**1. Introduction**

The safety managemen<sup>t</sup> of the high formwork is one of the important tasks of construction safety management, and the majority of failures occur due to inadequate site supervision and poor design [1]. Undoubtedly, the real-time monitoring of near-miss accidents provides an insight into possible accidents and can significantly improve safety performance by appropriate action being taken before potentially impending accidents occur [2]. In view of this, many high formwork systems have installed health monitoring systems of different sizes around the world, and these have accumulated a large amount of data over a long period of time [3,4]. Therefore, how to process this huge monitoring data accurately and timely has become the key process of the high formwork condition assessment and performance prediction [5].

Structural damage will lead to changes in the physical properties of the structure, and these changes are often reflected in the monitoring sequence [6,7]. In the previous evaluation of structural safety performance, most of its data were related to time [8–12].

**Citation:** Yang, Y.; Yang, L.; Yao, G. Post-Processing of High Formwork Monitoring Data Based on the Back Propagation Neural Networks Model and the Autoregressive— Moving-Average Model. *Symmetry* **2021**, *13*, 1543. https://doi.org/ 10.3390/sym13081543

Academic Editor: Basil Papadopoulos

Received: 31 July 2021 Accepted: 16 August 2021 Published: 23 August 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The methods of dealing with stress sequences from health monitoring include the time e-frequency analysis method, modal analysis method, analytic hierarchy process, genetic algorithm, neural network analysis method, time series analysis method, and reliability theory analysis method [13–15]. Among them, the autoregressive–moving-average (ARMA) is a method widely used to model time series [16]. Two steps of the ARMA model that have an important impact on the establishment process are order estimation and coefficient estimation [17]. Coefficient estimation is based on an accurate estimate of the order. The most widely used methods of order estimation are the Akaike information criterion (AIC) [18] and the Bayesian information criterion (BIC) [19]. AIC and BIC are based on the concept of entropy and provide criteria for weighing the complexity of the estimation model against the goodness of the fitted data. In recent years, with the popularity of deep learning, ARMA models and artificial neural network associations have been used to analyze time series [20–24]. These studies prove that the combination of the two is effective for time series analysis [25]. However, in previous literature, the simulation performance of neural networks lacks comprehensive research on different models.

As widely used time series models, ARMA and BPNN are used less in structural monitoring. This paper applies these two methods to the template system for the first time. In previous studies, we have successfully applied neural networks to the safety assessment of long-span bridges and achieved good results [26]. The timeliness of the structure monitoring sequence is more obvious than the safety assessment of long-span bridges, so we model this using the ARMA model and the BPNN model [27]. Compared with the previous literature, we have improved the network structure of BPNN to a better mathematical model. The paper has mainly improved three aspects: (a) Designed and improved the BPNN structure according to the characteristics of the monitoring system; (b) Improved the judgment criteria of the ARMA model order and the model fitting effect; (c) Considering the influence of bias in BPNN modeling, the recognition of discrete data is enhanced.

In this paper, we use neural networks and ARMA to model and analyze the high formwork monitoring sequences with the goal of data-driven modeling. We use back propagation neural network (BPNN) [28] structures for the identification of the order of the ARMA model. The order estimation of the model uses the mean square error (MSE) of the neural network as the basis for the judgment of the order [29]. For neural network structures with low MSE, the structure of the input layer corresponds to the order of the ARMA model. Through the Monte Carlo simulation, a series of model simulation data is obtained. We used the defined neural network model to make a comprehensive comparative analysis of the sequences of different coefficients, sequence lengths, and orders.

This paper is organized as follows. The theoretical basis of the two methods is described in Section 2. In Section 3, the setting of neural network parameters is introduced and verified by the Monte Carlo method. In Sections 4 and 5, the methods proposed in this paper are compared with the typical existing methods, and the effect of using the actual observation data is also analyzed. Finally, the conclusions are provided in Section 6.
