A Sliding Windows Singular Decomposition Model of Monitoring Data for Operational Tunnels

: In order to extract the valuable information from massive and usually unstructured datasets, increasingly, a novel nonparametric approach is proposed for detecting early signs of structural deterioration in civil infrastructure systems from vast field-monitoring datasets. The process adopted six-sample sliding window overtime at one-hour time increments to overcome the fact that the sampling times were not precisely consistent at all monitoring points. After data processing by this method, the eigenvalues and eigenvectors were obtained for each moving window, and then an evaluation index was constructed. Monitored tunnel data were analyzed using the proposed method. The required information extracted from an individual moving window is represented by a set of principal components, which become the new orthogonal variables. The resulting evaluation indicator was strongly correlated with measured and calculated values up to 0.89, even for tiny monitoring datasets. Experiments have verified the rationality and effectiveness of the algorithm, which provides a reference for the application of the method in the monitoring data processing.


Introduction
The current condition of operating tunnels is monitored by various agencies, and continually advancing monitoring techniques are widely used in tunnel engineering and geo-engineering. Globally, most medium and long tunnels are equipped with structural monitoring systems to ensure the tunnel's safety; these include technologically advanced methods such as machine vision and monitoring through measurement reference networks [1,2]. However, commonly in practice, the monitoring system cannot efficiently use the recorded data-for instance, if it does not include intelligent analysis or other capabilities [3].
Displacement and deformation monitoring systems have been used to judge the stability of the rock surrounding tunnels, and fitted algorithms have been developed to measure the deformation of the rock and supporting structures. The movement of the soil and liner displacement between tunnels was analyzed and discussed, and the importance of a tunnel's efficient data management being equipped with an information system was emphasized [4][5][6][7][8]. Sun et al. [9] developed a high-precision multi-point optical grating displacement monitoring technique that has subsequently been widely used in physical model tests (e.g., [10,11]).
Convergence monitoring systems comprising several measuring points with laserranging sensors have been reported, but the measured precision was not always assured. Farahani et al. [12] used 3D laser coupled technology and the digital imaging systems to monitor tunnel deformation and establish tunnel contour models. As digital image correlation (DIC) systems need to be installed in front of the suspect region, however, the practical value was poor in actual tunnels. Kallinikidou et al. [13] reported an analysis of a monitoring database of the Vincent Thomas Bridge's structural health. However, it is questionable whether this algorithm's calculation scheme and accuracy are appropriate for tunnels.
A novel monitoring method based on "on-site visualization" (OSV) was proposed by Izumi et al. [14] to conduct real-time data processing. In particular, they addressed the limitations of OSV in cases of incomplete coverage of construction-site accident monitoring. They considered the need to improve the reliability of the power supply to the equipment and how to take equipment failure into account.
In the present study, "data-driven" refers to the implementation of various valuable features such as forecasting, monitoring, evaluation, diagnosis, decision making, scheduling, and optimization based on continuously updated data (Salami et al. [15]). The rapid progress in big data-driven technology and associated software for extracting information from massive and usually unstructured datasets increasingly compensates for the defects cited above. He et al. [16] reported, using computer intelligence technology to compare and analyze the predictive capabilities of the support vector machine, a wavelet neural network and Gaussian process (GP) to determine deformation sequences in the rock surrounding underground openings. They concluded that, of these approaches, GP provides the best predictions. An automatic, highly accurate real-time wireless tunnel deformation monitoring system based on machine-vision and laser technology was proposed by Qiu et al. [17] for predicting tunnel collapse accidents. The system is also suitable for monitoring in low-visibility tunnels.
Examples of proposed extensive data-driven monitoring in non-geotechnical contexts include a modeling strategy proposed by Wang et al. [18] for optimizing BioH2 production during dark fermentation. Chronological multiphase partition analysis and differential monitoring technology are used in a proposed uneven-duration process monitoring scheme for batch processes based on the sequential moving principal component analysis (Guo and Zhang,[19]).
Related to all of these, Shlens et al. [20] explained the working method and procedures of principal component analysis (PCA) using fundamental mathematical cognition. Yun et al. [21] developed a data-driven monitoring framework based on PCA, which they used in various mine monitoring applications to determine the effect of nearby tunnels on the monitored tunnel. Harley and Moura [22] used data-driven matching field processing technology to significantly reduce the positioning error and improve the resolution of structural health monitoring systems. He et al. [23] proposed a flexible, robust PCA method that uses two matrices for error correction, one of which obtains data representation to improve performance and practicability.
Long-term deformation monitoring systems for tunnels accumulate enormous datasets that are difficult to analyze and process in a way that reflects the deformation of the tunnel in real-time. Google GFS, MapReduce, and Impala based on Hadoop (real-time) are variously used for big data processing (Isard et al., [24]; Leibiusky et al., [25]). Xiang et al. [26] proposed a design and optimization method for an extensive data processing system based on orthogonal decomposition, which has the advantage of performing efficient and hierarchical calculations such that the primary target and secondary factors are interlayered, and the primary target is prioritized. It dramatically accelerates the processing of large datasets and improves its efficiency (Li et al. [27]; Li et al. [28]; Suzuki et al. [29]).
The abovementioned deformation monitoring methods are essentially parametric approaches to the problem. They, therefore, suffer from limitations, such as unknown structural characteristics, unknown material properties, and the strange behavior of the surrounding rock. Precisely, the responses are measured using a range of sensors, and no input information is obtained. It may prevent the verification of the correctness of the model, so such approaches do not necessarily reflect the actual structural failure progression. By contrast, nonparametric approaches do not suffer from these drawbacks, making them more feasible for analyzing the large datasets accumulated during monitoring.
This study reviews recent developments in nonparametric approaches, and a datadriven detection model is proposed for long-term deformation monitoring of an operational tunnel adopting the orthogonal decomposition approach.

Site Description and Engineering Background
Long-term deformation monitoring was carried out at a light subway tunnel retrofit project in Chongqing, China. Horizontal and vertical displacements, structural stress, structural fractures, and settlement of the track-bed and rail structure were monitored at a location between the rail platform and the station. Comprehensive analysis of the impact on the structure and operational safety of the subway was the basis for the decision making of the safe operation of the rail transit system.
The monitoring system comprised an independent coordinate system with eight reference points (two at each end of the left-and right-hand tunnels) set at each end of the interval to be monitored and inside the hazardous area affected by construction. The coordinates of all eight reference points were the reference data for the entire monitoring system. Work base points 300-600 m apart were arranged at appropriate positions inside the tunnel and within the interval to be monitored. Precision wires between the reference and work base point served as the first-level control network. A measuring robot automatically and accurately aligned the operational base or reference point. It used quasistable adjustment to calculate the precise coordinates and elevation of the station in realtime to monitor each point. The cross-section of the monitoring tunnel is shown in Figure  1.  Each displacement monitoring section contained one arch-top settlement monitoring point, two arch waist settlement monitoring points (also used as horizontal convergence monitoring points), and two points for monitoring the track-bed settlement. Each stress monitoring section contained three stress/strain monitoring points. The arrangement of the measuring points is shown in Figure 2. There were seven underground passages within the monitoring interval, with twenty-one deformation monitoring sections in the left-hand tunnel and forty-two in the right-hand tunnel. A stress/strain monitoring section comprising three monitoring points was located in the central area of the surface passage. In total, 14 stress/strain monitoring sections were located not more than 100 m apart in the left-hand and right-hand tunnel, and 42 deformation monitoring sections were located in each. Overall, 82 displacement monitoring sections and 14 stress/strain monitoring sections were positioned in the tunnels in this project. The track-bed and arch point arrangement images are shown in Figure  3.

Continuous Monitoring of Tunnel Deformation
The monitoring system was commissioned on 14 March 2015 and the first monitoring round was completed on 16 March. Automatic deformation monitoring was conducted every two hours. The average of the observations for each round was considered the daily observation. The monitoring during the construction period began on 26 December 2014 and ended on 4 January 2016; the monitoring of the operational period commenced in January 2016 and ceased in December 2017. Monitoring during the operational period was carried out fortnightly, monthly, and once every two months in the first, second, third, and fourth quarters of 2016. After long-term monitoring, gross errors in the recorded observations were eliminated to obtain accurate and reliable data for tunnel deformation analysis.
The recorded data were grouped into primary divisions to obtain the time-displacement relationships for the five monitoring points at monitoring sections Z10, Z11, Z12, and Z13. Figures 4 and 5 show the displacement variation with time at Sections Z10 and Z11.  Monitoring progressed from Z10 to Z11, 20 m apart (Benchmark no. DK0+637 for Section Z10; benchmark no. DK0+657 for Section Z11). It is clear in Figure 4 that large displacement variations at section Z10 began on 1 July 2015 (commencement of track construction work) and ended on 7 December 2015 when construction work ceased. Displacements fluctuated between 2.94 and 0.04 mm (average 1.70 mm; standard deviation 0.90 mm). Similarly, Figure 5 shows large displacement variations at section Z11 corresponding to the period of track construction work from 5 July to 10 December 2015. Displacements fluctuated between 3.80 and 0.06 mm (average 1.76 mm; standard deviation 0.85 mm). Figures 6 and 7 show displacement vs. time at sections Z12 and Z13, 80 m apart (Benchmark no. DK0+737 for Section Z12; benchmark no. DK0+817 for Section Z13).  Monitoring activity progressed from Z12 towards Z13. Figures 6 and 7 show significant, abrupt deformation changes at sections Z12 and Z13, corresponding with the period of track construction overhead (8 July-14 December 2015 for Z12, and 8 July-20 December 2015 for Z13). At section Z12, displacement fluctuated between 0.04 and 2.87 mm (average 1.17 mm; standard deviation 0.53 mm), and from 0.10 to 2.81 mm at section Z13 (average 0.98 mm; standard deviation 0.58 mm). These indicate that the long-term monitored displacements at sections Z12 and Z13 were highly accurate and sensitive, and few errors were apparent.

Fundamentals of Principal Component Analysis and Singular Value Decomposition Method
The following review of nonparametric approaches for tunnel structure analysis includes some implications for resolving the challenges. Proper orthogonal decomposition (POD) methods are increasingly adopted in engineering contexts. In data analysis, POD is a powerful and elegant method to obtain low-dimensional approximations of high-dimensional processes. The main objective of POD is to seek ordered orthonormal basis vectors within a subspace.

The Basic Principle of Principal Component Analysis
The purpose of PCA is usually to identify the structure of multivariate stochastic observations, either by variance maximization or least mean squares techniques. The POD method based on PCA is described below.
Assuming that ∈ m x R is a random vector and X is the original dataset, each X column is a single sample. PCA is a linear combination of its primary vectors to the revealed internal law of the data. Consider Y an × m n matrix related by the linear transformation P, and Y is a new representation of the original dataset X. This is defined as: PX = Y (1) Consider P a rotation and stretch tensor, and P is a matrix which transforms X into Y. The columns of X are expressed by the rows of P: Without a loss of generality, the 1st, 2nd,…, mth principal components are ∈  If the right-hand side of Equation (6) is zero, then the equation can be simplified as α λα where λ 1 is the eigenvalue and α 1 is the corresponding eigenvector of the covariance matrix Σ x . Multiplying both sides of Equation (7) Note that λ 1 must be as large as possible since it was selected as the maximum eigenvalue of Σ x . Similarly, the second principal component 2 y is obtained from where α 2 is α α α Note that λ 2 is selected as the second eigenvalue of Σ x , such that λ λ ≥ ≥ 1 2 0 .
The principal components, which are uncorrelated and unordered, closely retain the variation in the original dataset. In particular, the first several principal components retain almost all the variation in all the original variables, and the first principal component y1 is uncorrelated with the second principal component y2.
The variances of B are defined as: Considering the degree of linearity of the relationship between A and B, the covariance function is given by where the degree of redundancy is the magnitude or absolute value of σ 2 Without a loss of generality, it is generalized to an arbitrary number; that is, a new m × n matrix X and row vectors  1 2 m x , x , , x is defined by Therefore, the CX represents the covariance matrix and is defined as 1 = T X C XX n (18) where CX is a square symmetrical m × m matrix, the variance of particular measurement types, and the covariance between measurement types, are the diagonal and off-diagonal terms of CX, respectively. In this study, the covariance values reflected the redundancy of measurements.

Singular Value Decomposition Method
The dataset X is an m × n matrix, where The sampling number is n and the number of sampling types is m. Therefore, when the number of samplings is greater than the sampling types, let where  1 2 , , , n x x x represent n samples.
To obtain the principal components of X, two objectives should be fulfilled: CY is a diagonal matrix, and the matrix P in Y = PX is an orthonormal matrix. The expression for Substituting PX into Equation (20) gives For any symmetrical matrix A, there is a diagonal matrix D and a matrix E of eigenvectors of A arranged as columns, such that = T A EDE and each row of matrix P is an eigenvector of 1 T XX n , such that CY may be written as To summarize, the variance of X along with pi is the ith diagonal value of CY, and the eigenvectors of CX are the principal components of X.
Let the eigenvalues of T XX be arranged in a decreasing order: The singular values of T X are σ Generally, there are orthonormal or base sets of eigenvectors The rank-ordered set of singular values is σ σ σ ≥ ≥ ≥ 

Model Construction Using Data-Driven Analysis
Monitored data were recorded in stages, so that sampling times were not precisely consistent at all monitoring points. The proposed method for analyzing the tunnel deformation dataset adopted six-sample sliding window overtime at one-hour time increments to overcome this limitation. After data processing, the eigenvectors and eigenvalues were obtained for each moving window.
The proposed model adopted the spectral proper orthogonal decomposition (SPOD) approach (described step by step below) for the sampling sequence at each monitoring point [30][31][32]. Then an evaluation index was constructed from the datasets.
Step 1 (data matrix): the data matrix , ( ) i j X x , where i is the number of samples and j is the number of measurement types, is sliced in the direction of increasing data.
Step 2 (determining the moving window): A 'moving window' is a sample clip of a fixed amount of monitoring data. It is written Wi , j where i = 1, 2, …, n (window size), and j is the number of measurement types.
Step 3 (efficient moving mechanism): The moving window moves in steps with a fixed number of samples on matrix X to obtain the moving matrix Wk,j . An example of a moving window is: After the following new sample is added, the current moving window slices in the direction of the added data to form a new moving window.
Step 4 (forming an evaluating indicator): For each moving window Wk,j , the model was established by the POD method. The evaluating indicator k is given by σ σ where σ 2 1 is the first eigenvalue and σ 2 i is the sum of all eigenvalues.

Dataset Matrix of Real-Time Monitoring Data
This section evaluates the proposed method and compares it with previous methods at different monitoring stages. The evaluation was performed for four datasets to confirm the effectiveness of the proposed method. The datasets represent displacements in the x-, y-, and z-directions in the tunnel wall, extracted from five monitoring points in each case. The SPOD procedure was then conducted for a six-sample moving window at one-hour intervals. The detailed compositions of the evolving indices in the moving window are listed in Table 1.

Discussion and Analysis of Real-Time Monitoring Data
Each moving window was decomposed by the SPOD algorithm, then the eigenvalues (Σ 2 ) and eigenvectors (V) of each dataset were obtained for each moving window. The magnitude of the normalized eigenvector is measured by the eigenvalue, as described in Equation (27). For each moving unit, Mode 1 was the dominant mode during the monitoring period ( σ σ σ σ σ → The peak values in the ratio curves of Z10-2, Z10-3, and Z10-4 ( Figures A1-A3) appear about two months before Z10-1 ( Figure 8) and Z10-5 ( Figure A4), when the construction work was taking place closest to the left-hand side of the tunnel. The recorded changes at the monitoring section Z10 were probably due to the effect of construction. Figure 9 shows the monitoring results for monitoring point 1 at Section 11 (Z11-1). No significant change was observed from 15 March to 30 May, as the tunnel was not noticeably affected by construction. The values showed a significant decrease and increase on 30 May and 20 July, respectively, and reached 0.7883 on 9 September, followed by a decrease and then an increase. The maximum ratio of 0.8072 was observed on 27 November.

Monitored Results at Section 11
The peak values in the ratio curves of Z11-2, Z11-3, and Z11-4 appear ahead of Z11-1 and Z11-5 (see Appendix B) by approximately one week, probably because the construction area was near the left-hand side of the tunnel.
The peak eigenvalue ratios of Mode 1 of Section 12 (Z12-1) was observed on 5 June. The result shows that the peak value of Section 12 (Z12-1) occurred three days before Section 11 (Z11-1). It is found that the closer the monitoring points were to the displacementaffected zone, the larger the peak response was.  Figure 11 shows the Mode 1 eigenvalue ratios and relevant dates for Section 13. The eigenvalue ratios of Mode 1 for monitoring point Z13-1 were 0.8303, 0.7581, and 0.7198 on 4 June, 9 September, and 24 November 2015, respectively. The peak value of Mode 1 for monitoring point Z13-1 decreases gradually, attributable to the effect of the redistribution of the surrounding rock pressure. All the values were greater than 0.7, which tends to confirm the high sensitivity of the proposed method.

Comparison with Traditional Analytical Method
To improve the precision of analysis in conventional geotechnical engineering approaches to assess or predict tunnel deformation, it is usually necessary to have as much information available about the relevant variables as possible, including their minimum and mean values and standard deviation.
The orthogonal decomposition method was employed for processing large datasets in this study of monitoring the deformation of operational tunnels. The results shown in Figures 8-11 and Appendices A to D demonstrate that the proposed method adequately interprets the monitoring data. The Mode 1 eigenvalue ratios were found to be highly sensitive to fluctuations in tunnel deformation in all cases.
This finding may provide insight into the phenomenon that higher eigenvalue ratios are valuable indicators of value fluctuations in large monitoring datasets.

Conclusions
In tunnel monitoring systems and associated geotechnical engineering activities, it is crucial to extract helpful information from the vast datasets generated by tunnel displacement monitoring to evaluate the current performance of the tunnel. The task of analyzing and managing tunnel deformation data is the core of the long-term deformation monitoring of operating tunnels.
This study proposes an orthogonal decomposition method to construct a data processing model for datasets from current tunnel monitoring methods. As discussed in Section 2, the actual monitoring process was studied by considering four groups of five monitoring points, each of which collected tunnel deformation data over the same 10-month period. The SPOD technique analyzed the datasets in which observations were described using several intercorrelated quantitative dependent variables. The goal of this technique was to extract meaningful information from vast amounts of raw data.
Subsequently, the information was represented as a set of principal components that were new orthogonal variables. The quality was evaluated using cross-validation techniques. Additionally, data fusion processing was performed on the data to make them more detailed, intelligent, and accurate.
In order to confirm the feasibility and effectiveness of the proposed method on the four datasets, each moving unit was decomposed and its principal features were extracted by the proposed algorithm. For each monitored period, four sets of eigenvectors and eigenvalues were obtained using Scikit-learn machine learning tools. The experimental results show that the evaluating k was between 0.7065 and 0.8888. The proposed algorithm revealed the changes and trends of the critical variables in tunnel deformation monitoring. Such methods will help achieve the goal of the early detection of possible structural deterioration.  Institutional Review Board Statement: "Not applicable" for studies not involving humans or animals.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.

Data Availability Statement:
The sample data collection sheet, the data used for analysis, and the associated computer programs can be made available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The changes of deformation and eigenvalues at the measurement date for points 2-5, Section 10, and the left-hand tunnel are shown in Figures A1-A4.

Appendix B
The changes of deformation and eigenvalues at the measurement date for points 2-5, Section 11, and the left-hand tunnel are shown in Figures A5-A8.