Probabilistic graphical models like Bayesian networks have been widely used in process monitoring and fault diagnosis, however, their application is mostly limited to discrete variables or continuous Gaussian variables due to the difficult in estimation of multivariate joint density. In order to deal with the estimation problem of multivariate joint density for continuous variables, this paper decomposes the graphical model into hierarchical structure so that the problem of joint density estimation can be transferred to estimation of several conditional probability densities and low-dimensional probability densities. The conditional densities can be effectively estimated from data by a nonparametric kernel method and the low-dimensional densities can be estimated using the kernel density estimation (KDE). Based on the estimated densities, process faults can be detected by examining which probability is lower than the cutoff value. Application to the blast furnace ironmaking process is used to illustrate the advantages of the proposed method.
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