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Sparse Bayesian Learning: Abstract & Introduction

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This article proposes a novel fault diagnosis method: clustering spatially correlated sparse Bayesian learning.
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This paper is available on arxiv under CC 4.0 license.


(1) Jihoon Chung;

(2) Zhenyu (James) Kong.


Sensor technology developments provide a basis for effective fault diagnosis in manufacturing systems. However, the limited number of sensors due to physical constraints or undue costs hinders the accurate diagnosis in the actual process. In addition, time-varying operational conditions that generate nonstationary process faults and the correlation information in

the process require to consider for accurate fault diagnosis in the manufacturing systems. This article proposes a novel fault diagnosis method: clustering spatially correlated sparse Bayesian learning (CSSBL), and explicitly demonstrates its applicability in a multistation assembly system that is vulnerable to the above challenges. Specifically, the method is based on a practical assumption that it will likely have a few process faults (sparse). In addition, the hierarchical structure of CSSBL has several parameterized prior distributions to address the above challenges. As posterior distributions of process faults do not have closed form, this paper derives approximate posterior distributions through Variational Bayes inference. The proposed method’s efficacy is provided through numerical and real-world case studies utilizing an actual autobody assembly system. The generalizability of the proposed method allows the technique to be applied in fault diagnosis in other domains, including communication and healthcare systems.

Note to Practitioners—This article proposes a new process fault diagnosis method: clustering spatially correlated sparse Bayesian learning. This method effectively diagnoses time-varying defects by leveraging the correlation structures in the process when sensor measurements are insufficient. The actual autobody assembly process is utilized to show the proposed method’s effectiveness. The proposed method performs superior to the benchmark methods in fault detection capability. In addition, the proposed method accurately estimates the severity of the process faults, providing significant information to the practitioners for their decision-making in the maintenance schedule. Specifically, the error between the estimation from the proposed method and the actual severity of the process faults achieves less than 10% of error of all the benchmark methods when there exists a high correlation between the variations of the fixture locators in the autobody assembly system.

Index Terms—Sparse Bayesian Learning, Spatially Correlated Faults, Nonstationary Faults, Variational Bayes Inference, MultiStage Assembly Systems.


The sparse estimation has received considerable attention in signal processing due to its ability to reconstruct a high-dimensional sparse source signal from a low-dimensional measurement [1]. Specifically, sparse estimation has broad applications in a wide range of industries in identifying the sources of sensor measurements. These applications include channel estimation in wireless systems [2], Electroencephalography (EEG) source localization in neuroimaging [3], radar detection [4], and fault diagnosis in manufacturing systems [5]. However, time-varying operational conditions in the manufacturing systems cause nonstationary process faults, hindering the accurate sparse estimation. For example, the component degradation (e.g., fixture wearing in assembly systems) will vary over time, violating the stationary process faults assumptions of most existing sparse estimation methods [6], [7]. In addition, the correlation information that occurred due to the structure of the manufacturing system should consider for effective fault diagnosis [8], but it is often neglected in sparse estimation [9], [10].

One motivating example to address the above challenges is the process fault diagnosis in the multistation assembly systems. The systems perform assembly operations from multiple stations to assemble a final product. The final product’s quality relies on several factors known as key control characteristics (KCCs) [8]. The positioning accuracy of fixture locators is KCCs in the multistation assembly [5]. Fixture locators carry out the clamping of parts during the assembly process. Therefore, any deviations from their nominal positions can lead to dimensional quality issues in the final product. Hence, fault diagnosis in multistation assemblies estimates the mean and variance of KCCs, namely, the variations of fixture locators [9]. This article focuses on process faults due to excessive variance, which is a more challenging task to diagnose than mean shifts [7].

Since monitoring the dimensional variation of KCCs is not feasible due to physical constraints in the process [10], the key product characteristics (KPCs), which are essentially measurements obtained from the final product, can be utilized to estimate the variance of KCCs. Specifically, a fault-quality linear model represents the relationship between KPCs and KCCs as follows [11], [12]:

Numerous studies have employed Eq. (1) on fault diagnosis in multistation assembly systems [13], [14]. However, they assume that the number of measurements (M) is greater than the number of KCCs (N) which may not hold in actual manufacturing applications. This is because using an excessive number of sensors (measurements) can result in undue costs [8]. If this assumption is violated, Eq. (1) becomes an underdetermined system that has a nonunique solution. To address this challenge, the sparse solution assumption [15] that x in Eq. (1) has a minimal number of non zero elements is required. In the fault diagnosis problem, sparsity refers to the small number of process faults in the fault-quality linear model, which is a reasonable assumption since there are typically only a few process faults in practice [5]. Among the several sparse estimation methods, the Bayesian approach called sparse Bayesian learning has received much attention recently because of its superior estimation performance guaranteed from the several theoretical properties [16]–[18].

Sparse Bayesian learning has been applied for fault diagnosis in multistation assembly systems [10], [19]. These studies successfully identified process faults by providing prior distribution of KCCs (i.e., x in Eq. (1)) to encourage the sparsity of process faults. Especially the work in [10] applied Bayesian learning to diagnose the process faults based on the following multiple measurements vectors (MMV) model in sparse Bayesian learning [20]:

Beyond the spatial correlation between KCCs, the non stationarity of process faults also needs to be considered for accurate fault diagnosis. In practice, the external time-varying operational condition, including temperature and noise, may cause nonstationary process faults over time in the manufacturing systems [6]. For example, the fixture locators in the assembly process can be deviated because of the thermal expansions of pins caused by the ambient temperature of the process that varies over time [22]. In addition, the variation of some locators will be propagated to other locators over time if the process faults are not mitigated immediately [23]. It results in nonstationary process faults along the KPCs samples collected over time. In other words, the KCCs with excessive variance differ depending on the columns of X in Eq. (2). Therefore, the stationary process faults assumption applied in the previous studies [8], [10] needs to be addressed.

To consider both the spatial correlation of KCCs and the nonstationary process faults along the KPCs samples, this paper proposes a novel sparse Bayesian learning method,

namely, clustering and spatially correlated sparse Bayesian learning (CSSBL). Given KPCs samples, the proposed method clusters the samples into groups sharing the same process faults. At the same time, our method estimates the variance of KCCs of each group to identify the process faults accurately. The following summarizes the contributions of this study:

• From the methodological point of view, this paper proposes a novel sparse Bayesian learning that can consider both the spatial correlation of KCCs and the nonstationary process faults. Since the posterior distribution of the sparse solution in the proposed method is computationally intractable, this article derives the approximate posterior distribution of the sparse solution via Variational Bayes inference [24].

• From the application perspective, the proposed method is applied to fault diagnosis in the actual auto-body assembly process. The method performs better for process fault detection capability than the benchmark methods. Furthermore, the proposed method accurately estimates locators’ variance representing the severity of the process faults to assist practitioners’ decision-making in their maintenance policy.

The subsequent sections of this paper are structured in the following manner. Section II presents an overview of relevant literature, while Section III introduces the proposed methodology. The methodology’s effectiveness is evaluated through numerical case studies in Section IV. Section V provides realworld case studies which are fault diagnosis problems in the multistation assembly system. Finally, Section VI discusses conclusions and future work.