## 4 Experimental Validation
### The Efficacy of the LSTM-RNN-Oriented RLU Prediction  
The LSTM-RNN-based RLU predictor is then trained with previously constructed HI curves, which are segmented into a series of time-steps. The data can be considered a univariate time series due to it being a sequence of one value per time step. The RUL predictor utilizes past HI values to predict the future ones until a predefined threshold is met. In order to maximize the number of time series to train the model, a shifting window of size 50 is used in the segmentation process. This procedure is shown in Figure 13.  
At this step the LSTM predictor is already capable of forecasting the next time-step value. However, with an alteration in the procedure, it can also predict more than just one future state. The next prediction can be added to the inputs (acting as if this predicted value had occurred) to further predict the following ones until the end. Our model is trained to forecast at every time-step instead of just the final time-step. By following this technique, the loss can contain a term for every time-step output rather than just the output at the last time step. This allows more stabilization and faster training as more error gradients are able to flow through the model [22]. One input layer of the same size as the number of time-steps is utilized for the construction of the applied LSTM-RNN. Following this layer are two hidden LSTM layers of size 20. A linear activation function is utilized in the dense output layer of one neuron. Early stopping and checkpoint techniques are also used during training to construct a better LSTM-RNN model. The developed method is implemented with degradation data collected from three concrete beams A, B, and C. Generally, RUL estimations toward the end of a degradation process are more important than earlier ones because this is usually when maintenance \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of HI** & **Min Value** & **Max Value** & **M** & **R** \\ \hline RMS & 0.0005 & 0.1529 & 0.037 \(\pm\) 0.022 & 0.347 \(\pm\) 0.064 \\ Kurtosis & 2.7496 & 5629.1 & 0.004 \(\pm\) 0.021 & 0.458 \(\pm\) 0.050 \\ Crest factor & 3.0803 & 145.91 & 0.0013 \(\pm\) 0.183 & 0.7416 \(\pm\) 0.024 \\ Skewness & \(-\)19.279 & 10.855 & 0.0102 \(\pm\) 0.0215 & 0.1077 \(\pm\) 0.1020 \\ Entropy & 0.0883 & 4.8324 & 0.0019 \(\pm\) 0.0234 & 0.1886 \(\pm\) 0.1740 \\ SAE-DNN & 0.0125 & 1 & 0.6788 \(\pm\) 0.079 & 0.6801 \(\pm\) 0.0489 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Metric comparison of different types of HIs. Figure 13: Time series construction from training degradation data. decisions are made. Therefore, two major time stamps at cycle 350 (minor crack initialization) and cycle 450 (major crack initialization) are chosen for the average prediction error calculation. The definition of the RUL prediction error is:  
\[\overline{e}=\left|LC_{i}-\widehat{LC_{i}}\right| \tag{16}\]  
with \(\overline{e}\) being the RUL prediction error, and \(LC_{i}\), \(\widehat{LC_{i}}\) are the actual cycles and the estimated cycles, respectively. In this study, 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. In this case, the fixed HI threshold should be set to 1. However, experimental results have shown that the output of DNN rarely reaches the value of 1. Hence, this study has set the fixed HI threshold of 0.95 to mark the specimen's breakdown. Figure 14 shows specimen A's RUL prediction error with the prediction calculated at cycle 350, and the prediction error calculated at cycle 450. The details of statistics concerning specimen A's RUL prediction error can be found in Table 3 along with the other two specimens; a minor prediction error with a small standard deviation is preferred.