fourier transform (FFT) is utilized to determine the signal segment's spectrum. Since each segment is \(5\times 10^{6}\) in length, after FFT there are \(2.5\times 10^{6}\) data points which are too big to feed into the SAE. Therefore, the number of inputs is reduced by splitting the spectrum of the AE signals into a suitable amount of frequency bands and then computing their root mean square (RMS), which represents an approximation of each band's energy. Following the data preprocessing, the SAE model is constructed and trained. Signal spectrum vectors of size 2000 are fed to the encoder, which are then processed by three size-diminishing dense layers (1000 to 200 to 10 units with Xavier initialization and the ELU activation function). The encoder's output (size of 10) is then fed to the decoder and processed by three size-increasing dense layers (200 to 1000 to 2000 units). Dropout layers with rate of 0.1 are added before the dense layers to improve the SAE's regularization. Afterwards, Adam optimization is utilized for SAE training with unlabeled signal spectra as both the inputs and targets. Different levels (from 0.1-0.5) of the fractions of masked zero are tested, which shows the best performance at 0.1. Then the encoder's layers are reused in the DNN model as the hidden layers. After a logistic regression layer addition, this DNN model is fine-tuned in a supervised way. The DNN output layer's size is 1 and its according label is the normalized number (in the range of [0, 1]) of AE hits detected in each degradation cycle. The DNN model is designed so that the outputs, which are the HI values, are in the range of [0, 1], therefore the sigmoid activation is chosen as the activation function of the output layer. During the training process, the reused layers are frozen so that they retain the learning ability of high-level features from the low-level input features of the SAE model. In addition, the early stopping and checkpoint techniques are applied during training SAE and DNN to get the best parameters of SAE and DNN structures. Following the completion of SAE-DNN-based HI constructor's training, the run-to-failure data is harnessed for the concrete beam's HI construction. Half of the signals are utilized here for the evaluation of the HI constructor's performance. Figure 11 shows the  
Figure 10: Illustration of the degradation process of concrete beams using RMS values, kurtosis values and AE hits. HIs of the three tests from sensor 5. In comparison to HIs constructed from conventional features like RMS and kurtosis with larger scales, the proposed method's HIs ranges between 0 and 1, with 1 being the failure condition. Consequently, the threshold definition has no need for such HI employment. An HI exceeding a fixed FT triggers the obtain of RUL. Two other metrics, monotonicity and trendability, [24] are also used in this study for HI fitness validation. Monotonicity is the characterization of the underlying increasing or decreasing trend:  
\[M=\left|\frac{no.of.d/dx>0}{n-1}-\frac{no.of.d/dx<0}{n-1}\right| \tag{14}\]  
with \(n\) being the number of observations for a specific feature. Monotonicity \(M\) can be calculated with the absolute difference between a feature's \"positive\" and \"negative\" derivatives. It ranges from 0-1, with 0 emphasizing a non-monotonic feature and 1 showing a highly monotonic feature. The second metric, trendability, is related to an extracted feature's functional form and correlation with time. In another words, it shows how an asset's state varies with time. For example, a constant function presents zero correlation with time, while a high correlation can be found with a linear function. Similarly, non-linearity also causes a variation in correlation.