مدل سازی شبکه های بیزی از متغیرهای تصادفی همبسته گرفته شده از یک میدان تصادفی گاوسی

In many civil engineering applications, it is necessary to model vectors of random variables drawn from a random field. Furthermore, it is often of interest to update the random field model in light of available or assumed observations on the random field or related variables. The Bayesian network (BN) methodology is a powerful tool for such updating purposes. However, there is a limiting characteristic of the BN that poses a challenge when modeling random variables drawn from a random field: due to the full correlation structure of the random variables, the BN becomes densely connected and inference can quickly become computationally intractable with increasing number of random variables. In this paper, we develop approximation methods to achieve computationally tractable BN models of correlated random variables drawn from a Gaussian random field. Using several generic and systematic spatial configuration models, numerical investigations are performed to compare the relative effectiveness of the proposed approximation methods. Finally, the effects of the random field approximation on estimated reliabilities of example spatially distributed systems are investigated. The paper concludes with a set of recommendations for BN modeling of random variables drawn from a random field. Highlights ► BN modeling of correlated random variables drawn from a Gaussian random field. ► Reducing the density of connections in a BN by eliminating nodes and/or links. ► Methods based on classical decomposition techniques and numerical optimization. ► Effects of the approximation methods on estimates of system failure probability. ► Optimization methods achieve best trade-off of accuracy versus computational efficiency.

In civil engineering applications, it is often necessary to model vectors of random variables drawn from a random field. For example, in investigating the seismic risk of a lifeline, the earthquake-induced ground motion intensities at the locations of the system components constitute a vector of random variables drawn from the ground motion random field. Similarly, factors determining the progress of deterioration in elements of concrete surfaces are random variables drawn from environmental and material property random fields. Proper modeling of the dependence structure of vectors of random variables is essential for accurate probabilistic analysis. In the special case when the field is Gaussian, or derived from a Gaussian field, the spatial dependence structure of the field is completely defined by the autocorrelation function and the correlation matrix fully defines the dependence structure of the random vector drawn from the field. Typically, this correlation matrix is fully populated. Although this paper only deals with Gaussian random fields, the methods developed are equally applicable to non-Gaussian fields that are derived from Gaussian fields, e.g., [1]. In some applications, including the aforementioned examples, it is of interest to update a probabilistic model in light of available or assumed observations of the random field. For example, in the case of a lifeline subjected to an earthquake, one might be interested in updating the reliability of the system when ground motion intensities at one or more locations are observed, or when evidence is available on the performance of individual components based on the output from structural health monitoring sensors or observations made by inspectors [2]. In the case of a concrete surface subject to deterioration, the reliability of the system can be updated, e.g., when cracking (or no cracking) of the concrete in some of the elements is observed. The Bayesian network (BN) methodology is a powerful tool for such updating purposes, particularly when the available information evolves in time and the updating must be done in (near) real time, see, e.g., [3] and [4]. However, there is a limiting characteristic of the BN that poses a challenge when modeling random variables drawn from a random field: due to the full correlation structure of the random variables, the BN becomes densely connected. When combining these random variables with system models that involve additional random variables, the computational and memory demands of the resulting BN rapidly grow with the number of points drawn from the random field. In this paper, we develop approximate methods to overcome this difficulty. Specifically, we present methods for reducing the density of the BN model of the random field by selectively eliminating nodes and links. The aim is to minimize the number of links in the BN while limiting the error in the representation of the correlation structure of the random variables drawn from the Gaussian random field. When the random field as well as the observed random variables are jointly Gaussian, a well-known analytical solution exists for computing the conditional probabilistic model. However, the random field model often is part of a larger problem involving mixtures of continuous and discrete random variables and fields. For example, in seismic risk assessment of a lifeline, a random field may define the ground motion intensity across a geographic region, while discrete random variables define the performance or damage states of the lifeline and its constituent components. When evidence is entered on non-Gaussian or discrete random variables in such a model, e.g., the observed damage state of a component, the existing analytical solution for updating the distribution of the Gaussian variables is no longer applicable. It is in this context that the BN is useful for modeling and updating of Gaussian random fields. The paper begins with a brief introduction to BNs as a means for probabilistic inference and describes their advantages and limitations. Next, BN models of random variables drawn from a Gaussian random field are described. Approximation methods are then developed to achieve computationally tractable BN models. Using several generic and systematic spatial configuration models, numerical investigations are performed to compare the relative effectiveness of the proposed approximation methods. Finally, the effects of the random field approximation on estimated reliabilities of example spatially distributed systems are investigated. The paper ends with a set of recommendations for BN modeling of random variables drawn from a random field. More details on development of BN models for random fields and application to infrastructure seismic risk assessment can be found in [2].

Methods for efficient Bayesian network (BN) modeling of correlated random variables drawn from a Gaussian random field, such as those arising in seismic risk assessment of spatially distributed infrastructure systems, are investigated. The modeling of broadly dependent random variables results in a BN that is densely connected. Because exact inference algorithms in densely connected BNs are demanding of computer memory, approximate methods are necessary to make the BN computationally tractable for large systems. This paper develops methods for reducing the density of connections in a BN by eliminating nodes and/or links, while minimizing the error in the representation of the correlation structure. Methods based on classical decomposition techniques as well as numerical optimization are developed and compared. It is found that optimization methods are able to achieve the best trade-off of accuracy versus computational efficiency. The effects of the approximation methods on estimates of failure probability for idealized infrastructure systems are also considered. It is found that the optimization-based approaches offer significant increases in efficiency when modeling the performance of parallel systems. For series systems, which are known to be less sensitive to the correlation structure of component demands, classical decomposition approaches may suffice. While the work done in this paper has been performed as part of an effort to develop a BN-based framework for seismic infrastructure risk assessment, it is believed that the findings are useful for more general applications involving correlated Gaussian random variables.