Modeling risks in dependent systems: A Copula-Bayesian approach
Introduction
The underground infrastructure, as a significant part of physical infrastructure systems, is one of the essential foundations for supporting human life and social activities around the world. Owning to the issues, including aging and inadequate management, the existing infrastructure systems are experiencing server structural deterioration without sustainable assessment and maintain, bringing about great property loss and high operational risks [1]. Especially for the urban metro systems, many considerable potential risks induced by the complex environment during the metro operation have been highlighted. Such risks generated from structural diseases, including cracks, water leakage, settlement, and others, will inevitably occur with evidence of various minor defects in the complex underground environment [2]. Subsequently, the risks will destroy tunnel structures and reduce the safety reliability of the metro system significantly. Metro operation accidents have occurred occasionally, which tend to exert heavy casualties and numerous economic loss to the public. For example, the Baku metro accident in the Azerbaijan Republic in 1995 killed more than 340 people, the Daegu metro accident in Korea in 2004 killed more than 189 people [3], and the Washington metro collision in the United States in 2009 killed 9 people [4]. With respect to occurred tunnel accidents, only 10% of them utterly result from natural causes, whereas nearly 90% of accidents are more or less associated with insufficient risk handling and safety management [5]. Since accidents are highly related to the complexity of environment with uncertainty and interactions of risk factors, how to accurately define the critical risk factors and evaluate the safety of the operational tunnels has become an extremely complicated problem.
Most risks can be avoided under control by early detection and suitable measures [6], [7]. Various tunnel structural inspection systems have been developed to detect structural diseases automatically and using techniques for data acquisition and computer vision [8], [9], [10], [11]. Then, prospective solutions, which improve the metro system's performance and offer decision support for safety management in operational tunnels [12], [13], can come out based on the information from inspection systems. Innumerable studies have analyzed potential risks quantitatively and qualitatively in order to make proactive safety strategies for the metro tunnels. Eskesen et al. [14] set specific risk acceptance criteria and provided guidelines for risk management in tunneling. Anoop et al. [15] estimated the life service of reinforced concrete structural elements by a hybrid method combining the vertex method of fuzzy set theory with the Monte Carlo simulation technique. Liu et al. [16] evaluated the risk connotation of land subsidence by an improved fuzzy analytic hierarchy process method and applied it to metro safety operation in Shanghai. Šejnoha et al. [17] adopted fault tree analysis (FTA) and event tree analysis (ETA) to evaluate the risk in tunnel collapsing based on a probabilistic approach. Some artificial intelligence techniques, like artificial neural network (ANN), support vector machine (SVM), decision trees, and others, have been applied to perform the classification and prediction of the risk degree in the field of structural health monitoring [18], [19], [20]. However, the above-mentioned studies mainly focus on static risk evaluation and rarely concern about dynamic risk management, resulting in ineffective diagnosing and handling with possible structural damage for real-time safety control and decision making [21], [22]. To address the concern, some researches have attempted to make dynamic risk analysis in the context of Bayesian updating. For instance, Kalantarnia et al. [23] utilized Bayesian theory to update both the occurrence likelihood of events and failure probability of systems. Khakzad et al. [24] developed a dynamic risk assessment method based on a hierarchical Bayesian model and verified its effectiveness in near accident data of offshore blowouts. Wu et al. [25] proposed a systemic Bayesian network-based approach to dynamically analyze the risk of adjacent buildings in tunneling environments, providing real-time support in the entire life cycle of risk-prone events. Luque and Straub [26] estimated and updated the reliability of deteriorating structural systems with inspection and monitoring results based upon dynamic Bayesian networks. Thus, the Bayesian network can be considered to combine with the concept of the copula, which copies with relatively complex correlations. The hybrid copula-Bayesian approach is able to not only construct a model consistent with the actual conditions, but also conduct forward and backward analysis to predict the risk in a more dynamic and accurate way.
Risk modeling and analysis for an operational metro system is regarded as a complicated process on account of (1) a lot of uncertainties from inaccurate or ambiguous information about underground conditions and structural behavior [27], [28]; (2) a lack of sufficient and accurate measured data [21], [29]; (3) non-linear models to characterize risk factors and their relevant relationships [30]; and (4) continuous and dynamic changes in environment and tunnel structure during metro operation process. Moreover, more uncertainties will arise from incomplete expert knowledge and the value, variation trend and correlations of risk factors, making the model difficult to be established and validated [31]. Additionally, failures are usually caused by the confluence of a whole series, or chain of errors, rather than just a single failure or mistake [32], which certainly add the complexity in the metro system. For instance, water is more likely to seep from the cracked area once cracks appear in the tunnel structure, and then the interaction of crack area and leakage water impact the degree of risk. However, most existing risk assessment researches only focus on specific factor mostly in the scattered and repetitive conditions, which is not systematically and comprehensively organized [33], [34]. In general, it is necessary to explore the coupling risks from the systematic view, in order to address the environment complexity of metro systems with various uncertainties and interrelations of risk factors during the metro operation.
Bayesian networks (BNs) [35], [36], a probabilistic graphics model, could be utilized to construct the conditional probability distribution among bivariate or multivariate random variables by the combination of the prior information and measured data. Since notes in BNs could achieve real-time knowledgeable reasoning in a dynamic environment, BNs are usable for predictions in the fields which are fraught with uncertainty and support for decision making [37], [38], [39], [40]. To this end, some efforts have been made to implementing BNs in civil engineering projects. For example, Zhang [41] presented a novel and systemic decision support model based on BNs for safety control in dynamic complex project environments. Špačková [42] proposed a model for probabilistic prediction to evaluate tunnel construction performance using a dynamic Bayesian network (DBN). Sousa [43] provided a methodology using BNs to systematically assess and manage risks associated with tunnel construction. Wang [44] combined the fuzzy comprehensive evaluation method (FCEM) and a BN to make the risk evaluation more comprehensive in metro construction projects. Zhang [21] used the Fuzzy Bayesian Networks (FBN) to investigate influential variables based on the risk analysis concerning the tunnel leakage in the construction of a metro tunnel. However, most existing BN studies have the following problems: (1) They paid much attention to the risk assessment during the construction stage and ignored to the system safety risk assessment in the system operating stage, which should be equally crucial as the previous one [21]; (2) The risk probability is represented by a crisp value rather than a continuous distribution, however, probability distribution can better approximate the uncertainty [45]; and (3) They rarely considered the verification of the effectiveness and reliability for the developed BNs [46].
Discrete BNs are easily applied and widely used, which assigns marginal distributions to source nodes and conditional probability tables to child nodes. However, it is not suitable for the complex system due to its strict operation conditions: (1) The database should be complete; (2) Nodes could not have too many possible states; and (3) Child nodes could only own fewer parent nodes. If these conditions are not satisfied, a number of probabilities assigned to child nodes will experience exponentially increase with the parent nodes number, resulting in exponential execution times and complexity [47], [48]. Meanwhile, it is also impossible to populate large conditional probability tables (CPT) and satisfy child marginal constraints. In order to handle continuous variables in BNs, continuous BNs is developed without assigning a number of probabilities. However, it is mainly limited to joint normal distributions, which is likely to lead to significant deviations between prediction results and reality [49], [50]. Thus, it is essential to apply other theoretical models on the basis of the conventional BNs, for the purpose of solving the problems mentioned above and increasing the creditability of evaluation results. The copula theory is a useful and flexible tool to construct the joint distribution function of multivariate data, in order to model the dependence structure among variables, primarily to describe the nonlinear relationship appropriately [51]. Different kinds of copula functions have their own characteristics of the dependence structure, such as the symmetry and tail dependence [52]. Therefore, this research intends to investigate the possibility of merging the copula theory and BNs to support the system safety underlying dependence structures.
The main research questions are: (1) How to represent the dependence structure that couples the system outcome and influential risk factors? (2) How to validate the feasibility of the developed Coupla–Bayesian model? (3) How to update the system safety risk under given observations in a continuous manner in the full life cycle of the tunnel? In this research, a hybrid Copula-Bayesian-based risk assessment approach is proposed to dynamically estimate the structural risk in operational tunnels and assist in providing support for possible actions to reduce the risk. This research contributes to (a) The state of the knowledge by modeling the dependence structure of risk factors in a complex system more accurately under the combination of the copula theory and Bayesian networks, which helps to reduce the discrepancy of actual conditions. The results will be in the form of probability distribution to describe the uncertainty, rather than a simple crisp value. To conduct the predictive analysis based on correlation analysis, forward and backward reasoning, the most critical risk factors causing structural failure are identified as the main checking points. Furthermore, the safety status of these risk factors can be updated continuously through the overall life cycle of the tunnel. (b) The state of the practice by adopting the hybrid Copula-Bayesian-based risk assessment approach is deemed as a real-time decision support tool for safety control in highly complicated projects under complex and volatile environment, which is not just restricted to operational metro tunnels. Safety monitoring will be implemented through the whole life cycle of the system, and then related safety control measures are put forward to prevent possible structural failure ahead of time or lower the risk after accidents.
Overall, the structure of the paper is organized as follow. Section 2 reviews the basic concept of Bayesian networks and copula theory. Section 3 introduces the framework for the risk inference process based on the Copula-Bayesian model, including model design, model validation and model analytics. Section 4 applies the proposed method to a case, the Wuhan Yangtze Metro Tunnel, in order to provide guidelines for the risk control in operational metro tunnels and prove the validity of the novel approach. Section 5 discusses how the type of marginal distribution for each risk factors will influence the results of correlation analysis. Section 6 draws up the conclusions and future works.
Section snippets
Bayesian networks (BNs)
Bayesian networks (BNs), a robust probabilistic model, is able to perform quantitative analysis in the forward and backward reasoning under uncertainty. Based up the graph and probability theory, BNs present a compact expression for the joint probability distribution (JPD) of random variables and their associated conditional independence structure via the directed acyclic graph (DAG). Typically, BNs are defined by , in which G is the DAG with nodes and directed edges, P is the
Methodology
Aiming to dynamically assess the interacted risk in operational metro tunnels, Bayesian networks, and copula functions are combined to develop a novel Copula-Bayesian approach. It is consisting of three main stages: (1) model design, (2) model validation, and (3) model analytics. Fig. 1 illustrates the conceptual workflow for the proposed risk assessment method. It is effective to determine the most significant potential risk factors, which are remarkably correlative with the safety state in
Case background
On account of demonstrating the feasibility and effectiveness of the developed Copula-Bayesian model, Wuhan Yangtze Metro Tunnel (WYMT), an underwater tunnel, is utilized as a test case to evaluate its structural health in the operating state. WYMT, the first tunnel crossing through the Yangtze River, connects Hankou and Wuchang districts in the city of Wuhan, China. It is a double-spool tunnel with two lines and the length of each line is 3100 m. Its excavation depth is 78 m under the average
Discussions
As a matter of fact, the hybrid Copula–Bayesian model has more powerful ability to capture the reliable association in actual operational tunnels by means of determining the best-fitting marginal distribution, rather than just assuming that random variables all follow the widely used normal distribution. That is to say, the type of marginal distribution for each variable will affect the outcomes of risk assessment to some extent. Thus, the marginal distributions, including normal, Weibull,
Conclusions and future works
A novel structure risk assessment approach based on the hybrid Copula–Bayesian model is implemented with a step-by-step process, in order to mine the interaction of the latent risk factors threating structural health in operational metro tunnels comprehensively and systematically. The Copula–Bayesian-based risk assessment approach is typically a continuous and dynamic probability state with various potential risks. Of particular importance is the construction of the Bayesian network with
Acknowledgments
The National Key Research Projects of China (grant no. 2016YFC0800208), the start-up grant at Nanyang Technological University, Singapore (no. M4082160.030), the Ministry of Education grant, Singapore (no. M4011971.030), and the National Natural Science Foundation of China (grant nos. 51778262 and 71571078) are acknowledged for their financial support of this research.
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