Fault diagnosis based on incomplete sensor variables with a hierarchical semi-supervised Gaussian mixture classifier

We propose a hierarchical semi-supervised variational semi-Bayesian Gaussian mixture classifier based on the partially incomplete and unlabeled samples for the fault diagnosis of mechanical and electrical systems. These systems are typically complex in structure and are monitored by multiple sensors simultaneously. Some sensor variables in the collected data may be incomplete due to sensor malfunctions or transmission errors. Additionally, labeling the data can be a time-consuming and labor-intensive task, resulting in many unlabeled samples. To address these challenges, the missing sensor variables and the unavailable labels are treated as hidden values and handled within the framework of the Expectation Maximization algorithm. We employ a semi-Bayesian technique with variational inference to estimate the model parameters. Specifically, we introduce prior distribution to the mean and covariance matrix to address the possible singularity of the empirical covariance matrix. The weighting coefficients are left without putting prior distribution so that their values can decay to zero, effectively removing redundant components of the mixture model. The factorized distribution is utilized to approximate the posterior distribution of the model parameters, as well as the missing labels and other latent variables. Numerical and real case studies are carried out to verify the effectiveness of the proposed technique.