*5.2. Application in Stress Sequence*

In the past, the ARMA model was often used to remove the noise of the time series to obtain the trend of the series [39–41]. These documents select a number of different ARMA models to process the time series and obtain the optimal solution by comparing the results. Selecting a model through the results often takes more time when the amount of data is large, and an accurate estimation of the order can reduce the workload and facilitate the batch processing of data. In this subsection, we demonstrate that the proposed model order selection method based on BPNN can be used to analyze stress sequences. The time series we analyzed comes from a part of the stress change of the high formwork system. According to the loading status of the system, the stress change of the high formwork can be divided into the loading phase and load stabilization phase. For the stress time series in the use phase, the ARMA model generally satisfies the requirements of the series causality. If the time series contains data in the loading phase, then we often need to perform nonlinear processing (differential) on the data to meet the requirements of causality. The accurate processing of monitoring data results is an indispensable part

of the high formwork safety monitoring system. Based on the initial data collected by the high formwork safety monitoring system in the actual project, this paper discusses the specific application of the proposed method in the post-processing of monitoring data. In the following, we use two examples to verify the effect of the order selection method proposed earlier.

**Example 1.** *This example considers a time series that meets the requirements of causality, and the data is obtained from engineering field measurements. The data includes stress changes at 37 positions, and the sequence used in this example is one of the data. Since the increase in the length of the sequence has no adverse effects, we intercepted the sequence with a length of 5000, which is the stress change at the steady stage of the load. We use AIC, BIC, and BPNN to estimate the best order of the model. As before, regarding the setting of the BPNN, we use 80% of the data as the training set and the rest as the test set. The parameter settings are also the same as before. Finally, the ARMA parameters were estimated using the least-squares method for the model order estimated by the BPNN, the AIC and the BIC.*

Figure 12a plots the original data and the sequence obtained by different processing methods, and the residuals of the different methods are shown in Figure 12b. In order to better describe the distribution of the data, we draw the envelope of the obtained sequence and calculate the average width between the upper and lower envelopes. Compared with the AIC and the BIC, the BPNN method reduces the average width of the envelope by 83.08% and 9.16%. It can be found that the BPNN and BIC have similar and accurate judgments on sequence trends for the data in this example, while the AIC has a higher degree of dispersion. In addition, there is no obvious difference in residuals of the three judgment methods.

**Figure 12.** Simulation results of different methods: (**a**) Sequence trend (**b**) Residual sequence.

**Example 2.** *The data used in this example includes the loading phase and the load stabilization phase, and the length of the sequence is 29,610. The setup of the neural network is the same as in Example 1. The difference from Example 1 is that this time series does not meet the requirements of causality, so we have performed different processing on it. The differentiated sequence meets the requirements of causality. Similar to before, we estimated the order of the differenced sequence and used the ARMA process for fitting. Finally, we restored the sequence [42].*

Figure 13 shows the processing results of different judgment methods. The obvious difference between Figure 12a and 13a is that the sequence of the former is stable, while the latter has an obvious rising stage, which is why the latter needs to be processed by difference. Compared with the AIC and the BIC, the BPNN method reduces the average width of the envelope by 51.91% and 52.14%. Therefore, we find that the BPNN's analysis of data trends is more compact than the AIC and the BIC. This shows that the model obtained by the BPNN for order estimation has a better effect on noise extraction.

**Figure 13.** Simulation results of different methods: (**a**) Sequence trend (**b**) Residual sequence.
