# Development of a framework to estimate the condition of mechanical, electrical and plumbing systems with Bayesian Networks

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## Abstract

As the Western European building stock ages, attention is increasingly allocated to the maintenance of building components, particularly mechanical, electrical and plumbing (MEP) systems. Although the latter are essential in ensuring the correct operation of a building and the safety of its occupants, they remain the crafts where the most defects are observed, resulting in significant material costs. This phenomenon partly finds explanation in the shortcomings of current condition assessment methods for MEP systems, which often poorly describe the actual state of the components.

As a result, novel approaches to estimate the condition of these building elements are investigated by industry participants. Among them, Bayesian Networks (BNs) are probabilistic models that progressively gain momentum for real-life applications. In the context of the present research, their relevance is twofold: (i) their graphical structure allows to visually model influence between large sets of variables, and (ii) they robustly handle missing data. Unfortunately, like most probabilistic models, their quantification requires extensive amounts of empirical data which is extremely sparse for MEP systems. Therefore, this thesis attempts to answer the following question: "How can Bayesian Networks be applied to estimate the condition of mechanical, electrical, and plumbing systems in the absence of empirical data?"

In their `traditional' discrete form, BNs have a limited range of applications. First, they do not allow the integration of continuous variables, which for numerous physical problems is a major drawback. Second, the number of parameters to quantify discrete networks quickly becomes intractable as the number of states and parents increases, again limiting their implementation for complex systems. Therefore, Non-Parametric Bayesian Networks (NPBNs) are adopted in this research, whose formulation is based on (conditional) rank correlations (dependence) and marginal distributions associated to each of the network's variables.

To overcome the challenge imposed by the limited availability of empirical data, several studies have investigated the use of field experts' judgments for the quantification of BNs. While the elicitation of univariate distributions has been thoroughly studied, the assessment of dependence remains an emerging topic in structured expert judgment (SEJ) literature. Consequently, this thesis focuses on the development of a method for the assessment of rank correlations by field experts, whereas a lesser effort is allocated to the elicitation of the marginal distributions.

Existing research has delved into the use of two approaches for the elicitation of dependence: statistical and conditional fractile estimates. Here, the suitability of probabilities of concordance, a third type of probabilistic assessment, is investigated. Under the normal copula assumption, common in the context of SEJ, unconditional rank correlations can be retrieved from probabilities of concordance using a set of closed-form relations. Then, the individual experts' opinions are aggregated using dependence-calibration, a performance-based aggregation method gaining momentum for NPBNs. The application of these approaches to MEP systems in discussed later in this summary.

The first step in the creation of a BN is the definition of a graph. Therefore, a classification of the MEP systems is developed and constitutes the foundation of the network. Subsequently, the factors influencing the condition of the sub-systems classified previously are identified. The literature reviewed suggests a distinction between two types of relationships: those between exogenous variables (e.g. maintenance or environmental conditions) and building components, and those between components themselves. Following the identification of these relationships, a `global' graph encompassing all MEP systems arose.

Before engaging in the quantification of this network, it is crucial to interrogate its feasibility given the time span of this research and the absence of empirical data. With 23 variables and over 30 edges, the assessment of all correlations (leave alone of the marginal distributions) is practically unrealistic solely based on experts' judgments. Therefore, the remainder of the report presents a case study on air handling units (AHUs), for which the elicitation method is implemented. The graph defined for AHUs is illustrated in the figure below.

Questions for the assessment of probabilities of concordance related to the newly created graph are then formulated, taking a similar form as follows: "Two buildings A and B are randomly selected among all non-residential buildings in the Netherlands. Given that the AHU in building A is maintained more regularly than in building B, what is the probability that the coils are in better condition in building A than building B ?"

Similar questions were created for each of the network's edges and presented to a panel of five experts, resulting in five individual correlation matrices. As mentioned previously, the experts were then evaluated using seed questions in the context of dependence-calibration. Additional questions were thus asked to the participants with regards to precipitation in the Netherlands, a choice motivated by the absence of data related to AHUs and mechanical systems for calibration. The respondents' calibration scores were then calculated using their assessments on the seed questions and the correlation matrix retrieved from empirical data. Finally, a combination of the experts' dependence structures was built using their calibration scores in a weighted average, resulting in a unique set of correlations which were implemented in the NPBN.

Lastly, two of the five experts consulted previously participated in the elicitation of the marginal distributions, either by the direct provision of the distribution or through answers to qualitative statements. The resulting model includes both marginal distributions and (conditional) rank correlations, and is ready to be validated.

To conclude, the NPBN is validated. While the lack of empirical data prevents the quantification of the model's predictive validity, a scenario analysis is performed to observe its output under different input combinations. It reveals that the exclusion of the environmental conditions from the network results in unrealistic outcomes, thus refuting an assumption made earlier in this research. Moreover, a global sensitivity analysis is conducted based on Sobol's method, which demonstrates the high contributions of all inputs to the outputs' variances. Consequently, evidence on any of the inputs substantially reduces the uncertainty in the output distributions, a comforting conclusion on the relevance of the chosen factors.

The final result of this thesis is a flowchart illustrating the construction process of a Non-Parametric Bayesian Network. It provides academics and practitioners with a foundational framework for the creation of Bayesian Networks, irrespective of the quantification method selected. While this thesis proposes the implementation of a particular expert-based elicitation method, the most suitable approach should be chosen with regards to the system modelled.