This paper is available on arxiv under CC 4.0 license.

Authors:

(1) Jihoon Chung;

(2) Zhenyu (James) Kong.

## Table of Links

- Abstract & Introduction
- Review of Related Work
- Proposed Research Methodology
- Numerical Case Studies
- Real-World Simulation Case Studies
- Conclusion
- References

## II. REVIEW OF RELATED WORK

The related existing studies of fault diagnosis in manufacturing systems are reviewed in Section II A. Then, the literature related to sparse Bayesian learning is provided in Section II-B. Afterward, the research gaps in the current work are identified in Section II-C.

*A.* Fault Diagnosis Methodologies in Manufacturing Systems Numerous studies have focused on fault diagnosis methodologies for manufacturing systems, utilizing the fault-quality model outlined in Eq. (1). [14] developed a PCA-based orthogonal diagonalization strategy to transform the measurement data. It enabled the estimation of the variance of KCCs in a multistation assembly system. [13] presented a fault diagnosis method in the multistation assembly systems integrating the state space model of the process and matrix perturbation theory. [25], [26] proposed a fault diagnosis method in the machining process considering the process physics regarding how fixtures generate the patterns. Using this method, root cause identification was conducted sequentially. The approaches mentioned above assume that the number of measurements is greater than the number of KCCs (i.e., M > N in Eq. (1)). However, this assumption may not always be consistent with industrial practice. These approaches become ineffective when this assumption is violated because the fault-quality linear model leads to an underdetermined system, resulting in a nonunique solution.

To overcome an underdetermined system in the fault-quality linear model, sparse learning can be utilized, which has gained considerable attention in fault diagnosis and detection within manufacturing systems. For fault diagnosis in the manufacturing system, [27] developed a fault diagnosis method using dictionary learning and sparse representationbased classification. [28] proposed a novel root cause diagnostic framework satisfying the assumption that sparse inputs affect the process output. [29] proposed a group-sparsity learning approach for bearing fault diagnosis. In addition to fault diagnosis and detection in the general manufacturing system, sparse learning has been widely utilized to address the issue of an underdetermined system in the multistation assembly system. In particular, sparse Bayesian learning has been widely utilized to incorporate the sparsity of process faults as the prior distribution. [5] proposed a fault diagnosis approach by enhancing the relevant vector machine to detect process faults using the sparse estimate of the variance change of KCCs. [8] developed a spatially correlated sparse Bayesian learning to consider the spatial correlation of KCCs in sparse estimation. The work identifies the KCCs with mean shifts. [10] proposed a Bayesian model to identify the KCCs that have variance increases in a multistation manufacturing system using the sparse variance component prior. [9] developed a novel sparse Bayesian learning to figure out the KCCs with mean shifts by considering the temporal correlation of KCCs and the prior knowledge of process faults.

*B. Sparse Bayesian Learning*

After the introduction of sparse Bayesian learning (SBL) by [30], many researchers have extended this approach significantly. For instance, [31] was the first to apply SBL to sparse estimation for the single measurement vector model given in Eq. (1). Subsequently, [32] further extended it to the MMV model (Eq. (2)) by developing the MSBL algorithm under the common support assumption. The notable advantage of SBL and MSBL is that their global minimum is always the sparsest solution, while that of the minimization-based sparse algorithms [17] [33] is usually not the sparsest solution [16], [34]. Based on the MMV model, many previous studies exploit the spatial correlation in solution vectors (i.e., rows in X in Eq. (2)). [35] proposed a block structure to exploit the intra-block correlation for sparse estimation. [36] developed a Bayesian method for recovery of block-sparse solution whose block-sparse structures are entirely unknown. [37] modeled the spatial structure of the solution as Markov dependency by the Beta process. Besides considering the spatial correlation of sparse solutions, work that considers nonstationary sparse solutions has been studied recently under the MMV model in the SBL framework. [38] developed a method using the Dirichlet process to cluster the measurements into groups with common sparsity patterns. Compared to [38], [2] proposed a more general method having two sparsity components: a commonly shared sparsity and an individual sparsity to deal with outliers that deviated from the uniform sparsity pattern in each group.

*C. Research Gap Analysis*

The research presented in Section II-A focuses on utilizing the sparsity of process faults for accurate fault diagnosis when low dimensional measurements exist in actual manufacturing systems. However, there is a lack of efforts to identify process faults by considering the spatial correlation and the nonstationary process faults, which is common in industrial practice. Section II-B introduces methods in SBL. It introduces the work concerning the spatial correlation in the solution vector and dealing with the nonstationary sparse solution individually. However, it still lacks the work that uses both properties simultaneously. Therefore, this paper proposes a novel SBL method considering both properties for accurate fault diagnosis.