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Faults scenarios and interpretation in the TEP benchmark. Table 2 can distinguish between an approximation A;(i = 1, 2, .,n) function with deterministic attributes and a detailed Dj(i=1,2,.,n) function with stochastic features (Ali et al., 2022a; Nounou and Nounou, 2006). It enables a more accurate evaluation of the temporal resonant frequencies of process characteristics (Watanabe et al., 2022).

Table 2 Faults scenarios and interpretation in the TEP benchmark. Table 2 can distinguish between an approximation A;(i = 1, 2, .,n) function with deterministic attributes and a detailed Dj(i=1,2,.,n) function with stochastic features (Ali et al., 2022a; Nounou and Nounou, 2006). It enables a more accurate evaluation of the temporal resonant frequencies of process characteristics (Watanabe et al., 2022).

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Fig. 1. Traditional PCA monitoring strategy.
Fig. 1. Traditional PCA monitoring strategy.
Fig. 2. A simple information entropy Venn-diagram.
Fig. 2. A simple information entropy Venn-diagram.
Fig. 3. IE-driven identification framework. The structural body of the paper is organized as follows. Section 2 Correlation with an appropriate time lag across each set of variables may also demonstrate causal logic across process variables. However, it is a linear correlation methodology that is unable to be used to gather non-linear connections and does not account for the time lag. Several data-driven-based causal logical machine learning approaches, such as
Fig. 3. IE-driven identification framework. The structural body of the paper is organized as follows. Section 2 Correlation with an appropriate time lag across each set of variables may also demonstrate causal logic across process variables. However, it is a linear correlation methodology that is unable to be used to gather non-linear connections and does not account for the time lag. Several data-driven-based causal logical machine learning approaches, such as
Fig. 5. A systematic illustration of SDG. Fig. 4. Multiscale wavelet transformation framework.
Fig. 5. A systematic illustration of SDG. Fig. 4. Multiscale wavelet transformation framework.
Fig. 6. Proposed methodological machine learning framework.
Fig. 6. Proposed methodological machine learning framework.
Fig. 7. Systematic flowsheet of Tennessee Eastman Process (TEP) Benchmark.
Fig. 7. Systematic flowsheet of Tennessee Eastman Process (TEP) Benchmark.
TEP system variables. Table 1
TEP system variables. Table 1
FDR and FAR detecting results of TEP benchmark. Table 3
FDR and FAR detecting results of TEP benchmark. Table 3
Fig. 8. Traditional PCA-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) PCA-T? statistical plot (b) PCA-SPE statistical plot.
Fig. 8. Traditional PCA-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) PCA-T? statistical plot (b) PCA-SPE statistical plot.
Fig. 9. Multiscale (PCA- D2)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) D.-T” statistical plot (b) D2-SPE statistical plot.
Fig. 9. Multiscale (PCA- D2)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) D.-T” statistical plot (b) D2-SPE statistical plot.
Fig. 10. Multiscale (PCA- D3)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) D3-T? statistical plot (b) D3-SPE statistical plot.
Fig. 10. Multiscale (PCA- D3)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) D3-T? statistical plot (b) D3-SPE statistical plot.
Fig. 11. Multiscale (PCA- A3)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) A3-T? statistical plot (b) A3-SPE statistical plot.
Fig. 11. Multiscale (PCA- A3)-aided detection statistical plot of step fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) A3-T? statistical plot (b) A3-SPE statistical plot.
Fig. 12. Traditional PCA-aided detection statistical plot of step fault scenario in condenser cooling water inlet temperature (a)PCA-T” statistical plot (b) PCA-SPE statistical plot.
Fig. 12. Traditional PCA-aided detection statistical plot of step fault scenario in condenser cooling water inlet temperature (a)PCA-T” statistical plot (b) PCA-SPE statistical plot.
Fig. 13. Multiscale (PCA- D2)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) D.-T? statistical plot (b) Do-SPE statistical plot.
Fig. 13. Multiscale (PCA- D2)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) D.-T? statistical plot (b) Do-SPE statistical plot.
fig. 14. Multiscale (PCA- D3)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) D;-T° statistical plot (b) Ds-SPE statistical plot.
fig. 14. Multiscale (PCA- D3)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) D;-T° statistical plot (b) Ds-SPE statistical plot.
Fig. 15. Multiscale (PCA- A3)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) Az-T? statistical plot (b) A3-SPE statistical plot.
Fig. 15. Multiscale (PCA- A3)-aided detection statistical plot of step fault scenario in condenser cooling water temperature (a) Az-T? statistical plot (b) A3-SPE statistical plot.
Fig. 16. Contribution Plots for step Fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) Traditional PCA aided T? & SPE Contribution (b) Multiscale (PCA-D2) aided T? & SPE Contribution (c) Multiscale (PCA-D3) aided T? & SPE Contribution (d) Multiscale (PCA-A3) aided T? & SPE Contribution.
Fig. 16. Contribution Plots for step Fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) Traditional PCA aided T? & SPE Contribution (b) Multiscale (PCA-D2) aided T? & SPE Contribution (c) Multiscale (PCA-D3) aided T? & SPE Contribution (d) Multiscale (PCA-A3) aided T? & SPE Contribution.
Fig. 17. Wavelet-IE Value Plots for step Fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) Traditional PCA aided Wavelet-IE Plot (b) Multiscale (PCA-D2) aided Wavelet-IE Plot (c) Multiscale (PCA-D3) aided Wavelet-IE Plot (d) Multiscale (PCA-A3) aided Wavelet-IE Plot.
Fig. 17. Wavelet-IE Value Plots for step Fault scenario in A/C input-feed proportion, stream 4 - constant (B) component composition (a) Traditional PCA aided Wavelet-IE Plot (b) Multiscale (PCA-D2) aided Wavelet-IE Plot (c) Multiscale (PCA-D3) aided Wavelet-IE Plot (d) Multiscale (PCA-A3) aided Wavelet-IE Plot.
Table 4 PCA-T? and PCA-SPE detection monitoring plots, indicate a notable change around the data observation point 160th. However, when the fault has been recognized and detected, a part of the statistical moni- toring values on both plots stays away from the fault initiation data observation point. As illustrated in Figs. 9-11, the proposed multiscale-PCA (MSPCA) framework detects abnormalities and faults early and efficiently, accompanied by detailed (D2, D3) and approximation (A3) functions. Figs. 9-10(a) and (b) demonstrate the MSPCA detection monitoring plots results based on DzT”, D2-SPE and, D;T”, D3-SPE that identify the abnormalities and faults at the 167th and 164th data observation points, respectively. The fault detection results based on detailed functions (Dg, D3) demonstrate that there is still a limitation of early detection of ab- normalities and faults on monitoring plots, but significant than the traditional PCA approach. 5.1.1. Fault detection scenario 01 In this fault scenario 01, astep trend is inducted in the A/C feed ratio, B composition constant (stream 4). Figs. 8-11 represent the monitoring detection results for this fault scenario using the traditional single-origin PCA technique and the new proposed wavelet-entropy-based frame- work. All the proposed and traditional algorithms can detect the ab- normality and fault within a few samples after it is initiated. The fault is noticeable on both PCA-based detection monitoring statistical plots at the 169th data observation point, as illustrated in Fig. 8(a) and (b). The
Table 4 PCA-T? and PCA-SPE detection monitoring plots, indicate a notable change around the data observation point 160th. However, when the fault has been recognized and detected, a part of the statistical moni- toring values on both plots stays away from the fault initiation data observation point. As illustrated in Figs. 9-11, the proposed multiscale-PCA (MSPCA) framework detects abnormalities and faults early and efficiently, accompanied by detailed (D2, D3) and approximation (A3) functions. Figs. 9-10(a) and (b) demonstrate the MSPCA detection monitoring plots results based on DzT”, D2-SPE and, D;T”, D3-SPE that identify the abnormalities and faults at the 167th and 164th data observation points, respectively. The fault detection results based on detailed functions (Dg, D3) demonstrate that there is still a limitation of early detection of ab- normalities and faults on monitoring plots, but significant than the traditional PCA approach. 5.1.1. Fault detection scenario 01 In this fault scenario 01, astep trend is inducted in the A/C feed ratio, B composition constant (stream 4). Figs. 8-11 represent the monitoring detection results for this fault scenario using the traditional single-origin PCA technique and the new proposed wavelet-entropy-based frame- work. All the proposed and traditional algorithms can detect the ab- normality and fault within a few samples after it is initiated. The fault is noticeable on both PCA-based detection monitoring statistical plots at the 169th data observation point, as illustrated in Fig. 8(a) and (b). The
Fig. 18. Contribution Plots for step Fault scenario in condenser cooling water temperature (a) Traditional PCA aided T° & SPE Contribution (b) Multiscale (PCA-D2) aided T? & SPE Contribution (c) Multiscale (PCA-D3) aided T°? & SPE Contribution (d) Multiscale (PCA-D3) aided T°? & SPE Contribution.
Fig. 18. Contribution Plots for step Fault scenario in condenser cooling water temperature (a) Traditional PCA aided T° & SPE Contribution (b) Multiscale (PCA-D2) aided T? & SPE Contribution (c) Multiscale (PCA-D3) aided T°? & SPE Contribution (d) Multiscale (PCA-D3) aided T°? & SPE Contribution.
Fig. 19. Wavelet-IE Value Plots for step Fault scenario in condenser cooling water temperature (a) Traditional PCA aided Wavelet-IE Plot (b) Multiscale (PCA-D2) aided Wavelet-IE Plot (c) Multiscale (PCA-D3) aided Wavelet-IE Plot (d) Multiscale (PCA-A3) aided Wavelet-IE Plot.
Fig. 19. Wavelet-IE Value Plots for step Fault scenario in condenser cooling water temperature (a) Traditional PCA aided Wavelet-IE Plot (b) Multiscale (PCA-D2) aided Wavelet-IE Plot (c) Multiscale (PCA-D3) aided Wavelet-IE Plot (d) Multiscale (PCA-A3) aided Wavelet-IE Plot.
Table 5 diagnostic propagating root path findings of TEP faults scenarios computed by the proposed framework and some previous researchers’ studies (Kim et al., 2019) and (Hu et al., 2022). The comprehensive description of fault diagnosis propagating root path and cause results based on the proposed framework is explained using fault scenarios 01 and 05. A;-IE value monitoring plot variables are consistent and indicate the actual fault contribution interpretation shown in Table 5. When the actual fault contribution variables are found and selected based on the proposed wavelet-entropy-based multiscale PCA framework, then there are utilized for building the SDG framework for finding the actual faulty propagating root path. 5.3.1. Fault diagnosis propagating root path/cause scenario 01 5.3. Fault diagnosis propagating root path/cause results and outcomes - ee - When the actual faulty contribution variables are detected and identified, classify the faulty variable’s correlation and propagation causality based on the proposed wavelet-entropy-aided multiscale PCA- SDG framework. Then, the proposed framework is utilized to establish the SDG model framework for determining the actual diagnostic prop- agation root path. Finally, the fault diagnostic propagating root cause, path results, and outcomes are discussed based on the proposed wavelet-entropy-aided multiscale PCA-SDG framework. The primary reason for the fault is represented as a reddish colour variable node. The blueish nodes reflect the cause that changes as the fault spreads, while the yellowish variable node depicts the fault’s outcomes. Table 6 summarizes the fault In previous studies, some researchers, like (Kim et al., 2019) and (Hu
Table 5 diagnostic propagating root path findings of TEP faults scenarios computed by the proposed framework and some previous researchers’ studies (Kim et al., 2019) and (Hu et al., 2022). The comprehensive description of fault diagnosis propagating root path and cause results based on the proposed framework is explained using fault scenarios 01 and 05. A;-IE value monitoring plot variables are consistent and indicate the actual fault contribution interpretation shown in Table 5. When the actual fault contribution variables are found and selected based on the proposed wavelet-entropy-based multiscale PCA framework, then there are utilized for building the SDG framework for finding the actual faulty propagating root path. 5.3.1. Fault diagnosis propagating root path/cause scenario 01 5.3. Fault diagnosis propagating root path/cause results and outcomes - ee - When the actual faulty contribution variables are detected and identified, classify the faulty variable’s correlation and propagation causality based on the proposed wavelet-entropy-aided multiscale PCA- SDG framework. Then, the proposed framework is utilized to establish the SDG model framework for determining the actual diagnostic prop- agation root path. Finally, the fault diagnostic propagating root cause, path results, and outcomes are discussed based on the proposed wavelet-entropy-aided multiscale PCA-SDG framework. The primary reason for the fault is represented as a reddish colour variable node. The blueish nodes reflect the cause that changes as the fault spreads, while the yellowish variable node depicts the fault’s outcomes. Table 6 summarizes the fault In previous studies, some researchers, like (Kim et al., 2019) and (Hu
Comparison of fault diagnostic propagating root path findings of TEP benchmark faults scenarios with previsous study. Table 6
Comparison of fault diagnostic propagating root path findings of TEP benchmark faults scenarios with previsous study. Table 6
Table 7 Fig. 20. Wavelet-entropy-aided multiscale PCA-SDG Fault Diagnostic propagating Root Path for step fault scenario in A/C feed ratio, B composition constant (Stream 4). represented in a matrix, with rows reflecting the root cause system variables and columns expressing the effective system variables. A dark colour unity in the table shows a causal connection between the two variables. In contrast, a light colour zero entry denotes no causal rela- tionship and linkages between the two system variables. The failure root cause effects are found to be propagated to downstream functions, and the unusual aberrant operation influences multiple process system var- iables in the process. The proposed monitoring algorithm approach determines and finds out the process network and identifies the vital diagnostic propagation root cause variables using preliminary process information. et al. (2022), determined the diagnostic propagation root path of the TEP application case study. Kim’s analyses identified the fault scenario 01 diagnostic propagation path as (20, 16, 13, 7, 47, 44) (Kim et al., 2019). Similarly, Hu’s studies classified the fault scenario 01 diagnostic propagation path as (1, 16, 18, 39, 7, 13, 4, 6, 3, 20, 44, 9) (Hu et al., 2022). According to their interpretations, the failure is caused by an irregularity in reactor pressure and the compressor work, indicating that the fault propagation to variable 20 is doubtful. The compressor is also not the closest point where this failure occurs. Although the A/C input feed proportion is still not regulated, the process diagram shows that the A/C input feed comes from (stream 4) and could be transmitted to the stripper. As a result, the fault is injected into (stream 4), and the stripper is the nearest associated equipment. Fig. 20 illustrates the fault diagnostic propagating root cause result. This fault scenario’s primary propagating root cause is the variable (01), A feed (stream1) represented by a reddish colour variable node. The blueish nodes reflect the cause that changes as the fault spreads, while the yellowish variable node depicts the fault’s outcomes. This study identified the fault scenario 01 diagnostic propagation path as (1-> (4,44) > (7,13,18,39,31) +2016). Table 7 summarizes the wavelet-information entropy-aided causality monitoring matrix of TEP benchmark fault scenario 01. This approach estimates the causal relationship among each pair of system variables based on parameters and data measurement to locate all feasible diag- nostic propagation root paths. The causal relationship and linkages are
Table 7 Fig. 20. Wavelet-entropy-aided multiscale PCA-SDG Fault Diagnostic propagating Root Path for step fault scenario in A/C feed ratio, B composition constant (Stream 4). represented in a matrix, with rows reflecting the root cause system variables and columns expressing the effective system variables. A dark colour unity in the table shows a causal connection between the two variables. In contrast, a light colour zero entry denotes no causal rela- tionship and linkages between the two system variables. The failure root cause effects are found to be propagated to downstream functions, and the unusual aberrant operation influences multiple process system var- iables in the process. The proposed monitoring algorithm approach determines and finds out the process network and identifies the vital diagnostic propagation root cause variables using preliminary process information. et al. (2022), determined the diagnostic propagation root path of the TEP application case study. Kim’s analyses identified the fault scenario 01 diagnostic propagation path as (20, 16, 13, 7, 47, 44) (Kim et al., 2019). Similarly, Hu’s studies classified the fault scenario 01 diagnostic propagation path as (1, 16, 18, 39, 7, 13, 4, 6, 3, 20, 44, 9) (Hu et al., 2022). According to their interpretations, the failure is caused by an irregularity in reactor pressure and the compressor work, indicating that the fault propagation to variable 20 is doubtful. The compressor is also not the closest point where this failure occurs. Although the A/C input feed proportion is still not regulated, the process diagram shows that the A/C input feed comes from (stream 4) and could be transmitted to the stripper. As a result, the fault is injected into (stream 4), and the stripper is the nearest associated equipment. Fig. 20 illustrates the fault diagnostic propagating root cause result. This fault scenario’s primary propagating root cause is the variable (01), A feed (stream1) represented by a reddish colour variable node. The blueish nodes reflect the cause that changes as the fault spreads, while the yellowish variable node depicts the fault’s outcomes. This study identified the fault scenario 01 diagnostic propagation path as (1-> (4,44) > (7,13,18,39,31) +2016). Table 7 summarizes the wavelet-information entropy-aided causality monitoring matrix of TEP benchmark fault scenario 01. This approach estimates the causal relationship among each pair of system variables based on parameters and data measurement to locate all feasible diag- nostic propagation root paths. The causal relationship and linkages are
Wavelet-information entropy-aided causality monitoring matrix of TEP benchmark fault scenario 05.
Wavelet-information entropy-aided causality monitoring matrix of TEP benchmark fault scenario 05.
Fig. 21. Wavelet-entropy-aided multiscale PCA-SDG Fault Diagnostic propagating Root Path for step fault scenario in condenser cooling water temperature.
Fig. 21. Wavelet-entropy-aided multiscale PCA-SDG Fault Diagnostic propagating Root Path for step fault scenario in condenser cooling water temperature.
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Statistical machine learningProcess MonitoringFault Detection and Diagnosis
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