Spayr

A Sparse Bayesian Learning for Diagnosis of Nonstationary and Spatially Correlated Faults:References

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 VI. CONCLUSIONS This paper proposes a novel sparse hierarchical Bayesian method, CSSBL, to effectively identify the sparse process faults in multistation assembly systems. The method identifies process faults by considering the spatial correlation of KCCs and nonstationary process faults among the multiple KPCs. Since posterior distributions of KCCs in the proposed method are computationally intractable, this paper derives approximate posterior distributions of KCCs via Variational Bayes inference. The proposed method’s effectiveness is validated by numerical cases and real-world simulation application using an actual auto-body assembly system. In this work, the temporal correlations of the KCCs are not considered. However, a temporal correlation exists among the KCCs because of the degradation of wear of production tooling over time [23] or the machine-tool thermal distortion [22]. Therefore, extending the proposed CSSBL to consider the temporal correlation of KCCs makes the method more realistic in the actual multistation assembly system. Since the proposed method is not designed for specific processes, it can be effectively applied to other domains, including healthcare and communication systems, for their process monitoring. APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References VI. CONCLUSIONS This paper proposes a novel sparse hierarchical Bayesian method, CSSBL, to effectively identify the sparse process faults in multistation assembly systems. The method identifies process faults by considering the spatial correlation of KCCs and nonstationary process faults among the multiple KPCs. Since posterior distributions of KCCs in the proposed method are computationally intractable, this paper derives approximate posterior distributions of KCCs via Variational Bayes inference. The proposed method’s effectiveness is validated by numerical cases and real-world simulation application using an actual auto-body assembly system. In this work, the temporal correlations of the KCCs are not considered. However, a temporal correlation exists among the KCCs because of the degradation of wear of production tooling over time [23] or the machine-tool thermal distortion [22]. Therefore, extending the proposed CSSBL to consider the temporal correlation of KCCs makes the method more realistic in the actual multistation assembly system. Since the proposed method is not designed for specific processes, it can be effectively applied to other domains, including healthcare and communication systems, for their process monitoring. APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F

A Sparse Bayesian Learning for Diagnosis of Nonstationary and Spatially Correlated Faults:Conclusion

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 V. REAL-WORLD SIMULATION CASE STUDIES An assembly operation from an actual auto-body assembly process is used as a real-world case study. Fig. 5 describes a car’s floor pan, which is the assembled product from this process. The assembled product consists of four parts, including the right bracket, left bracket, right floor pan, and left floor pan. Fig. 6 shows the process assembly procedure consisting of three stations. During the assembly process, the parts are held by fixtures, which are the KCCs in this process [7]. KPCs are measured from four points, namely, M1, M2, M3, and M4, respectively, as shown in Fig. 5. Every part has a designated location for measuring the KPCs. For instance, part 1 has M1, part 2 has M2, part 3 has M3, and part 4 has M4. These measurements can be taken at each station once the relevant part has been assembled in the preceding stations. For example, M3 on part 3 cannot be measured in station 1 because part 3 has not yet been assembled at station 1 [7]. In the designated location, KPCs are measured in three directions (X, Y, and Z). In this assembly process, there exists a total of 33 KCCs, which are dimensional errors of fixture locators [8]. The fault pattern matrix Φ is established based on the literature [7], [11], [47] and provided in Appendix F. Since the number of measurements (12) is less than that of KCCs (33), it causes an underdetermined system in the fault quality linear model. Therefore, sparse estimation is required to identify process faults. In this assembly process, two lists of correlated KCCs exist; the rest are assumed to be independent [8]. The first list comprises six KCCs, including KCC8, KCC9, KCC10, KCC11, KCC12, and KCC13. The second list is composed of KCC31, KCC32, and KCC33. The correlation structures of the two groups are described (for simplification, the lower triangular terms are removed) as follows [8]: To provide nonstationary process faults along the KPCs samples collected over time, two groups are provided. In the first group, the first correlated list is provided as process faults. For the process faults of the second group, all the correlated lists and one independent KCCs are included. Therefore, the case study represents the situation when the process faults evolve over time since the variation of certain KCCs will be propagated to other KCCs if the process faults are not mitigated immediately during the process. 50 KPCs samples are generated from each group. The variances of process faults, non-process faults, and noise of KPCs are determined to be the same value as in Section IV. In addition, performance evaluation measures and benchmark methods used in Section IV are still utilized in Section V. Table III shows the performance comparison in AUC and NMSE by varying the correlation coefficients. For convenience, the correlation coefficients k1 and k2 are equal and are selected from {0.1, 0.3, 0.6, 0.9, 0.95} in the case studies. Specifically, the AUC of the proposed method is close to 1.0 at every correlations levels, representing the perfect classification between the process faults and non-process faults in terms of estimated variance. The effectiveness of the proposed method is significant in NMSE when a high correlation exists. Specifically, the NMSE of the proposed method is less than 10% of that of all the benchmark methods when the correlation coefficient is 0.9 and 0.95. To demonstrate the generalizability of the proposed method, the correlated KCCs are randomly selected in further case studies. Assuming there exist two correlated lists with the size of three, respectively. Therefore, six KCCs are randomly selected from 33 KCCs for two correlated lists in each trial. As in the previous study, one correlated list is provided as process faults in the first group, while two correlated lists and two independent KCCs are used as process faults in the remaining second group. Fig. 7 shows the performances of all the methods in AUC and NMSE results in ten different spatial correlations (i.e., k1 = k2). From Fig. 7 (a), it is prominent that the proposed method successfully differentiates between process faults and non-process faults. Specifically, the AUC of the proposed method is nearly 1.0, while those of most benchmark methods are below 0.9 in the highly correlated case studies. In addition, Fig. 7 (b) represents that the proposed method provides valuable information to the practitioners to determine when to stop the operations to maintain the process when a high correlation exists between KCCs. All the benchmark methods illustrate similar trends compared to Section IV since the methods cannot consider both the spatial correlation of KCCs and the nonstationary process faults. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References V. REAL-WORLD SIMULATION CASE STUDIES An assembly operation from an actual auto-body assembly process is used as a real-world case study. Fig. 5 describes a car’s floor pan, which is the assembled product from this process. The assembled product consists of four parts, including the right bracket, left bracket, right floor pan, and left floor pan. Fig. 6 shows the process assembly procedure consisting of three stations. During the assembly process, the parts are held by fixtures, which are the KCCs in this process [7]. KPCs are measured from four points, namely, M1, M2, M3, and M4, respectively, as shown in Fig. 5. Every part has a designated location for measuring the KPCs. For instance, part 1 has M1, part 2 has M2, part 3 has M3, and part 4 has M4. These measurements can be taken at each station once the relevant part has been assembled in the preceding stations. For example, M3 on part 3 cannot be measured in station 1 because part 3 has not yet been assembled at station 1 [7]. In the designated location, KPCs are measured in three directions (X, Y, and Z). In this assembly process, there exists a total of 33 KCCs, which are dimensional errors of fixture locators [8]. The fault pattern matrix Φ is established based on the literature [7], [11], [47] and provided in Appendix F. Since the number of measurements (12) is less than that of KCCs (33), it causes an underdetermined system in the fault quality linear model. Therefore, sparse estimation is required to identify process faults. In this assembly process, two lists of correlated KCCs exist; the rest are assumed to be independent [8]. The first list comprises six KCCs, including KCC8, KCC9, KCC10, KCC11, KCC12, and KCC13. The second list is composed of KCC31, KCC32, and KCC33. The correlation structures of the two groups are described (for simplification, the lower triangular terms are removed) as follows [8]: To provide nonstationary process faults along the KPCs samples collected over time, two groups are provided. In the first group, the first correlated list is provided as process faults. For the process faults of the second group, all the correlated lists and one independent KCCs are included. Therefore, the case study represents the situation when the process faults evolve over time since the variation of certain KCCs will be propagated to other KCCs if the process faults are not mitigated immediately during the process. 50 KPCs samples are generated from each group. The variances of process faults, non-process faults, and noise of KPCs are determined to be the same value as in Section IV. In addition, performance evaluation measures and benchmark methods used in Section IV are still utilized in Section V. Table III shows the performance comparison in AUC and NMSE by varying the correlation coefficients. For convenience, the correlation coefficients k1 and k2 are equal and are selected from {0.1, 0.3, 0.6, 0.9, 0.95} in the case studies. Specifically, the AUC of the proposed method is close to 1.0 at every correlations levels, representing the perfect classification between the process faults and non-process faults in terms of estimated variance. The effectiveness of the proposed method is significant in NMSE when a high correlation exists. Specifically, the NMSE of the proposed method is less than 10% of that of all the benchmark methods when the correlation coefficient is 0.9 and 0.95. To demonstrate the generalizability of the proposed method, the correlated KCCs are randomly selected in further case studies. Assuming there exist two correlated lists with the size of three, respectively. Therefore, six KCCs are randomly selected from 33 KCCs for two correlated lists in each trial. As in the previous study, one correlated list is provided as process faults in the first group, while two correlated lists and two independent KCCs are used as process faults in the remaining second group. Fig. 7 shows the performances of all the methods in AUC and NMSE results in ten different spatial correlations (i.e., k1 = k2). From Fig. 7 (a), it is prominent that the proposed method successfully differentiates between process faults and non-process faults. Specifically, the AUC of the proposed method is nearly 1.0, while those of most benchmark methods are below 0.9 in the highly correlated case studies. In addition, Fig. 7 (b) represents that the proposed method provides valuable information to the practitioners to determine when to stop the operations to maintain the process when a high correlation exists between KCCs. All the benchmark methods illustrate similar trends compared to Section IV since the methods cannot consider both the spatial correlation of KCCs and the nonstationary process faults.

Sparse Bayesian Learning: Real-World Simulation Case Studies

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 III. PROPOSED RESEARCH METHODOLOGY This section proposes a novel sparse Bayesian hierarchical method: clustering and spatially correlated sparse Bayesian learning (CSSBL). The proposed CSSBL is described in Section III-A, followed by Bayesian inference in Section III-B. A. Proposed Methodology The proposed methodology is a sparse Bayesian hierarchical model considering the spatial correlations of KCCs and nonstationarity of process faults along the KPCs samples. From Eq. (2), fault quality linear model of k th KPCs sample (yk ) can be written as follows: The prior distributions in the proposed method consist of the following two hierarchical layers. The first layer provides several prior distributions. First, the prior distribution of precision variable α in Eq. (4) is provided. Second, the prior distribution exploiting the spatial correlation of KCCs is offered. Finally, the prior distribution in the first layer clusters the KPCS samples into groups sharing the same process faults. The second layer consists of prior distribution encouraging the sparsity of process faults. Fig. 2 describes a graphical representation of the hierarchical layers in the proposed method. In the first layer of the hierarchical model in Fig. 2, the Gamma distribution is provided as the prior distribution of α in Eq. (4) as follows: To handle the spatial correlation of KCCs, the proposed approach considers two types of correlation structures of fixture locators in multistation assembly systems. One is an independent locator whose deviation is uncorrelated to the deviation of other locators (i.e., independent tolerance mode in [21]). The other is correlated locators (i.e., composite tolerance mode in [21]). Specifically, the locators in the correlated list vary with a certain correlation with other locators in the list [21]. To consider this structure, the proposed method models the covariance matrix (Ck) of the prior distribution of xk in Eq. (5) as follows: B. Bayesian Inference of the Proposed Methodology To address this challenge, this paper employs Variational Bayes inference (VBI) to derive approximate posterior distributions of hidden variables. Specifically, Variational Bayes Expectation Maximization (VBEM) [41] estimates hidden variables and hyperparameters to identify KCCs with excessive variance in the proposed method. VBEM consists of Estep: Variational Bayesian expectation step to estimate hidden variables X, Z, Γ, α by approximating the posterior distribution of hidden variables; and M-step: Variational Bayesian maximization step to update hyperparameters B by maximizing the expected value of the logarithm of the complete likelihood  [37]. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References III. PROPOSED RESEARCH METHODOLOGY This section proposes a novel sparse Bayesian hierarchical method: clustering and spatially correlated sparse Bayesian learning (CSSBL). The proposed CSSBL is described in Section III-A, followed by Bayesian inference in Section III-B. A. Proposed Methodology A. Proposed Methodology The proposed methodology is a sparse Bayesian hierarchical model considering the spatial correlations of KCCs and nonstationarity of process faults along the KPCs samples. From Eq. (2), fault quality linear model of k th KPCs sample (yk ) can be written as follows: The prior distributions in the proposed method consist of the following two hierarchical layers. The first layer provides several prior distributions. First, the prior distribution of precision variable α in Eq. (4) is provided. Second, the prior distribution exploiting the spatial correlation of KCCs is offered. Finally, the prior distribution in the first layer clusters the KPCS samples into groups sharing the same process faults. The second layer consists of prior distribution encouraging the sparsity of process faults. Fig. 2 describes a graphical representation of the hierarchical layers in the proposed method. In the first layer of the hierarchical model in Fig. 2, the Gamma distribution is provided as the prior distribution of α in Eq. (4) as follows: To handle the spatial correlation of KCCs, the proposed approach considers two types of correlation structures of fixture locators in multistation assembly systems. One is an independent locator whose deviation is uncorrelated to the deviation of other locators (i.e., independent tolerance mode in [21]). The other is correlated locators (i.e., composite tolerance mode in [21]). Specifically, the locators in the correlated list vary with a certain correlation with other locators in the list [21]. To consider this structure, the proposed method models the covariance matrix (Ck) of the prior distribution of xk in Eq. (5) as follows: B. Bayesian Inference of the Proposed Methodology B. Bayesian Inference of the Proposed Methodology To address this challenge, this paper employs Variational Bayes inference (VBI) to derive approximate posterior distributions of hidden variables. Specifically, Variational Bayes Expectation Maximization (VBEM) [41] estimates hidden variables and hyperparameters to identify KCCs with excessive variance in the proposed method. VBEM consists of Estep: Variational Bayesian expectation step to estimate hidden variables X, Z, Γ, α by approximating the posterior distribution of hidden variables; and M-step: Variational Bayesian maximization step to update hyperparameters B by maximizing the expected value of the logarithm of the complete likelihood  [37].

Sparse Bayesian Learning: Proposed Research Methodology

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 IV. NUMERICAL CASE STUDIES This section provides numerical studies to compare the performance between the proposed method and benchmark methods. The studies evaluate the performance by varying the correlations (ki in Eq. (7)) between the KCCs in nonstationary process faults. All the numerical case studies consist of 20 independent trials. The code of the proposed algorithm is implemented in Matlab 2021. The CPU used in case studies is an Intel® Core™ Processor i7-8750H. This section compares the proposed method with the following benchmark methods. • User Grouping Sparse Bayesian Learning (i.e., UGSBL) proposed in [2] is an SBL method that clusters the KPCs samples into groups sharing the same process faults in sparse estimation. • Spatially Correlated Bayesian Learning (i.e., SCBL) proposed in [8] is an SBL method to diagnose process faults by exploiting the spatial correlation of KCCs. • [10] proposed the SBL method that considers prior knowledge of the process faults in sparse estimation. • MSBL proposed in [32] is a basic SBL method that assumes the independence of KCCs and stationarity of process faults. Performance Evaluation: To evaluate the effectiveness of the proposed method, two performance measures are used in this paper. The first measure evaluates the process fault detection capability, and the second measure is related to estimation accuracy. Each measure is evaluated by varying the correlation coefficients of spatially correlated KCCs. Fault Detection Capability: First, the proposed method requires correctly identifying process faults whose KCCs have excessive variance. In order to assess the capability of detecting process faults under sparse conditions, the area under the receiver operating characteristics (ROCs) curve (AUC) is utilized [46]. AUC is widely used for evaluating the accuracy of binary classification. Specifically, AUC measures the quality of the classification accuracy between process faults and non process faults irrespective of the threshold of variance. In other words, AUC accounts for the trade-off between the type 1 and type 2 errors. When the method is capable of achieving perfect classification results, the resulting AUC value will be 1. However, the AUC with a value close to 0.5 denotes that the estimated variances of KCCs from the method are similar between the process faults and non-process faults. Table I represents that the proposed method always achieves the best performance of AUC in the various spatial correlation between KCCs. Specifically, the AUC of the proposed method is close to 1.0 at every correlations levels, representing the perfect classification between the process faults and nonprocess faults in terms of estimated variance. Since UGSBL clusters the KPCs samples that share the same process faults, the method shows a comparable result to the proposed method  when the spatial correlation of KCCs is relatively low and moderate. However, the performance degrades in the high spatial correlation since the method assumes independence between all KCCs. The performance of SSBL is robust to correlations. This is because SSBL utilizes the correlation of correlated KCCs in its estimation. However, SSBL performs worse than the proposed method since this method assumes the same process faults between all KPCs samples, which are unsuitable for dealing with nonstationary process faults. MSBL and [10] show poor performances since these methods assume the independence of KCCs and stationarity of process faults. 2) Estimation Accuracy: In addition to correctly identifying the process fault, the method needs to accurately estimate the variance of process faults. After the process faults are identified, the practitioners are interested in how large the variances are to make decisions for the maintenance of the process. To achieve this objective, this paper provides the normalized mean squared error (NMSE) between true and estimated variance. Table II represents that the proposed method achieves the best performance in the estimation accuracy of the variance in most of the correlations. It implies that the proposed method is useful not only for the process faults identification but also for measuring how severe the process faults are. Compared to benchmark methods, the effectiveness of the proposed method in the NMSE results is noticeable in the presence of high spatial correlations between KCCs. Unlike AUC performance, the NMSE of SSBL increases as the correlation increases. It implies that SSBL successfully identifies the process faults utilizing the correlation information but cannot accurately estimate the variance since SSBL assumes the stationary process faults over KPC samples. The performance of UGSBL, MSBL, and [10] represent similar trends to the performance of AUC. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References IV. NUMERICAL CASE STUDIES This section provides numerical studies to compare the performance between the proposed method and benchmark methods. The studies evaluate the performance by varying the correlations (ki in Eq. (7)) between the KCCs in nonstationary process faults. All the numerical case studies consist of 20 independent trials. The code of the proposed algorithm is implemented in Matlab 2021. The CPU used in case studies is an Intel® Core™ Processor i7-8750H. This section compares the proposed method with the following benchmark methods. • User Grouping Sparse Bayesian Learning (i.e., UGSBL) proposed in [2] is an SBL method that clusters the KPCs samples into groups sharing the same process faults in sparse estimation. • Spatially Correlated Bayesian Learning (i.e., SCBL) proposed in [8] is an SBL method to diagnose process faults by exploiting the spatial correlation of KCCs. • [10] proposed the SBL method that considers prior knowledge of the process faults in sparse estimation. • MSBL proposed in [32] is a basic SBL method that assumes the independence of KCCs and stationarity of process faults. Performance Evaluation: To evaluate the effectiveness of the proposed method, two performance measures are used in this paper. The first measure evaluates the process fault detection capability, and the second measure is related to estimation accuracy. Each measure is evaluated by varying the correlation coefficients of spatially correlated KCCs. Performance Evaluation: Fault Detection Capability: First, the proposed method requires correctly identifying process faults whose KCCs have excessive variance. In order to assess the capability of detecting process faults under sparse conditions, the area under the receiver operating characteristics (ROCs) curve (AUC) is utilized [46]. AUC is widely used for evaluating the accuracy of binary classification. Specifically, AUC measures the quality of the classification accuracy between process faults and non process faults irrespective of the threshold of variance. In other words, AUC accounts for the trade-off between the type 1 and type 2 errors. When the method is capable of achieving perfect classification results, the resulting AUC value will be 1. However, the AUC with a value close to 0.5 denotes that the estimated variances of KCCs from the method are similar between the process faults and non-process faults. Fault Detection Capability: First, the proposed method requires correctly identifying process faults whose KCCs have excessive variance. In order to assess the capability of detecting process faults under sparse conditions, the area under the receiver operating characteristics (ROCs) curve (AUC) is utilized [46]. AUC is widely used for evaluating the accuracy of binary classification. Specifically, AUC measures the quality of the classification accuracy between process faults and non process faults irrespective of the threshold of variance. In other words, AUC accounts for the trade-off between the type 1 and type 2 errors. When the method is capable of achieving perfect classification results, the resulting AUC value will be 1. However, the AUC with a value close to 0.5 denotes that the estimated variances of KCCs from the method are similar between the process faults and non-process faults. Table I represents that the proposed method always achieves the best performance of AUC in the various spatial correlation between KCCs. Specifically, the AUC of the proposed method is close to 1.0 at every correlations levels, representing the perfect classification between the process faults and nonprocess faults in terms of estimated variance. Since UGSBL clusters the KPCs samples that share the same process faults, the method shows a comparable result to the proposed method  when the spatial correlation of KCCs is relatively low and moderate. However, the performance degrades in the high spatial correlation since the method assumes independence between all KCCs. The performance of SSBL is robust to correlations. This is because SSBL utilizes the correlation of correlated KCCs in its estimation. However, SSBL performs worse than the proposed method since this method assumes the same process faults between all KPCs samples, which are unsuitable for dealing with nonstationary process faults. MSBL and [10] show poor performances since these methods assume the independence of KCCs and stationarity of process faults. 2) Estimation Accuracy: In addition to correctly identifying the process fault, the method needs to accurately estimate the variance of process faults. After the process faults are identified, the practitioners are interested in how large the variances are to make decisions for the maintenance of the process. To achieve this objective, this paper provides the normalized mean squared error (NMSE) between true and estimated variance. Table II represents that the proposed method achieves the best performance in the estimation accuracy of the variance in most of the correlations. It implies that the proposed method is useful not only for the process faults identification but also for measuring how severe the process faults are. Compared to benchmark methods, the effectiveness of the proposed method in the NMSE results is noticeable in the presence of high spatial correlations between KCCs. Unlike AUC performance, the NMSE of SSBL increases as the correlation increases. It implies that SSBL successfully identifies the process faults utilizing the correlation information but cannot accurately estimate the variance since SSBL assumes the stationary process faults over KPC samples. The performance of UGSBL, MSBL, and [10] represent similar trends to the performance of AUC.

Sparse Bayesian Learning: Numerical Case Studies

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. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References 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. A. 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 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 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.

Sparse Bayesian Learning: Review of Related Work

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 Abstract 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. I. INTRODUCTION 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. This paper is available on arxiv under CC 4.0 license. Authors: (1) Jihoon Chung; (2)  Zhenyu (James) Kong. 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 Abstract & Introduction Abstract & Introduction Review of Related Work Review of Related Work Proposed Research Methodology Proposed Research Methodology Numerical Case Studies Numerical Case Studies Real-World Simulation Case Studies Real-World Simulation Case Studies Conclusion Conclusion References References Abstract 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. Note to Practitioners Index Terms —Sparse Bayesian Learning, Spatially Correlated Faults, Nonstationary Faults, Variational Bayes Inference, MultiStage Assembly Systems. Index Terms I. INTRODUCTION 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.

Sparse Bayesian Learning: Abstract & Introduction

2 weeks, 4-5 hours of Sleep each day, and the Birth of Miraki: A Savage Hackathon Tale Let's be real, most hackathons are a chaotic circus. You've got sleep-deprived devs chugging energy drinks like it's water, half-baked ideas flying around, and enough tech jargon to make your head spin. But hey, that's where the magic (or madness) happens. My team decided to dive headfirst into the Web3 online hackathon with a mission to make life less miserable for DAO contributors. You know the drill – tools scattered across the internet like lost socks, communication breakdowns that would give couples therapy a run for its money, and a general sense of "Where the heck is anything? Introducing Miraki: VS Code for DAO tools This is just a small explainer. More details about Miraki and the team behind will be published in later blogs. Miraki is our brainchild, a browser-based devtool that aims to bring order to the wild world of DAOs. Think of it as VSCode, but with a Web3 twist. Here's the breakdown: Workspaces: Your customizable command center for a specific DAO or project. Imagine all those DAO tools finally living in harmony, under one glorious roof. Public or private spaces – you choose the level of chaos you're comfortable with. Anyone who wants to participate in your DAO or project can join your workspace (if public) and they’ll see all the apps the workspace uses in the dashboard along with all the activities within.



Apps: Like VS Code extensions, but for all your DAO needs. Plug in Trello for project management, Snapshot for voting, or even Confluence for collectively losing your minds over documentation. All these apps can be found on miraki app store, and in case you have a specific app that hasn’t been made yet, you can make it yourself.


Universal Search: Because let's face it, nobody has time to play hide-and-seek with vital info across a million different tools. One single search box for entire workspace. Your contributors will be able to search across different apps within the workspace. The Hackathon Hustle: Caffeine, Code, and Controlled Chaos The frontend was a React party, with TypeScript and Vite adding that extra bit of class. Our MVP sported three must-have apps: Trello, Snapshot, and DappSuit (swagger UI for Solana programs) to show that integration of such diverse app into one platform is possible. Shoutout to Turbo monorepo for keeping our sprawling codebase from turning into a spaghetti monster. We ideated fancy isolation techniques for our apps: iframes, web workers, we read blogs from figma to understand how they implemented plugins, we digged deep into vs-codes original source code, the whole nine yards. But pragmatism won the day – in the true spirit of hackathons, we embraced the "it works (for now)" mantra to avoid disappearing down that rabbit hole. The backend? Django and Postgres, a classic combo for building the foundation of our little empire. React Pluggable gave us that sweet extensibility we craved, without the crippling overengineering headaches. And that’s how we managed to wrap up the project just a day before last submission date. The Savage Truth Was it smooth sailing? Pfft. More like navigating a storm in a leaky boat. But amidst the sleep deprivation, the caffeine-induced jitters, and enough bugs to rival an Amazon rainforest, a truly useful tool emerged. Miraki ain't perfect (yet), but it's got the potential to make DAOs less of a soul-sucking experience for everyone involved. And that, my friends, is what hackathons are all about – taking a problem, caffeinating yourself into oblivion, and emerging with something that might just make the tech world a slightly better place. So, what's next for Miraki? We're aiming to unleash this beast into the wild, gather feedback, and iterate like there's no tomorrow. If you're a fellow DAO dweller tired of feeling like a lost explorer in a digital jungle, stay tuned. Miraki might just be your salvation. PS: The blog is rephrased using Gemini AI. Because the writer was on Leet-code when he should be taking English writing classes. Follow Miraki on https://twitter.com/Miraki_io. All feedback is most-welcomed. You can contact us through email at iampiyushjaiswal103@gmail.com. 2 weeks, 4-5 hours of Sleep each day, and the Birth of Miraki: A Savage Hackathon Tale 2 weeks, 4-5 hours of Sleep each day, and the Birth of Miraki: A Savage Hackathon Tale Let's be real, most hackathons are a chaotic circus. You've got sleep-deprived devs chugging energy drinks like it's water, half-baked ideas flying around, and enough tech jargon to make your head spin. But hey, that's where the magic (or madness) happens. My team decided to dive headfirst into the Web3 online hackathon with a mission to make life less miserable for DAO contributors. You know the drill – tools scattered across the internet like lost socks, communication breakdowns that would give couples therapy a run for its money, and a general sense of "Where the heck is anything? Introducing Miraki: VS Code for DAO tools Introducing Miraki: VS Code for DAO tools This is just a small explainer. More details about Miraki and the team behind will be published in later blogs. This is just a small explainer. More details about Miraki and the team behind will be published in later blogs. Miraki is our brainchild, a browser-based devtool that aims to bring order to the wild world of DAOs. Think of it as VSCode, but with a Web3 twist. Here's the breakdown: Workspaces: Your customizable command center for a specific DAO or project. Imagine all those DAO tools finally living in harmony, under one glorious roof. Public or private spaces – you choose the level of chaos you're comfortable with. Anyone who wants to participate in your DAO or project can join your workspace (if public) and they’ll see all the apps the workspace uses in the dashboard along with all the activities within. Apps: Like VS Code extensions, but for all your DAO needs. Plug in Trello for project management, Snapshot for voting, or even Confluence for collectively losing your minds over documentation. All these apps can be found on miraki app store, and in case you have a specific app that hasn’t been made yet, you can make it yourself. Universal Search: Because let's face it, nobody has time to play hide-and-seek with vital info across a million different tools. One single search box for entire workspace. Your contributors will be able to search across different apps within the workspace. Workspaces: Your customizable command center for a specific DAO or project. Imagine all those DAO tools finally living in harmony, under one glorious roof. Public or private spaces – you choose the level of chaos you're comfortable with. Anyone who wants to participate in your DAO or project can join your workspace (if public) and they’ll see all the apps the workspace uses in the dashboard along with all the activities within. Workspaces: Your customizable command center for a specific DAO or project. Imagine all those DAO tools finally living in harmony, under one glorious roof. Public or private spaces – you choose the level of chaos you're comfortable with. Anyone who wants to participate in your DAO or project can join your workspace (if public) and they’ll see all the apps the workspace uses in the dashboard along with all the activities within. Workspaces: Apps: Like VS Code extensions, but for all your DAO needs. Plug in Trello for project management, Snapshot for voting, or even Confluence for collectively losing your minds over documentation. All these apps can be found on miraki app store, and in case you have a specific app that hasn’t been made yet, you can make it yourself. Apps : Like VS Code extensions, but for all your DAO needs. Plug in Trello for project management, Snapshot for voting, or even Confluence for collectively losing your minds over documentation. All these apps can be found on miraki app store, and in case you have a specific app that hasn’t been made yet, you can make it yourself. Apps Universal Search: Because let's face it, nobody has time to play hide-and-seek with vital info across a million different tools. One single search box for entire workspace. Your contributors will be able to search across different apps within the workspace. Universal Search: Because let's face it, nobody has time to play hide-and-seek with vital info across a million different tools. One single search box for entire workspace. Your contributors will be able to search across different apps within the workspace. Universal Search: The Hackathon Hustle: Caffeine, Code, and Controlled Chaos The Hackathon Hustle: Caffeine, Code, and Controlled Chaos The frontend was a React party, with TypeScript and Vite adding that extra bit of class. Our MVP sported three must-have apps: Trello, Snapshot, and DappSuit (swagger UI for Solana programs) to show that integration of such diverse app into one platform is possible. Shoutout to Turbo monorepo for keeping our sprawling codebase from turning into a spaghetti monster. We ideated fancy isolation techniques for our apps: iframes, web workers, we read blogs from figma to understand how they implemented plugins, we digged deep into vs-codes original source code, the whole nine yards. But pragmatism won the day – in the true spirit of hackathons, we embraced the "it works (for now)" mantra to avoid disappearing down that rabbit hole. figma The backend? Django and Postgres, a classic combo for building the foundation of our little empire. React Pluggable gave us that sweet extensibility we craved, without the crippling overengineering headaches. And that’s how we managed to wrap up the project just a day before last submission date. React Pluggable The Savage Truth The Savage Truth Was it smooth sailing? Pfft . Pfft More like navigating a storm in a leaky boat. But amidst the sleep deprivation, the caffeine-induced jitters, and enough bugs to rival an Amazon rainforest, a truly useful tool emerged. Miraki ain't perfect (yet), but it's got the potential to make DAOs less of a soul-sucking experience for everyone involved. And that, my friends, is what hackathons are all about – taking a problem, caffeinating yourself into oblivion, and emerging with something that might just make the tech world a slightly better place. So, what's next for Miraki? So, what's next for Miraki? We're aiming to unleash this beast into the wild, gather feedback, and iterate like there's no tomorrow. If you're a fellow DAO dweller tired of feeling like a lost explorer in a digital jungle, stay tuned. Miraki might just be your salvation. PS: The blog is rephrased using Gemini AI. Because the writer was on Leet-code when he should be taking English writing classes. PS: The blog is rephrased using Gemini AI. Because the writer was on Leet-code when he should be taking English writing classes. Follow Miraki on https://twitter.com/Miraki_io . https://twitter.com/Miraki_io All feedback is most-welcomed. You can contact us through email at iampiyushjaiswal103@gmail.com. iampiyushjaiswal103@gmail.com.

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