# Modified DEMATEL Method Based on Objective Data Grey Relational Analysis for Time Series

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Evaluate Correlation between Categories Using Grey Relational Analysis

**Definition 1.**

- (i)
- Obtain the initial image of each sequence:$${Y}_{j}{}^{\prime}=\left({y}_{j}{}^{\prime}\left(1\right),{y}_{j}{}^{\prime}\left(2\right),\dots ,{y}_{j}{}^{\prime}\left(n\right)\right)={Y}_{j}/{y}_{j}\left(1\right),j=0,1,2,\dots ,m.$$
- (ii)
- Find the absolute value sequence of the difference between the corresponding components of the initial image of ${Y}_{0}$ and ${Y}_{j}$, denoted by ${\alpha}_{j}=\left({\alpha}_{j}\left(1\right),{\alpha}_{j}\left(2\right),\dots ,{\alpha}_{j}\left(n\right)\right)$, where$${\alpha}_{j}\left(k\right)=\left|{y}_{0}{}^{\prime}\left(k\right)-{y}_{j}{}^{\prime}\left(k\right)\right|,k=1,2,\dots ,n,j=1,2,\dots ,m.$$
- (iii)
- Find the maximum $\mathsf{\Phi}$ and minimum $\varphi $ of ${\alpha}_{j}\left(k\right)$, $k=1,2,\dots ,n$, $j=1,2,\dots ,m$:$$\mathsf{\Phi}=\underset{j}{\mathrm{max}}\underset{k}{\mathrm{max}}{\alpha}_{j}\left(k\right),$$$$\varphi =\underset{j}{\mathrm{min}}\underset{k}{\mathrm{min}}{\alpha}_{j}\left(k\right).$$
- (iv)
- Compute the k-point relation coefficient:$$\delta \left({y}_{0}{}^{\prime}\left(k\right),{y}_{j}{}^{\prime}\left(k\right)\right)=\frac{\varphi +\lambda \mathsf{\Phi}}{{\alpha}_{j}\left(k\right)+\lambda \mathsf{\Phi}},k=1,2,\dots ,n,j=1,2,\dots ,m.$$
- (v)
- Find the GRD:$$\delta \left({Y}_{0},{Y}_{j}\right)=\delta \left({Y}_{0}{}^{\prime},{Y}_{j}{}^{\prime}\right)=\frac{1}{n}{\displaystyle \sum _{k=1}^{n}\delta \left({y}_{0}{}^{\prime}\left(k\right),{y}_{j}{}^{\prime}\left(k\right)\right)},j=1,2,\dots ,m.$$

#### 2.2. Determine Priority of Categories Using Modified DEMATEL

- (i)
- Obtain the direct relation matrix $Q$ by GRD between sequences ${Y}_{i}$ and ${Y}_{j}$ ($i,j=0,1,2,\dots ,m,i\ne j$). Specifically, the direct relation matrix $Q$ satisfies

- (ii)
- Find the normalized direct relation matrix $D$:

^{−5}[36,37].

- (iii)
- Obtain the total relation matrix $\Gamma ={\left({\tau}_{ij}\right)}_{\left(m+1\right)\times \left(m+1\right)}$ , which satisfies:

- (iv)
- Obtain the prominence and relation:

## 3. Experiment

#### 3.1. Simulation Experiment

_{4}is more important than F

_{2}, while in our proposed method, F

_{2}is more important than F

_{4}. The differences can be explained by the strength of the connections with F

_{5}, which is as follows: F

_{5}is a very important factor. From the total relation matrix $\Gamma $, we find that ${\tau}_{25}$ is larger than ${\tau}_{45}$, which means that F

_{2}allocates more influence to important factor F

_{5}than to F

_{4}. Consequently, our method considers F

_{2}to be more important.

#### 3.2. Case Study

#### 3.2.1. Evaluate Correlation between Data Categories Using GRA

_{1}and ES

_{j}(j = 2, 3,..., 14) to get the following:

#### 3.2.2. Determine Priority of Data Categories Using Modified DEMATEL

_{10}> ES

_{9}> ES

_{5}> ES

_{6}> ES

_{8}> ES

_{1}> ES

_{4}> ES

_{2}> ES

_{11}> ES

_{12}> ES

_{7}> ES

_{3}> ES

_{14}> ES

_{13}.

#### 3.2.3. Results of Remaining Useful Life Prediction

_{13}and ES

_{14}, optimizes the accuracy of predictions, resulting in improved RMSE values of 0.310 and 0.163, respectively. Overall, the results demonstrate the effectiveness of our proposed method in identifying the critical sensors for predicting the remaining engine life accurately.

_{10}and ES

_{9}, are removed, while removing less important sensors, such as ES

_{13}and ES

_{14}, can improve the model’s predictive performance. When other sensors are removed, the prediction results also change. These results highlight the importance of extracting pertinent data from a large dataset for better decision-making.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

ES_{1} | ES_{2} | ES_{3} | ES_{4} | ES_{5} | ES_{6} | ES_{7} | ES_{8} | ES_{9} | ES_{10} | ES_{11} | ES_{12} | ES_{13} | ES_{14} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ES_{1} | 0.0000 | 0.8377 | 0.7334 | 0.8200 | 0.9115 | 0.9015 | 0.7893 | 0.8721 | 0.9125 | 0.8904 | 0.8210 | 0.8286 | 0.6679 | 0.6877 |

ES_{2} | 0.8761 | 0.0000 | 0.7780 | 0.8307 | 0.8696 | 0.8586 | 0.8607 | 0.8407 | 0.8697 | 0.8651 | 0.8584 | 0.8530 | 0.7377 | 0.7471 |

ES_{3} | 0.8280 | 0.8207 | 0.0000 | 0.7711 | 0.8057 | 0.8320 | 0.8474 | 0.7890 | 0.8060 | 0.8119 | 0.8625 | 0.8629 | 0.7001 | 0.7090 |

ES_{4} | 0.8542 | 0.8217 | 0.7107 | 0.0000 | 0.9044 | 0.8322 | 0.7745 | 0.9284 | 0.9039 | 0.8857 | 0.7830 | 0.7825 | 0.8018 | 0.8204 |

ES_{5} | 0.9177 | 0.8402 | 0.7187 | 0.8879 | 0.0000 | 0.8815 | 0.7872 | 0.9279 | 0.9986 | 0.9166 | 0.8054 | 0.8143 | 0.7193 | 0.7390 |

ES_{6} | 0.9284 | 0.8630 | 0.7978 | 0.8458 | 0.9071 | 0.0000 | 0.8306 | 0.8906 | 0.9076 | 0.9304 | 0.8594 | 0.8708 | 0.7244 | 0.7421 |

ES_{7} | 0.8490 | 0.8721 | 0.8237 | 0.8007 | 0.8371 | 0.8391 | 0.0000 | 0.8097 | 0.8371 | 0.8354 | 0.8828 | 0.8692 | 0.7255 | 0.7325 |

ES_{8} | 0.8945 | 0.8275 | 0.7256 | 0.9261 | 0.9371 | 0.8774 | 0.7788 | 0.0000 | 0.9369 | 0.9137 | 0.7972 | 0.8026 | 0.7586 | 0.7798 |

ES_{9} | 0.9186 | 0.8402 | 0.7189 | 0.8872 | 0.9986 | 0.8821 | 0.7872 | 0.9277 | 0.0000 | 0.9167 | 0.8056 | 0.8146 | 0.7188 | 0.7385 |

ES_{10} | 0.9150 | 0.8610 | 0.7629 | 0.8885 | 0.9307 | 0.9254 | 0.8165 | 0.9183 | 0.9308 | 0.0000 | 0.8371 | 0.8438 | 0.7472 | 0.7654 |

ES_{11} | 0.8695 | 0.8659 | 0.8361 | 0.8031 | 0.8475 | 0.8626 | 0.8791 | 0.8211 | 0.8477 | 0.8499 | 0.0000 | 0.8863 | 0.7122 | 0.7230 |

ES_{12} | 0.8684 | 0.8530 | 0.8279 | 0.7928 | 0.8465 | 0.8666 | 0.8574 | 0.8170 | 0.8468 | 0.8482 | 0.8797 | 0.0000 | 0.6941 | 0.7069 |

ES_{13} | 0.7808 | 0.7842 | 0.7001 | 0.8497 | 0.8086 | 0.7678 | 0.7563 | 0.8197 | 0.8082 | 0.7990 | 0.7505 | 0.7459 | 0.0000 | 0.9003 |

ES_{14} | 0.7877 | 0.7848 | 0.6997 | 0.8587 | 0.8163 | 0.7752 | 0.7546 | 0.8297 | 0.8160 | 0.8070 | 0.7520 | 0.7487 | 0.8959 | 0.0000 |

ES_{1} | ES_{2} | ES_{3} | ES_{4} | ES_{5} | ES_{6} | ES_{7} | ES_{8} | ES_{9} | ES_{10} | ES_{11} | ES_{12} | ES_{13} | ES_{14} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ES_{1} | 0.0000 | 0.0733 | 0.0642 | 0.0718 | 0.0798 | 0.0789 | 0.0691 | 0.0764 | 0.0799 | 0.0780 | 0.0719 | 0.0725 | 0.0585 | 0.0602 |

ES_{2} | 0.0767 | 0.0000 | 0.0681 | 0.0727 | 0.0761 | 0.0752 | 0.0754 | 0.0736 | 0.0761 | 0.0757 | 0.0752 | 0.0747 | 0.0646 | 0.0654 |

ES_{3} | 0.0725 | 0.0719 | 0.0000 | 0.0675 | 0.0705 | 0.0728 | 0.0742 | 0.0691 | 0.0706 | 0.0711 | 0.0755 | 0.0755 | 0.0613 | 0.0621 |

ES_{4} | 0.0748 | 0.0719 | 0.0622 | 0.0000 | 0.0792 | 0.0729 | 0.0678 | 0.0813 | 0.0791 | 0.0775 | 0.0686 | 0.0685 | 0.0702 | 0.0718 |

ES_{5} | 0.0803 | 0.0736 | 0.0629 | 0.0777 | 0.0000 | 0.0772 | 0.0689 | 0.0812 | 0.0874 | 0.0803 | 0.0705 | 0.0713 | 0.0630 | 0.0647 |

ES_{6} | 0.0813 | 0.0756 | 0.0698 | 0.0740 | 0.0794 | 0.0000 | 0.0727 | 0.0780 | 0.0795 | 0.0815 | 0.0752 | 0.0762 | 0.0634 | 0.0650 |

ES_{7} | 0.0743 | 0.0764 | 0.0721 | 0.0701 | 0.0733 | 0.0735 | 0.0000 | 0.0709 | 0.0733 | 0.0731 | 0.0773 | 0.0761 | 0.0635 | 0.0641 |

ES_{8} | 0.0783 | 0.0725 | 0.0635 | 0.0811 | 0.0820 | 0.0768 | 0.0682 | 0.0000 | 0.0820 | 0.0800 | 0.0698 | 0.0703 | 0.0664 | 0.0683 |

ES_{9} | 0.0804 | 0.0736 | 0.0629 | 0.0777 | 0.0874 | 0.0772 | 0.0689 | 0.0812 | 0.0000 | 0.0803 | 0.0705 | 0.0713 | 0.0629 | 0.0647 |

ES_{10} | 0.0801 | 0.0754 | 0.0668 | 0.0778 | 0.0815 | 0.0810 | 0.0715 | 0.0804 | 0.0815 | 0.0000 | 0.0733 | 0.0739 | 0.0654 | 0.0670 |

ES_{11} | 0.0761 | 0.0758 | 0.0732 | 0.0703 | 0.0742 | 0.0755 | 0.0770 | 0.0719 | 0.0742 | 0.0744 | 0.0000 | 0.0776 | 0.0624 | 0.0633 |

ES_{12} | 0.0760 | 0.0747 | 0.0725 | 0.0694 | 0.0741 | 0.0759 | 0.0751 | 0.0715 | 0.0741 | 0.0743 | 0.0770 | 0.0000 | 0.0608 | 0.0619 |

ES_{13} | 0.0684 | 0.0687 | 0.0613 | 0.0744 | 0.0708 | 0.0672 | 0.0662 | 0.0718 | 0.0708 | 0.0700 | 0.0657 | 0.0653 | 0.0000 | 0.0788 |

ES_{14} | 0.0690 | 0.0687 | 0.0613 | 0.0752 | 0.0715 | 0.0679 | 0.0661 | 0.0726 | 0.0714 | 0.0707 | 0.0658 | 0.0655 | 0.0784 | 0.0000 |

ES_{1} | ES_{2} | ES_{3} | ES_{4} | ES_{5} | ES_{6} | ES_{7} | ES_{8} | ES_{9} | ES_{10} | ES_{11} | ES_{12} | ES_{13} | ES_{14} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ES_{1} | 1.1725 | 1.1989 | 1.0888 | 1.2065 | 1.2596 | 1.2274 | 1.1601 | 1.2339 | 1.2598 | 1.2429 | 1.1799 | 1.1836 | 1.0595 | 1.0796 |

ES_{2} | 1.2601 | 1.1464 | 1.1067 | 1.2232 | 1.2729 | 1.2403 | 1.1809 | 1.2477 | 1.2730 | 1.2573 | 1.1984 | 1.2011 | 1.0790 | 1.0985 |

ES_{3} | 1.2153 | 1.1738 | 1.0069 | 1.1786 | 1.2264 | 1.1977 | 1.1415 | 1.2028 | 1.2265 | 1.2121 | 1.1597 | 1.1627 | 1.0408 | 1.0596 |

ES_{4} | 1.2540 | 1.2092 | 1.0975 | 1.1513 | 1.2712 | 1.2339 | 1.1701 | 1.2501 | 1.2713 | 1.2546 | 1.1884 | 1.1914 | 1.0803 | 1.1005 |

ES_{5} | 1.2753 | 1.2264 | 1.1125 | 1.2393 | 1.2145 | 1.2538 | 1.1864 | 1.2663 | 1.2950 | 1.2733 | 1.2056 | 1.2095 | 1.0878 | 1.1083 |

ES_{6} | 1.2902 | 1.2418 | 1.1311 | 1.2498 | 1.3023 | 1.1961 | 1.2030 | 1.2774 | 1.3024 | 1.2884 | 1.2233 | 1.2273 | 1.1003 | 1.1209 |

ES_{7} | 1.2443 | 1.2041 | 1.0982 | 1.2075 | 1.2565 | 1.2253 | 1.0980 | 1.2316 | 1.2566 | 1.2413 | 1.1873 | 1.1893 | 1.0662 | 1.0854 |

ES_{8} | 1.2732 | 1.2252 | 1.1128 | 1.2419 | 1.2900 | 1.2532 | 1.1854 | 1.1909 | 1.2901 | 1.2727 | 1.2047 | 1.2083 | 1.0907 | 1.1113 |

ES_{9} | 1.2754 | 1.2264 | 1.1126 | 1.2393 | 1.2949 | 1.2539 | 1.1864 | 1.2663 | 1.2146 | 1.2733 | 1.2057 | 1.2095 | 1.0878 | 1.1083 |

ES_{10} | 1.2938 | 1.2461 | 1.1324 | 1.2575 | 1.3087 | 1.2756 | 1.2062 | 1.2841 | 1.3088 | 1.2177 | 1.2259 | 1.2296 | 1.1060 | 1.1267 |

ES_{11} | 1.2552 | 1.2127 | 1.1074 | 1.2168 | 1.2668 | 1.2363 | 1.1783 | 1.2418 | 1.2669 | 1.2518 | 1.1245 | 1.1995 | 1.0732 | 1.0928 |

ES_{12} | 1.2453 | 1.2022 | 1.0981 | 1.2064 | 1.2567 | 1.2269 | 1.1674 | 1.2317 | 1.2569 | 1.2418 | 1.1866 | 1.1181 | 1.0634 | 1.0829 |

ES_{13} | 1.1926 | 1.1526 | 1.0478 | 1.1664 | 1.2075 | 1.1741 | 1.1167 | 1.1865 | 1.2076 | 1.1923 | 1.1330 | 1.1356 | 0.9672 | 1.0581 |

ES_{14} | 1.1989 | 1.1582 | 1.0528 | 1.1727 | 1.2139 | 1.1803 | 1.1219 | 1.1930 | 1.2139 | 1.1986 | 1.1385 | 1.1412 | 1.0448 | 0.9900 |

## Appendix B

_{12}> ES

_{8}> ES

_{4}> ES

_{1}> ES

_{7}> ES

_{13}> ES

_{14}> ES

_{11}> ES

_{3}> ES

_{2}> ES

_{9}> ES

_{5}> ES

_{10}> ES

_{6}, and the importance ranking of the proposed method is ES

_{10}> ES

_{9}> ES

_{5}> ES

_{6}> ES

_{8}> ES

_{1}> ES

_{4}> ES

_{2}> ES

_{11}> ES

_{12}> ES

_{7}> ES

_{3}> ES

_{14}> ES

_{13}. The results show that there is a significant difference between the results of the two methods.

_{10}> ES

_{9}> ES

_{5}> ES

_{6}> ES

_{8}> ES

_{1}> ES

_{4}> ES

_{2}= ES

_{11}> ES

_{12}> ES

_{7}> ES

_{3}> ES

_{14}> ES

_{13}. It can be seen from the accuracy ranking of the RUL prediction results that the ranking obtained by our proposed method is more in line with reality.

Engine Sensor | GRA-DEMATEL | Proposed Method | ||
---|---|---|---|---|

$\mathit{P}\left(\mathit{i}\right)$ | Ranking | $\mathit{P}\left(\mathit{i}\right)$ | Ranking | |

ES_{1} | 22.4991 | 4 | 2.0338 | 6 |

ES_{2} | 20.9809 | 10 | 2.0139 | 8 |

ES_{3} | 21.3425 | 9 | 1.9068 | 12 |

ES_{4} | 22.5051 | 3 | 2.0176 | 7 |

ES_{5} | 18.7684 | 12 | 2.0642 | 3 |

ES_{6} | 11.6702 | 14 | 2.0507 | 4 |

ES_{7} | 22.1287 | 5 | 1.9774 | 11 |

ES_{8} | 22.5209 | 2 | 2.0468 | 5 |

ES_{9} | 18.7684 | 11 | 2.0643 | 2 |

ES_{10} | 12.1468 | 13 | 2.0664 | 1 |

ES_{11} | 22.0440 | 8 | 1.9974 | 9 |

ES_{12} | 22.6625 | 1 | 1.9926 | 10 |

ES_{13} | 22.1159 | 6 | 1.8749 | 14 |

ES_{14} | 22.0922 | 7 | 1.8931 | 13 |

## References

- Zhang, J.; Cui, S.; Xu, Y.; Li, Q.; Li, T. A novel data-driven stock price trend prediction system. Expert Syst. Appl.
**2018**, 97, 60–69. [Google Scholar] [CrossRef] - Jia, R.; Jiang, P.; Liu, L.; Cui, L.; Shi, Y. Data driven congestion trends prediction of urban transportation. IEEE Internet Things J.
**2017**, 5, 581–591. [Google Scholar] [CrossRef] - Poongodi, M.; Nguyen, T.N.; Hamdi, M.; Cengiz, K. Global cryptocurrency trend prediction using social media. Inf. Process. Manag.
**2021**, 58, 102708. [Google Scholar] - Huang, K.; Jiao, Z.; Cai, Y.; Candidate; Zhong, Z. Artificial intelligence-based intelligent surveillance for reducing nurses’ working hours in nurse–patient interaction: A two-wave study. J. Nurs. Manag.
**2022**, 30, 3817–3826. [Google Scholar] [CrossRef] - Lee, H.; Aydin, N.; Choi, Y.; Lekhavat, S.; Irani, Z. A decision support system for vessel speed decision in maritime logistics using weather archive big data. Comput. Oper. Res.
**2018**, 98, 330–342. [Google Scholar] [CrossRef] - Ba’Its, H.A.; Puspita, I.A.; Bay, A.F. Combination of program evaluation and review technique (PERT) and critical path method (CPM) for project schedule development. Int. J. Integr. Eng.
**2020**, 12, 68–75. [Google Scholar] - Kroll, A.; Moynihan, D.P. The design and practice of integrating evidence: Connecting performance management with program evaluation. Public Adm. Rev.
**2018**, 78, 183–194. [Google Scholar] [CrossRef] - Shen, X.; Fu, X.; Zhou, C. A combined algorithm for cleaning abnormal data of wind turbine power curve based on change point grouping algorithm and quartile algorithm. IEEE Trans. Sustain. Energy
**2018**, 10, 46–54. [Google Scholar] [CrossRef] - Zuckermann, M.; Hovestadt, V.; Knobbe-Thomsen, C.B.; Zapatka, M.; Northcott, P.A.; Schramm, K.; Belic, J.; Jones, D.T.W.; Tschida, B.; Moriarity, B.; et al. Somatic CRISPR/Cas9-mediated tumour suppressor disruption enables versatile brain tumour modelling. Nat. Commun.
**2015**, 6, 7391. [Google Scholar] [CrossRef] - Wang, G.; Zhao, B.; Wu, B.; Zhang, C.; Liu, W. Intelligent prediction of slope stability based on visual exploratory data analysis of 77 in situ cases. Int. J. Min. Sci. Technol.
**2023**, 33, 47–59. [Google Scholar] [CrossRef] - Wang, Y.; Chen, Q.; Hong, T.; Kang, C. Review of smart meter data analytics: Applications, methodologies, and challenges. IEEE Trans. Smart Grid
**2018**, 10, 3125–3148. [Google Scholar] [CrossRef] - Shortreed, S.M.; Ertefaie, A. Outcome-adaptive lasso: Variable selection for causal inference. Biometrics
**2017**, 73, 1111–1122. [Google Scholar] [CrossRef] - Gregorutti, B.; Michel, B.; Saint-Pierre, P. Correlation and variable importance in random forests. Stat. Comput.
**2017**, 27, 659–678. [Google Scholar] [CrossRef] - Polson, N.G.; Sokolov, V.O. Deep learning for short-term traffic flow prediction. Transp. Res. Part C Emerg. Technol.
**2017**, 79, 1–17. [Google Scholar] [CrossRef] - Krawczyk, B. Learning from imbalanced data: Open challenges and future directions. Prog. Artif. Intell.
**2016**, 5, 221–232. [Google Scholar] [CrossRef] - Hossin, M.; Sulaiman, M.N. A review on evaluation metrics for data classification evaluations. Int. J. Data Min. Knowl. Manag. Process
**2015**, 5, 1. [Google Scholar] - Kastouni, M.Z.; Lahcen, A.A. Big data analytics in telecommunications: Governance, architecture and use cases. J. King Saud Univ.-Comput. Inf. Sci.
**2022**, 34, 2758–2770. [Google Scholar] [CrossRef] - Paudel, S.; Elmitri, M.; Couturier, S.; Nguyen, P.H.; Kamphuis, R.; Lacarrière, B.; Le Corre, O. A relevant data selection method for energy consumption prediction of low energy building based on support vector machine. Energy Build.
**2017**, 138, 240–256. [Google Scholar] [CrossRef] - Kuo, R.J.; Wang, Y.C.; Tien, F.C. Integration of artificial neural network and MADA methods for green supplier selection. J. Clean. Prod.
**2010**, 18, 1161–1170. [Google Scholar] [CrossRef] - Moro, S.; Cortez, P.; Rita, P. A data-driven approach to predict the success of bank telemarketing. Decis. Support Syst.
**2014**, 62, 22–31. [Google Scholar] [CrossRef] - Lei, H.; Huang, K.; Jiao, Z.; Tang, Y.; Zhong, Z.; Cai, Y. Bayberry segmentation in a complex environment based on a multi-module convolutional neural network. Appl. Soft Comput.
**2022**, 119, 108556. [Google Scholar] [CrossRef] - Yu, Y.; Zhang, K.; Yang, L.; Zhang, D. Fruit detection for strawberry harvesting robot in non-structural environment based on Mask-RCNN. Comput. Electron. Agric.
**2019**, 163, 104846. [Google Scholar] [CrossRef] - Cheng, C.H.; Tsai, M.C.; Chang, C. A time series model based on deep learning and integrated indicator selection method for forecasting stock prices and evaluating trading profits. Systems
**2022**, 10, 243. [Google Scholar] [CrossRef] - Kapetanakis, D.S.; Mangina, E.; Finn, D.P. Input variable selection for thermal load predictive models of commercial buildings. Energy Build.
**2017**, 137, 13–26. [Google Scholar] [CrossRef] - Guo, L.; Li, N.; Jia, F.; Lei, Y.; Lin, J. A recurrent neural network based health indicator for remaining useful life prediction of bearings. Neurocomputing
**2017**, 240, 98–109. [Google Scholar] [CrossRef] - Yuan, T.; Zhu, N.; Shi, Y.; Chang, C.; Yang, K.; Ding, Y. Sample data selection method for improving the prediction accuracy of the heating energy consumption. Energy Build.
**2018**, 158, 234–243. [Google Scholar] [CrossRef] - Khan, N.M.; Abraham, N.; Hon, M. Transfer learning with intelligent training data selection for prediction of Alzheimer’s disease. IEEE Access
**2019**, 7, 72726–72735. [Google Scholar] [CrossRef] - Kuo, Y.; Yang, T.; Huang, G.W. The use of grey relational analysis in solving multiple attribute decision-making problems. Comput. Ind. Eng.
**2008**, 55, 80–93. [Google Scholar] [CrossRef] - Si, S.L.; You, X.Y.; Liu, H.C.; Zhang, P. DEMATEL technique: A systematic review of the state-of-the-art literature on methodologies and applications. Math. Probl. Eng.
**2018**, 2018, 3696457. [Google Scholar] [CrossRef] - Frederick, D.K.; DeCastro, J.A.; Litt, J.S. User’s Guide for the Commercial Modular Aero-Propulsion System Simulation (C-MAPSS); NASA: Cleveland, OH, USA, 2007. Available online: https://ntrs.nasa.gov/api/citations/20070034949/downloads/20070034949.pdf (accessed on 20 January 2023).
- Song, Q.; Shepperd, M. Predicting software project effort: A grey relational analysis based method. Expert Syst. Appl.
**2011**, 38, 7302–7316. [Google Scholar] [CrossRef] - Liu, S.; Lin Forrest, J.Y. Grey Systems: Theory and Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Costa, F.; Granja, A.D.; Fregola, A.; Picchi, F.; Staudacher, A.P. Understanding relative importance of barriers to improving the customer–supplier relationship within construction supply chains using DEMATEL technique. J. Manag. Eng.
**2019**, 35, 04019002. [Google Scholar] [CrossRef] - Liu, H.C.; You, J.X.; Shan, M.M.; Su, Q. Systematic failure mode and effect analysis using a hybrid multiple criteria decision-making approach. Total Qual. Manag. Bus. Excell.
**2019**, 30, 537–564. [Google Scholar] [CrossRef] - Li, P.; Xu, Z.; Wei, C.; Bai, Q.; Liu, J. A novel PROMETHEE method based on GRA-DEMATEL for PLTSs and its application in selecting renewable energies. Inf. Sci.
**2022**, 589, 142–161. [Google Scholar] [CrossRef] - Li, P.; Xu, Z.; Wei, C.; Bai, Q.; Liu, J. Revised DEMATEL: Resolving the infeasibility of DEMATEL. Appl. Math. Model.
**2013**, 37, 6746–6757. [Google Scholar] - Wang, Q.; Jia, G.; Song, W. Identifying critical factors in systems with interrelated components: A method considering heterogeneous influence and strength attenuation. Eur. J. Oper. Res.
**2022**, 303, 456–470. [Google Scholar] [CrossRef] - Fang, H.; Li, J.; Song, W. A new method for quality function deployment based on rough cloud model theory. IEEE Trans. Eng. Manag.
**2020**, 69, 2842–2856. [Google Scholar] [CrossRef] - Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Dodge, J.; Ilharco, G.; Schwartz, R.; Farhadi, A.; Hajishirzi, H.; Smith, N. Fine-tuning pretrained language models: Weight initializations, data orders, and early stopping. arXiv
**2020**, arXiv:2002.06305. [Google Scholar] - Reimers, N.; Gurevych, I. Reporting score distributions makes a difference: Performance study of LSTM-networks for sequence tagging. arXiv
**2017**, arXiv:1707.09861. [Google Scholar]

Factor | DEMATEL | Modified DEMATEL | ||
---|---|---|---|---|

$\mathit{P}\left(\mathit{i}\right)$ | Ranking | $\mathit{P}\left(\mathit{i}\right)$ | Ranking | |

F_{1} | 1.9557 | 4 | 1.7415 | 4 |

F_{2} | 2.5704 | 3 | 2.2887 | 2 |

F_{3} | 1.6218 | 5 | 1.2752 | 5 |

F_{4} | 2.7364 | 1 | 2.2372 | 3 |

F_{5} | 2.7312 | 2 | 2.4574 | 1 |

**Table 2.**Model outputs: inlink importance $\mu \left(i\right)$, outlink importance $\nu \left(i\right)$, Prominence $P\left(i\right)$, and Relation $R\left(i\right)$ for the 14 engine sensors.

Engine Sensor | $\mathit{\mu}\left(\mathit{i}\right)$ | $\mathit{\nu}\left(\mathit{i}\right)$ | $\mathit{P}\left(\mathit{i}\right)$ | $\mathit{R}\left(\mathit{i}\right)$ |
---|---|---|---|---|

ES_{1} | 1.0395 | 0.9942 | 2.0338 | −0.0453 |

ES_{2} | 1.0079 | 1.0060 | 2.0139 | −0.0019 |

ES_{3} | 0.9306 | 0.9762 | 1.9068 | 0.0456 |

ES_{4} | 1.0147 | 1.0029 | 2.0176 | −0.0118 |

ES_{5} | 1.0496 | 1.0147 | 2.0642 | −0.0349 |

ES_{6} | 1.0259 | 1.0248 | 2.0507 | −0.0011 |

ES_{7} | 0.9814 | 0.9961 | 1.9774 | 0.0147 |

ES_{8} | 1.0324 | 1.0144 | 2.0468 | −0.0180 |

ES_{9} | 1.0496 | 1.0147 | 2.0643 | −0.0350 |

ES_{10} | 1.0383 | 1.0281 | 2.0664 | −0.0101 |

ES_{11} | 0.9946 | 1.0028 | 1.9974 | 0.0083 |

ES_{12} | 0.9968 | 0.9957 | 1.9926 | −0.0011 |

ES_{13} | 0.9123 | 0.9626 | 1.8749 | 0.0503 |

ES_{14} | 0.9264 | 0.9668 | 1.8931 | 0.0404 |

Rank of Proposed Method | Engine Sensor | Optimal RMSE | RMSE |
---|---|---|---|

1 | ES_{10} | 16.051 | [16.051, 18.746] |

2 | ES_{9} | 14.847 | [14.847, 16.491] |

3 | ES_{5} | 14.778 | [14.778, 15.741] |

4 | ES_{6} | 14.693 | [14.693, 15.010] |

5 | ES_{8} | 14.134 | [14.134, 14.244] |

6 | ES_{1} | 13.991 | [13.991, 14.139] |

7 | ES_{4} | 13.860 | [13.860, 13.970] |

8 | ES_{2} | 13.797 | [13.797, 14.643] |

9 | ES_{11} | 13.797 | [13.797, 13.943] |

10 | ES_{12} | 13.502 | [13.502, 13.899] |

11 | ES_{7} | 13.478 | [13.478, 13.893] |

12 | ES_{3} | 13.398 | [13.398, 13.559] |

13 | ES_{14} | 13.387 | [13.387, 13.523] |

14 | ES_{13} | 13.240 | [13.240, 13.473] |

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## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Huang, K.; Goh, M.; Jiao, Z.; Jia, G.
Modified DEMATEL Method Based on Objective Data Grey Relational Analysis for Time Series. *Systems* **2023**, *11*, 267.
https://doi.org/10.3390/systems11060267

**AMA Style**

Wang Q, Huang K, Goh M, Jiao Z, Jia G.
Modified DEMATEL Method Based on Objective Data Grey Relational Analysis for Time Series. *Systems*. 2023; 11(6):267.
https://doi.org/10.3390/systems11060267

**Chicago/Turabian Style**

Wang, Qun, Kai Huang, Mark Goh, Zeyu Jiao, and Guozhu Jia.
2023. "Modified DEMATEL Method Based on Objective Data Grey Relational Analysis for Time Series" *Systems* 11, no. 6: 267.
https://doi.org/10.3390/systems11060267