在現實的工業製程中，一旦故障產生，可能會對整個製程造成影響，使得生產線的產量或是品質降低。如果無法及時地找出故障的元件，故障將會透過產物流向及回流路徑，影響製程中的其他元件，對整個程序造成負面的影響。為了避免此問題，對故障進行診斷並分析出其根本原因有其必要性，因此許多因果分析(causal analysis)方法開始被重視。格蘭傑因果關係(Granger causality)是一種已被廣泛運用於各種領域的因果分析方法。其基礎建立於迴歸分析中的自迴歸模型(Autoregressive model)，在一組時間序列變數X的自迴歸模型中引入另一變數Y的落後期項，並使用統計假設檢定，判斷其對於迴歸結果是否具有顯著改善，具有顯著差異時，判定兩組時間序列間有「格蘭傑因果關係」。但是格蘭傑因果關係具有很大的侷限性，例如它並沒有考慮時間序列間非線性的因果關係，也無法使用在非定態的時間序列，為了改善其對於不同類型故障診斷的表現，本研究使用高斯過程迴歸(Gaussian process regressian)，來建立非定態(nonstationary)或非線性系統的自迴歸模型，高斯過程迴歸對於非線性的系統擁有強大的擬和能力，另外為了避免其他正常的變數，對故障變數間因果關係的分析造成影響，還有過於複雜的因果關係，本文使用貢獻圖篩選出的故障變數群，再使用由高斯過程迴歸建模的格蘭傑因果關係對此變數群進行診斷，找出故障根本原因，以及對線性格蘭傑因果關係分析的結果使用最大生成樹，找到故障傳遞路線。
最後以模擬的非定態模型與田納西伊士曼製程(Tennessee Eastman process)作為研究案例，來展現原始與透過高斯過程迴歸建模的格蘭傑因果關係檢驗間對於故障發生根本原因診斷結果的差異，從模擬數據的分析結果中可看出本文的研究方法比原始的方法較能診斷出正確的故障根本原因，在田納西伊士曼製程的案例中，本文在原始的分析結果上使用最大生成樹，找出明確的故障根源以及傳遞路徑。

In industrial plants, productivity and product quality are often impacted by different types of faults. Specifically, oscillations commonly exist in many close-loop controlled processes. An oscillation generated in a single unit may propagate along process flows and feedback loops, affecting the performance of the entire plant. Therefore, it is critical to diagnose such oscillation-type plant faults and find out the root cause, so as to achieve fast recovery from abnormalities. In recent research, Granger causality (GC) test, which uses a statistical hypothesis test to judge whether a time series is useful in forecasting another, has been adopted to discover the root cause of plant-wide oscillations. However, the conventional GC is based on linear autoregressive (AR) models and cannot accurately handle the nonlinear causal relationship between time series. To solve this problem, the faulty variables are isolated by the contribution plot technique. Then, the nonlinear GC test based on Gaussian process regression (GPR) is conducted on the isolated process variables to discover the path of fault propagation and find out the root cause of the fault. Additionally, in process fault diagnosis, a large number of variables are often selected to be analyzed, making the causality map messy. Hence, it is not easy to identify the root cause of the fault from the analysis results. To solve this problem, the maximum spanning tree (MST) is adopted in this work to simplify the results of Granger causality analysis. The effectiveness of the proposed methods are illustrated using a simulated nonlinear model and the benchmark Tennessee Eastman process.

目錄
摘要 I
Abstract II
誌謝辭 III
目錄 IV
圖目錄 V
表目錄 V
第一章 緒論 1
1.1 前言 1
1.2 研究背景(文獻與動機) 2
1.3 文章結構 5
第二章 研究理論 6
2.1 主成分分析(Principal component analysis, PCA) 6
2.2 貢獻圖(Contribution plot) 10
2.3 格蘭傑因果關係(Granger causality) 11
2.4 單根檢定 15
2.5 高斯過程迴歸的因果關係 17
2.6 具體因果分析步驟 21
2.7 最大生成樹 22
第三章 非定態案例討論 24
第四章 非線性案例討論 30
4.1 田納西伊斯曼製程 30
4.2 震盪類型故障因果分析 34
4.2.1 IDV 1(A/C feed ratio, B composition constant) 34
4.2.2 IDV 7(C header pressure loss-reduced availability) 40
4.3 使用最大生成樹簡化因果關係 46
4.3.1 IDV 1(A/C feed ratio, B composition constant) 46
4.3.2 IDV 7(C header pressure loss-reduced availability) 51
第五章 結論 56
第六章 參考文獻 57

[1] C.W.J. Granger. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica, 37(3):424–438, 1969.
[2] C.W.J. Granger. Testing for causality: A personal viewpoint. Journal of Economic Dynamics and Control, 2(0):329–352, 1980.
[3] C.E. Rasmussen, C. Williams, Gaussian Processes for Machine Learning, The MIT Press, Cambridge, MA, USA, 2006.
[4] P. Duan, S. Shah, T. Chen, and F. Yang. Methods for root cause diagnosis of plant-wide oscillations, AIChE Journal, vol. 60, no. 6, 2019–2034, 2014.
[5] P.-O. Amblard, O.J.J. Michel, C. Richard, P. Honeine, A Gaussian process regression approach for testing Granger causality between time series data, in: Proc. ICASSP’2012 Osaka, 2012.
[6] T. Yuan, S.J. Qin. Root cause diagnosis of plant-wide oscillations using Granger causality. In: Proceedings of the 8th IFAC Symposium on Advanced Control of Chemical Processes. Furama Riverfront, Singapore, 160–165,2012.
[7] Zhengbing Yan. Variable Selection Method for Fault Isolation Using Least Absolute Shrinkage and Selection Operator (LASSO), Chemometrics and Intelligent Laboratory Systems, 2015.
[8] LIU, Yan and BAHADORI, Mohammad Taha (2012), “A Survey on Granger Causality: A Computational View” University of Southern California, http://www-bcf.usc.edu/~liu32/granger.pdf.
[9] Can-Zhong Yao. A study of industrial electricity consumption based on partial Granger causality network, Physica A: Statistical Mechanics and its Applications, Vol. 461, 629–646, 2016.
[10] F. Yang, D. Xiao, S.L. Shah. Signed directed graph-based hierarchical modelling and fault propagation analysis for large-scale systems. IET Control Theory Appl. 7:537–550, 2013.
[11] G. Weidl, A.L. Madsen, S. Israelson. Applications of object-oriented Bayesian networks for condition monitoring, root cause analysis and decision support on operation of complex continuous processes. Comput Chem Eng. 29:1996–2009, 2005.
[12] M. Bauer, J.W. Cox, M.H. Caveness, J.J. Downs, N.F. Thornhill. Finding the direction of disturbance propagation in a chemical process using transfer entropy. IEEE Trans Control Syst Technol. 15:12–21, 2007.
[13] C.W.J. Granger, Some recent development in a concept of causality, Journal of Econometrics 39:199–211, 1988.
[14] C. Hiemstra, J. Jones, Testing for linear and nonlinear granger causality in the stock price-volume relation, Journal of Finance 49:1639–1664, 2012.
[15] A.K. Seth, Causal connectivity of evolved neural networks during behavior, Network: Computation in Neural Systems 16:35–54, 2005.
[16] T.J. Mosedale,, D.B. Stephenson, M. Collins, T.C. Mills, Granger causality of coupled climate processes: Ocean feedback on the North Atlantic Oscillation, J. Clim., 19:1182–1194,2006.
[17] J.J. Downs, E.F. Vogel, A plant-wide industrial process control problem. Computers & Chemical Engineering, 17(3):245-255, 1993.
[18] S. Yin, S. Ding, A. Haghani, H. Hao, P. Zhang, A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process. Journal of Process Control, vol. 22, 1567-1581, 2012.
[19] D. Marinazzo, W. Liao, H. Chen, S. Stramaglia, Nonlinear connectivity by granger causality. Neuroimage, vol. 58, no. 2, 330-338, 2011.
[20] L. Barnett, A.K. Seth. The MVGC multivariate Granger causality toolbox: a new approach to Granger-causal inference J. Neurosci. Methods 223 50–68,2014.
[21] H. Jiang, R. Patwardhan, S.L. Shah. Root cause diagnosis of plantwide oscillations using the concept of adjacency matrix. J Process Control. 19:1347–1354, 2009.
[22] T. Schreiber. Measuring information transfer. Phys Rev Lett. 85:461–464, 2000.
[23] T.B. Marvell, C. E. Moody. Specification problems, police levels, and crime rates. Criminology, 34, 609-646,1996.
[24] C.W.J. Granger, P.Newbold. Spurious regressions in econometrics, Journal of Econometrics 2, 111–120,1974.
[25] F. Engle, C.W.J. Granger. Co-integration and error correction: representation, estimation and testing. Econometrica. 55:251–76, 1987.
[26] J. Edmonds, Optimum branchings, J. Research of the National Bureau of Standards, 71B, 233-240,1967.
[27] D. Peng, X. Gu, Y. Xu, and Q. Zhu. Integrating probabilistic signed digraph and reliability analysis for alarm signal optimization in chemical plant. Journal of Loss Prevention in the Process Industries, vol. 33, 279-288, 2015.
[28] C.Z. Yao, J.N. Lin, Q.W. Lin, X.Z. Zheng, X.F. Liu. A study of causality structure and dynamics in industrial electricity consumption based on Granger network, Physica A 462:297–320,2016.
[29] D,A, Dickey, W.A. Fuller.Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association，74，427–431,1979.
[30] Kwiatkowski, D.; Phillips, P. C. B.; Schmidt, P.; Shin, Y. (1992). "Testing the null hypothesis of stationarity against the alternative of a unit root". Journal of Econometrics. 54 (1–3): 159–178.
[31] Sofiane Brahim-Belhouari and Amine Bermak. Gaussian process for nonstationary time series prediction. Computational Statistics & Data Analysis, 47(4):705-712, Nov. 1 2004.
[32] J.S. Qin, S. Valle, M.J. Piovoso, On unifying multiblock analysis with application to decentralized process monitoring, Journal of Chemometrics 15 (2001) 715–742.
[33] E.L. Russell, L.H. Chiang, R.D. Braatz, Fault detection in industrial processes using canonical variate analysis and dynamic principal component analysis, Chemom. Intell. Lab. Syst. 51 (2000) 81–93.
[34] Geweke J. Measures of conditional linear dependence and feedback between time series. J Am Stat Assoc 1984;79:907–15.
[35] Cheng, C.Y., Hsu, C.C., Chen, M.C., 2010. Adaptive kernel principal component analysis (KPCA) for monitoring small disturbances of nonlinear processes. Industrial and Engineering Chemistry Research 49, 2254–2262.