Graphical models for probabilistic reasoning are now in widespread use. Many approaches have been developed such as Bayesian network. A newly developed approach named as dynamic uncertain causality graph (DUCG) is initially presented in a previous paper, in which only the inference algorithm in terms of individual events and probabilities is addressed. In this paper, we first explain the statistic basis of DUCG. Then, we extend the algorithm to the form of matrices of events and probabilities. It is revealed that the representation of DUCG can be incomplete and the exact probabilistic inference may still be made. A real application of DUCG for fault diagnoses of a generator system of a nuclear power plant is demonstrated, which involves variables. Most inferences take with a laptop computer. The causal logic between inference result and observations is graphically displayed to users so that they know not only the result, but also why the result obtained.

Probabilistic graphical models (PGMs) such as Bayesian network (BN) have been widely applied in uncertain causality representation and probabilistic reasoning. Dynamic uncertain causality graph (DUCG) is a newly presented model of PGMs, which can be applied to fault diagnosis of large and complex industrial systems, disease diagnosis, and so on. The basic methodology of DUCG has been previously presented, in which only the directed acyclic graph (DAG) was addressed. However, the mathematical meaning of DUCG was not discussed. In this paper, the DUCG with directed cyclic graphs (DCGs) is addressed. In contrast, BN does not allow DCGs, as otherwise the conditional independence will not be satisfied. The inference algorithm for the DUCG with DCGs is presented, which not only extends the capabilities of DUCG from DAGs to DCGs but also enables users to decompose a large and complex DUCG into a set of small, simple sub-DUCGs, so that a large and complex knowledge base can be easily constructed, understood, and maintained. The basic mathematical definition of a complete DUCG with or without DCGs is proved to be a joint probability distribution (JPD) over a set of random variables. The incomplete DUCG as a part of a complete DUCG may represent a part of JPD. Examples are provided to illustrate the methodology.

Intelligent systems for online fault diagnoses can increase the reliability, safety, and availability of large and complex systems. As an intelligent system, Dynamic Uncertain Causality Graph (DUCG) is a newly presented approach to graphically and compactly represent complex uncertain causalities, and perform probabilistic reasoning, which can be applied in fault diagnoses and other tasks. However, only static evidence was utilized previously. In this paper, the methodology for DUCG to perform fault diagnoses with dynamic evidence is presented. Causality propagations among sequential time slices are avoided. In the case of process systems, the basic failure events are classified as initiating, and non-initiating events. This classification can increase the efficiency of fault diagnoses greatly. Failure rates of initiating events can be used to replace failure probabilities without affecting diagnostic results. Illustrative examples are provided to illustrate the methodology.

The dynamic uncertain causality graph (DUCG) is a newly presented framework for uncertain causality representation and probabilistic reasoning. It has been successfully applied to online fault diagnoses of large, complex industrial systems, and decease diagnoses. This paper extends the DUCG to model more complex cases than what could be previously modeled, e.g., the case in which statistical data are in different groups with or without overlap, and some domain knowledge and actions (new variables with uncertain causalities) are introduced. In other words, this paper proposes to use i-mode, e-mode, and h-mode of the DUCG to model such complex cases and then transform them into either the standard i-mode or the standard e-mode. In the former situation, if no directed cyclic graph is involved, the transformed result is simply a Bayesian network (BN), and existing inference methods for BNs can be applied. In the latter situation, an inference method based on the DUCG is proposed. Examples are provided to illustrate the methodology.

A new approach for uncertain causality representation and probabilistic reasoning named as dynamic uncertain causality graph (DUCG) was presented previously, in which only the discrete variables and certain evidence were addressed. In this paper, the free mixtures of discrete and continuous variables as well as uncertain evidence are addressed. The general idea to deal with continuous variables is to transform them into fuzzy discrete variables along with their corresponding uncertain causalities, and then treat them as ordinary discrete variables, which involves how to deal with fuzzy evidence. It is pointed out that uncertain evidence is either: 1) fuzzy evidence that is an observed certain value of a continuous variable falling into a fuzzy area across two or more fuzzy discrete states of a variable or 2) soft evidence that can only be understood as a probability distribution over the states of a variable. The algorithm for utilizing uncertain evidence in inference is presented, in which uncertain evidence is treated as a virtual child variable of the observed variable without changing the knowledge and inference algorithm encoded in DUCG. It is proved that the two types of uncertain evidence are the same in nature and can be treated indiscriminatingly. Moreover, this method dealing with fuzzy evidence in DUCG can be used for failure forecasting of systems. Examples are provided to illustrate the methodology.