# Abnormality Detection and Failure Prediction Using Explainable Bayesian Deep Learning: Methodology and Case Study with Industrial Data

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## Abstract

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## 1. Introduction

#### 1.1. Artificial Intelligence

- (i)
- Machine learning: This is based on deep learning and predictive analytics.
- (ii)
- Natural language processing: This is related to translation, classification, and information extraction.
- (iii)
- Speech: This is visualized as speech to text and text to speech.
- (iv)
- Expert systems: This corresponds to inference engine and knowledge base.
- (v)
- Planning, schedule, optimization: This is associated with reduction (transforming complex planning challenges into other forms such as the Boolean satisfiability problem), classical (completely deterministic planning with one initial state), probabilistic (planning under uncertainty and incomplete information), as well as temporal (planning by the incorporation of duration and concurrency of actions and events).
- (vi)
- Robotic: This considers reactive machine, limited memory, theory of mind, and self-aware.
- (vii)
- Vision: This is based on image recognition and computer/machine vision.

- (i)
- Failures of robustness: The system is subjected to unusual or unforeseen inputs, causing failures.
- (ii)
- Failures of specification: The system is attempting to do something that is subtly different from what the developer anticipated, which might result in surprising behaviors or consequences.
- (iii)
- Failures of assurance: In operational mode, the system cannot be fully supervised or regulated.

- (i)
- Transparency: An AI system mechanism should be understood.
- (ii)
- Reliability and safety: An AI system should work as intended and be safe to use.
- (iii)
- Privacy and security: AI systems should respect confidentiality and be protected.
- (iv)
- Fairness: AI systems should behave equally toward all human beings.
- (v)
- Inclusiveness: AI systems should inspire and promote human participation.
- (vi)
- Accountability: Responsibility measures must be available when the AI system malfunctions.

- (i)
- Justifying the model’s decision, detecting its problems, especially during the trial period of the AI model, strengthening reliability and safety.
- (ii)
- Complying with the regulations, transparency that leads to accountability, enhanced security, and data privacy.
- (iii)
- Helping to understand AI reasoning and decrease problems related to fairness in AI use.
- (iv)
- Assisting practitioners in verifying the required proprieties of the AI system from the developer.
- (v)
- Promoting interactivity and expanding human creativity by discovering new perspectives on the model or the data.
- (vi)
- Allowing resources to be more optimized, avoiding wastage.
- (vii)
- Fostering collaboration between experts, data scientists, users, and stakeholders.

#### 1.2. Research Gaps and Opportunities

- (i)
- Lack of human involvement: Human engagement is crucial for assessing the generated explanation as the latter is meant for them. Furthermore, human–AI cooperation could contribute to the integration of human-related sciences and for the development of interactive AI, where experts and AI systems work hand in hand, providing more assurance in the AI system’s output.
- (ii)
- Explanation evaluation is practically absent: These measures are important for researchers and developers when evaluating explanation quality.
- (iii)
- Insufficiency in uncertainty management: Uncertainty quantification safeguards the system against adversarial examples where false explanations could be generated from unseen, new data. Moreover, it provides users with supplementary confidence in trusting AI methods prediction compared to point estimation statistical models. It is thus inconceivable for a working AI system to be devoid of this feature.

- (i)
- As shown in Figure 2b, model agnostic explainability, layer-wise relevance propagation, and logic analysis of data are less explored, but they possess great potential as they could be used with any black-box model without altering its performance.
- (ii)
- While Shapley additive explanations (SHAP) is an established method and employed in previous works, note that it was not used to improve PHM task’s performance.

- (iii)
- What are the important qualities of explanation issued from XAI methods, and how does one verify them?
- (iv)
- How does one distinguish between explanations of correct predictions and erroneous ones?
- (v)
- What are the other advantages of deep learning uncertainty quantification to promote its incorporation?
- (vi)
- As a flexible method, how can SHAP be exploited to enhance PHM performance?

- (i)
- To combine SHAP and deep learning uncertainty to constitute a wider explanation scope, where the first one explains the decision of the model, while the latter one describes its output confidence.
- (ii)
- To demonstrate the SHAP global explanation’s ability to improve prognostic task’s performance, which was absent from previous works.
- (iii)
- To conduct explanation evaluation, which is clearly deficient from previous PHM-XAI literature.
- (iv)
- To show the potential of deep learning uncertainty as an anomaly indicator for a real-world industrial dataset, which validates its capability.
- (v)
- To minimize deep learning uncertainties for enhancing anomaly detection and prognostic tasks.

- (i)
- To add model agnostic explainability to the collection of PHM-XAI articles, which is still lacking currently.
- (ii)
- To prove the local accuracy and consistency traits of the explanation. The former validates the efficiency property of Shapley values while the latter confirms the additivity and symmetry proprieties of these values.

#### 1.3. Related Works

- (i)
- In the first phase, both knowledge-driven and data-driven fault recognition and root cause analysis, using data from failure mode/effect analysis and fault tree analysis, are employed simultaneously. The data streams and case-specific context data are used as inputs. Faults from the knowledge-driven methods or outliers from the data-driven methods are produced with an interpretation of the detected anomalies and stored inside a knowledge graph.
- (ii)
- In the second phase, the detected anomalies are shown in a dynamic dashboard complete that contains the raw data and interpretation of results, where the user modification is authorized. This is also stored in the knowledge graph.
- (iii)
- Then, in the third phase, the information in the knowledge graph, which are anomalies, the feedback, and all contextual meta-information, are used to improve the techniques of anomaly detection, fault recognition, and root cause analysis of both methods (knowledge-driven and data-driven). The reported accuracy is good for anomaly detection, better than other standalone data-driven methods, partly because of the XAI approach.

## 2. Methodology

#### 2.1. Multi Output Bayesian LSTM and Uncertainty Quantification Layers

#### 2.2. Minimization of Uncertainties, Anomaly Detection, and RUL Estimation

#### 2.3. Model Performance Assesment and SHAP Explainability

#### 2.4. Explanation Visualization

- (i)
- Local: This is based on force and waterfall plots, which highlight the positive or negative forces of features influencing an instance’s output. On the one hand, the force plot shows successive colored bars, where each bar represents a feature contribution. The length of the colored bar represents its force amplitude or impact on the prediction, and the values associated with the features are the normalized values of the features. The red color means that the feature in question is pushing the prediction positively to increase the output value, $f\left(x\right)$, while the blue color means that the feature is dragging the prediction negatively to decrease the output. This plot was utilized for explaining anomalous instances. On the other hand, the waterfall plot arranges the feature contribution values in bar-like form according to their force amplitude, where the highest is in the top position, while the lowest is at the bottom spot, forming a waterfall-like pattern. Note that the color’s meaning is the same as before, that is, the direction of the force is clearly shown. This plot was used to verify the local accuracy and consistency properties of the explanation elaborated in the next subsection.
- (ii)
- Global: This is based on a summary plot, which highlights the most contributing features in a sequence. The plot arranges the features according to its contributing power and its forces’ directions. Here, the explanation was exploited to enhance the prognostic accuracy by employing only the most contributing features. The model was initially tested with all the features followed by only using 75% of the best of them. Therefore, the performances of the different settings were analyzed and compared with published results.

## 3. Results

#### 3.1. Case Study 1 from Industry: Real Gas Turbine Anomaly Detection

#### 3.2. Case Study 2: Turbofan Engines Failure Prognostic

- ${\sum}_{j=1}^{N}{\Phi}_{j}$ = 31.46 + 11.23 + 9.32 + 7.71 = 59.72.
- $f\left(x\right)-{\text{}\mathrm{E}}_{x}(\hat{f}\left(X\right))$ = 5131.426 − 5071.702 = 59.724 ≈ 59.72.

- ${\sum}_{j=1}^{N}{\Phi}_{j}$ = 1.69 + 0.63 + 0.12 = 2.44.
- $f\left(x\right)-{\text{}\mathrm{E}}_{x}(\hat{f}\left(X\right))$ = 464.511 − 462.076 = 2.435 ≈ 2.44.

- ${\sum}_{j=1}^{N}{\Phi}_{j}$ = 1.17 + 1.1 + 0.48 + 0.28 = 3.03.
- $f\left(x\right)-{\text{}\mathrm{E}}_{x}(\hat{f}\left(X\right))$ = 622.625 − 619.595 = 3.03.

- ${\sum}_{j=1}^{N}{\Phi}_{j}$ = 0.28 − 0.02 + 0.02 = 0.28.
- $f\left(x\right)-{\text{}\mathrm{E}}_{x}(\hat{f}\left(X\right))$ = 416.175 − 415.899 = 0.276 ≈ 0.28.

- ${f}_{x}^{\prime}\left({v}^{\prime}\right)$ = 5131.426; ${f}_{x}^{\prime}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$ = 464.511; ${f}_{x}^{\prime}\left({v}^{\prime}\right)$ − ${f}_{x}^{\prime}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$ = 4666.915.
- ${f}_{x}^{\u2033}\left({v}^{\prime}\right)$ = 622.625; ${f}_{x}^{\u2033}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$ = 416.175; ${f}_{x}^{\u2033}\left({v}^{\prime}\right)$ − ${f}_{x}^{\u2033}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$ = 206.45.

- ${f}_{x}^{\prime}\left({v}^{\prime}\right)$ − ${f}_{x}^{\prime}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$ > ${f}_{x}^{\u2033}\left({v}^{\prime}\right)$ − ${f}_{x}^{\u2033}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$.

- ${\Phi}_{j}\left({f}^{\prime},x\right)$ = 9.32; ${\Phi}_{j}\left({f}^{\u2033},x\right)$ = 1.1, thus ${\Phi}_{j}\left({f}^{\prime},x\right)$ > ${\Phi}_{j}\left({f}^{\u2033},x\right)$.

## 4. Discussion

#### 4.1. Anomaly Detection

#### 4.2. Failure Prognostic

#### 4.3. Safeguarding Security and Explanation Evaluation

#### 4.4. Other Aspects

#### 4.5. Impacts of the Current Work

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Parameters | Hidden Units | Fully Connected Layer Size | Mini Batch Size | Learning Rate |
---|---|---|---|---|

Space | 10 to 1000 | 10 to 500 | 26 to 130 | $5\times {10}^{-4}$ to $1\times {10}^{-3}$ |

Sensor | Description | Unit |
---|---|---|

S1 | Total temperature fan inlet | ^{0}R |

S2 | Total temperature at low pressure compressor (LPC) outlet | ^{0}R |

S3 | Total temperature at high pressure compressor (HPC) outlet | ^{0}R |

S4 | Total temperature at low pressure turbine (LPT) outlet | ^{0}R |

S5 | Pressure at fan inlet | PSIA |

S6 | Total pressure in bypass-duct | PSIA |

S7 | Total pressure at HPC outlet | PSIA |

S8 | Physical fan speed | RPM |

S9 | Physical core speed | RPM |

S10 | Engine pressure ratio (P50/P2) | N/A |

S11 | Static pressure at HPC outlet | PSIA |

S12 | Ratio of fuel flow to Ps30 | Pps/PSI |

S13 | Corrected fan speed | RPM |

S14 | Corrected core speed | RPM |

S15 | Bypass ratio | N/A |

S16 | Burner fuel-air ratio | N/A |

S17 | Bleed enthalpy | N/A |

S18 | Demanded fan speed | RPM |

S19 | Demanded corrected fan speed | RPM |

S20 | HPT coolant bleed | lbm/s |

S21 | LPT coolant bleed | lbm/s |

## Appendix B

**Figure A4.**CUSUM charts anomaly 20–21 March 2018 from (

**a**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2}$, (

**b**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4},$ and (

**c**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$.

**Figure A5.**Force plots anomaly 20–21 March 2018 from (

**a**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2},$ (

**b**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4}$, and (

**c**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$.

**Figure A9.**CUSUM charts anomaly 8–9 April 2018 from (

**a**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2},$ (

**b**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4}$, and (

**c**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$.

**Figure A10.**Force plots anomaly 8–9 April 2018 from (

**a**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2}$, (

**b**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4}$, and (

**c**) $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$.

## Appendix C

#### Appendix C.1. Bayes Theorem

#### Appendix C.2. Variational Inference

- (i)
- A probability, $q\left(w|\theta \right)$, is created over the weights $w$ and parameterized by $\theta $ as an approximation of $P(w|D)$ given by$$q\left(w|\theta \right)\approx P\left(w|D\right).$$

- (ii)
- From the above, a cost function, which seeks the minimum setting $\theta $, can be developed for$$\mathcal{L}\left(D,\theta \right)={\displaystyle \int}\left(\text{}q\left(w|\theta \right)\text{}\mathrm{log}\left(\frac{q\left(w|\theta \right)}{P\left(w\right)}\right)-q\left(w|\theta \right)\text{}\mathrm{log}\left(P\left(D|w\right)\right)\right)\mathrm{d}w.$$$$\mathcal{L}\left(D,\theta \right)=\mathrm{KL}[q(w|\theta )\left|\right|P\left(w\right)]-{E}_{q(w|\theta )}\left[\mathrm{log}\left(P\left(D|w\right)\right)\right]$$$$\mathcal{L}\left(D,\theta \right)\approx {\displaystyle \sum}_{i=1}^{n}(\mathrm{log}(q({w}^{i}|\theta ))-\mathrm{log}\left(P\left({w}^{i}\right)\right)-\mathrm{log}(P(D|{w}^{i}))).$$
- (iii)
- Then, during backpropagation, every time a forward pass is performed, this cost function is evaluated with sampled weights. In turn, backward pass updates the weights. This iteration is conducted until the training is over. To perform backpropagation through distribution, the local reparameterization trick introduced in [65] for the variational autoencoder is employed.

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**Figure 1.**Overview of AI incidents for (

**a**) incidents’ domain and (

**b**) incidents’ causes (data extracted from www.incidentdatabase.ai, accessed on 3 December 2021).

**Figure 2.**Overview of PHM-XAI domain [44] for (

**a**) PHM-XAI publications over the years and (

**b**) PHM-XAI published approaches.

**Figure 7.**CUSUM chart with $\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{N}1}$ predictions for anomalies: (

**a**) 20–21 March 2018 and (

**b**) 8–9 April 2018.

**Figure 9.**RUL targets for Turbofan 1 for (

**a**) linear degradation without FSP and (

**b**) transformed final RUL targets.

**Figure 15.**Waterfall plots of ${f}_{x}^{\prime}$ predictions for (

**a**) ${f}_{x}^{\prime}\left({v}^{\prime}\right)\text{}\mathrm{and}$ (

**b**) ${f}_{x}^{\prime}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$.

**Figure 16.**Waterfall plots of ${f}_{x}^{\u2033}$ predictions for (

**a**) ${f}_{x}^{\u2033}\left({v}^{\prime}\right)\mathrm{and}$ (

**b**) ${f}_{x}^{\u2033}({{v}^{\prime}}_{{\backslash}_{{N}_{1}}})$.

Notation | Input | Unit |
---|---|---|

${N}_{2}$ | Power turbine rotational speed | RPM |

${P}_{1}$ | Compressor inlet pressure | Bar |

${\U0001d4c2}_{\U0001d4bb}$ | Fuel mass flow rate | kg/s |

${T}_{1}$ | Compressor inlet temperature | K |

Notation | Output | Unit |
---|---|---|

${N}_{1}$ | Gas generator rotational speed | RPM |

${P}_{2}$ | Compressor outlet pressure | Bar |

${P}_{4}$ | Gas generator turbine outlet pressure | Bar |

${T}_{4}$ | Gas generator turbine outlet temperature | K |

Dataset | Date | Quantity (hour) |
---|---|---|

Training | 1 January 2018–23 October 2018 | 6672 |

Testing | 26 November 2018–30 December 2018 | 816 |

Validation | 23 October 2018–26 November 2018 | 816 |

Anomaly 1 | 20–21 March 2018 | 24 |

Anomaly 2 | 8–9 April 2018 | 24 |

Unused Data | 385 | |

Total | 8737 |

Model | Aleatoric Uncertainty | Epistemic Uncertainty |
---|---|---|

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{N}1}$ | 20.40 | 27.11 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2}$ | 702.49 | 787.87 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4}$ | 11.10 | 92.15 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$ | 32.68 | 49.74 |

Model | $\mathbf{A}{\mathbf{U}}_{\mathbf{s}\mathbf{t}\mathbf{d}\mathbf{m}\mathbf{a}\mathbf{x}}$ | $\mathbf{A}{\mathbf{U}}_{\mathbf{s}\mathbf{t}\mathbf{d}\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{n}}$ | $\mathbf{A}{\mathbf{U}}_{\mathbf{s}\mathbf{t}\mathbf{d}\mathbf{s}\mathbf{t}\mathbf{d}}$ | $\mathit{C}$ |
---|---|---|---|---|

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{N}1}$ | 52.10 | 47.92 | 1.53 | 2.72 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}2}$ | 87.35 | 82.70 | 1.87 | 2.49 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{P}4}$ | 22.90 | 21.46 | 0.47 | 3.07 |

$\mathrm{Bayes}\_{\mathrm{LSTM}}_{\mathrm{T}4}$ | 14.20 | 13.27 | 0.30 | 3.05 |

Dataset | Fault Mode | Operating Condition | Training Data | Testing Data |
---|---|---|---|---|

#1 | 1 | 2 | 100 | 100 |

RMSE with Aleatoric Uncertainty | RMSE with Epistemic Uncertainty | Score with Aleatoric Uncertainty | Score with Epistemic Uncertainty |
---|---|---|---|

17.94 | 18.41 | 1025.31 | 1231.10 |

Combination | Contribution Order |
---|---|

17 Features | S11, S13, S8, S12, S21, S4, S20, OC2, OC3, S7, OC1, S15, S2, S17, S9, S3, and S14 |

RMSE with Aleatoric Uncertainty | RMSE with Epistemic Uncertainty | Score with Aleatoric Uncertainty | Score with Epistemic Uncertainty |
---|---|---|---|

14.59 | 15.87 | 431.99 | 594.88 |

Year | Methods | RMSE | Score |
---|---|---|---|

2017 | variatioanal auto encoder + recurrent neural network | 14.80 | 419 |

2018 | convolutional neural network + feed forward neural network | 12.61 | 274 |

2019 | convolutional neural network + LSTM + feed forward neural network | 12.56 | 231 |

2021 | proposed method | 14.59 | 431 |

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## Share and Cite

**MDPI and ACS Style**

Nor, A.K.M.; Pedapati, S.R.; Muhammad, M.; Leiva, V.
Abnormality Detection and Failure Prediction Using Explainable Bayesian Deep Learning: Methodology and Case Study with Industrial Data. *Mathematics* **2022**, *10*, 554.
https://doi.org/10.3390/math10040554

**AMA Style**

Nor AKM, Pedapati SR, Muhammad M, Leiva V.
Abnormality Detection and Failure Prediction Using Explainable Bayesian Deep Learning: Methodology and Case Study with Industrial Data. *Mathematics*. 2022; 10(4):554.
https://doi.org/10.3390/math10040554

**Chicago/Turabian Style**

Nor, Ahmad Kamal Mohd, Srinivasa Rao Pedapati, Masdi Muhammad, and Víctor Leiva.
2022. "Abnormality Detection and Failure Prediction Using Explainable Bayesian Deep Learning: Methodology and Case Study with Industrial Data" *Mathematics* 10, no. 4: 554.
https://doi.org/10.3390/math10040554