## 2 Materials and Methods
### Regression Analysis
#### 2.3.2 Segmented Nonlinear Regression
### Segmented Nonlinear Regression

We discover the segmented nonlinear regression model considerably outperformed the linear regression model and the LSTM model based on the RMSE score. The model also rivals the Seq2 Seq LSTM model's result on the Crack A2 test set. From Table 1, we find the segmented nonlinear regression model significantly surpassed the linear regression model on any test set. Figure 7d shows that the model can approximate broken segments effectively, despite abrupt changes on the Crack B1 train set. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Prediction Model** & **Crack A1** & **Crack A2** & **Crack B1** & **Crack B2** \\ \hline LSTM & 0.00834 & 0.00439 & 0.07238 & 0.20018 \\ Seq2 Seq LSTM & **0.00064** & **0.00166** & **0.00253** & **0.00235** \\ Linear Regression & 0.00169 & 0.00178 & 0.12429 & 0.90953 \\ Segmented Nonlinear Regression & 0.00167 & 0.00158 & 0.02490 & 0.02920 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of all the models on the test sets. Scores in the table are from RMSE metric. A low score means high performance. The highest performances are in bold. Figure 7: Prediction of linear regression and segmented nonlinear regression on Crack A1 and Crack B1 train set. (**a**) Linear Regression on Crack A1 train set. (**b**) Segmented Nonlinear Regression on Crack A1 train set. (**c**) Linear Regression on Crack B1 train set. (**d**) Segmented Nonlinear Regression on Crack B1 train set. ### Long-Term Prediction Analysis

In addition to quantitative evaluations based on the RMSE metric, we analyze the long-term prediction capability of each model. We conduct experiments in which individual models produce two kinds of prediction: (1) 10 years and (2) eight months on each sensor. We present the prediction results on Crack A2 in Figure 8, where each model obtains similar performance in terms of the RMSE metric. We discover that the outputs of the LSTM model and the Seq2 Seq LSTM model are indistinguishable, as both models output constant values after a few steps. Figure 8b shows that the Seq2 Seq LSTM model produced its predictions in a logarithmic fashion in less than a month until it generated fixed outputs for the rest of its prediction. We observe that the LSTM model performed similarly, with an exception in which the model produced static outputs from the beginning. As a result, both models failed to emulate the crack's growth in the long term, as in our experiment for 10-year prediction. Figure 8a shows the results mentioned earlier on Crack A2 data. Furthermore, the training process has required a tremendous amount of time on both models. Similarly, the inference process for the 10-year prediction took a massive span of durations for the LSTM model and the Seq2 Seq LSTM models, as we report in Table 2. For these reasons, we consider that the LSTM model and the Seq2 Seq LSTM model may not be suitable for long-term predictions. Contrary to the LSTM model and the Seq2 Seq LSTM model, the linear regression requires a small amount of training and inference time. On average, the model only took a minute and three seconds to produce 10-year predictions. Although the linear regression model achieved the fastest training and inference time among the models, we discover the linear regression model produced negative increment outputs on 10-year and eight-month predictions, as shown in Figure 8a and Figure 8b, respectively. We conjecture that the decremental happened as a result of the inaccurate prediction. Figure 6c supports our \begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Model** & **Training Duration** & **Inference Duration** \\ \hline LSTM & 1:18:27 & 8:23:57 \\ Seq2 Seq LSTM & 1:15:34 & 1:25:11 \\ Linear Regression & **0:00:07** & **0:01:03** \\ Segmented Nonlinear Regression & 0:00:11 & 0:01:06 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average time consumption of the training process, and 10-year inference for each model on every sensor.