Modified Independent Component Analysis and Bayesian Network

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Modified Independent Component Analysis and Bayesian Network based Two-stage Fault Diagnosis of Process Operations Hongyang Yu, Faisal Khan, and Vikram Garaniya Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie503530v • Publication Date (Web): 26 Feb 2015 Downloaded from http://pubs.acs.org on March 8, 2015

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Industrial & Engineering Chemistry Research

Modified Independent Component Analysis and Bayesian Network based Two-stage Fault Diagnosis of Process Operations Hongyang Yu, Faisal Khan * and Vikram Garaniya National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, TAS, 7250, Australia

Abstract Statistical fault detection techniques are able to detect fault and diagnose root-cause(s) from the monitored process variables. For complex process operations, it is not feasible to screen all the process variables due to monitoring cost and flooding of alarms. Thus if a fault is originated from a process variable that is not monitored, conventional statistical techniques are incapable of locating the true rootcause. To relax this limitation, a two-stage fault diagnosis technique is proposed for process operations. In the first-stage, the modified independent component analysis is used for fault detection and to identify the faulty monitored variable. In the second-stage, a Bayesian Network model is constructed considering the process variables and their dependence obtained from the process flow diagram. Evidence is then generated at the network node corresponding to the faulty variable identified in the first-stage. Subsequently, the network is updated and analyzed using deductive and abductive reasoning to identify the true root-cause. To verify the applicability of the proposed technique it is tested on two process models. The results of both case studies have demonstrated the effectiveness of the proposed technique to diagnose the true root-cause originated from process variables that are not monitored. Once integrated with process loss functions, the proposed technique will serve as an important element of dynamic operational risk management framework.

Keywords: Process operations, fault diagnosis, modified independent component analysis, Bayesian network

*

Corresponding author: [email protected]

ACS Paragon Plus Environment

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Symbols and Acronyms Symbol

Description

A

Mixing Matrix

C

Projection matrix

D

Eigenvalue matrix

E

Residual Matrix

Q

Whitening matrix

S

Independent Component Matrix

Sˆ

Estimated Independent Component Matrix

t

PC score vector

V

Eigenvector matrix

W

Normalized demixing matrix

X

Process data matrix

y

Projected data vector

Z

Whitened data matrix

Θ

Bayesian network parameters

ˆ Θ

Estimated Bayesian network parameters

I2

I2 statistics

Q

SPE statistics Squared prediction error

CPDF

Conditional probability density functions

CPT

Conditional probability table

s0

Faulty state 0

s1

Normal state 1

µ x→ f

Sum-Product Message passed from node x to factorial node f

δ x→ f

Max-Product Message passed from node x to factorial node f


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Introduction Modern industrial processes are large-scale systems that comprise of many operating units and

multiple processing steps to produce high quality products. To ensure the safety of production and the personnel involved, industrial processes are monitored on a real-time basis. This requires the online measurement of a large number of process variables associated with various process components. Due to the complex nature of process operation, functions that govern the relationship amongst the process variables are often high-order and nonlinear, and are difficult to obtain explicitly. As a result, the conventional first-principle-model-based process monitoring techniques become less suitable 1. To relax this limitation, multivariate statistical process monitoring (MSPM) techniques have been proposed to extract features or latent variables from the highly-correlated and high-dimensional process data to detect and diagnose various faults of industrial processes 2-7. Principle Component Analysis (PCA) and Partial Least Square (PLS) are the most extensively used statistical feature extraction techniques for process monitoring 2, 8-11. These techniques implicitly assume that process variation follow a multivariate Gaussian distribution and determine a set of orthogonal projection vectors called loading vectors. Through these loading vectors, process data can be projected into a subspace or feature space with lower dimensionality

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directions of most significant variability of the process data

13-14

. These loading vectors represent the . In addition, due to the orthogonal

transformation, the cross-correlation (linear-correlation) between the process variables are removed so that a new set of pairwise independent variables known as the principal components (PCs) are formed into the feature space

15-16

. In this regard, PCA and PLS only manipulate the second-order statistics

(correlation and cross-correlation) of the process data

17-18

. For process monitoring, the Hotelling’s T2

statistics and squared prediction error (SPE) statistics of the PCs are computed to detect process abnormalities. The T2 and the SPE statistics have different physical meaning and cover different aspects of fault detection. T2 measures the correlated distance from the center of the feature space to the projected data sample. On the other hand, the SPE statistics is a L2 norm which measures the Euclidean distance from the residual space to the PC feature space. In other words, the T2 statistics measure the systematic variation among data while the SPE statistics measure the residual variation. The control limits for both statistics are derived based on the assumption that the PCs follow Gaussian distribution, in particular for SPE, a standard normal distribution 19. Multivariate contribution plots are also generated based on these two statistics to isolate the root-cause variable. However, in complex industrial processes, the behavior of a process variable can be affected by a number of other process variables. The


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second-order statistics that describes only pairwise relationship can become inadequate for feature extraction. Furthermore, the process data do not always follow Gaussian distribution due to process nonlinearity and external disturbance. As a result, PCA/PLS based techniques may produce misleading results for performance monitoring of complex industrial processes 20. Independent Component Analysis (ICA) has been proposed to address the limitation of PCA/PLS based techniques

21

. ICA determines a set of non-orthogonal demixing vectors through which the

process data can be transformed into a subset of independent components (ICs) that have minimum mutual information

22-23

. Mutual information is a measure of difference between the joint distribution

and the marginal distributions of the ICs

24

. Thus, mutual information takes into account the complete

dependence structure of the latent variables rather than second-order dependence of the PCA

22

.

Moreover, mutual information is equivalent to the well-known kullback-Leibler divergence that measures the difference in entropy between the joint distribution and marginal distributions of the latent variables 25. To minimize the entropy difference, the latent variables have to be not only as independent as possible but also as non-Gaussian as possible 22. In this regard, ICA explores high-order statistics and retains non-Gaussian features of the process; thus, it yields better results as compared to PCA for complex process monitoring 17, 21. However, one of the major drawbacks of the conventional ICA is that the extracted ICs are of the same importance. It is therefore difficult to determine the dominant ICs for dimensionality reduction. The modified ICA is then developed to solve this problem by preserving the ranking of PCs in the PCA whitening step 26. Subsequently, the ICA version of T2, also known as the I2 statistics, and the SPE statistics similar to PCA are computed for process monitoring. Since the ICs do not follow Gaussian distribution, the kernel density estimation is adopted to estimate the control limits for these two statistics 21. Similar to PCA, multivariate contribution plots can be generated based on the I2 and SPE statistics to locate the root-cause process variable. PCA, PLS and ICA do not require any prior knowledge of the process but rely heavily on availability of online monitored data. These techniques are inadequate to isolate root-cause from process variables that are not monitored. In industrial practice, abundant number of monitored variables may substantially increase rate of false alarms. In addition, some of the process variables are very costly to monitor as they may be associated with operating units that are located in congested space and require very sophisticated measuring instruments. Furthermore, the malfunction of these measuring instruments can produce misleading results that disguise the true root-cause of the process abnormality. To overcome the above problems, the number of monitored process variables is restricted, and therefore cannot cover full


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aspects of the process operation. In this case, statistical data-driven techniques may not be able to point to the true root-cause of the process abnormality. Bayesian Network (BN) can be used as an efficient tool to allay the limitation of statistical data-driven techniques. The BN utilizes the prior process knowledge to construct directed-acyclic-graphic representation of the process

27

. Unlike the

conventional model-based techniques, BN require only the causal relationship among the process variables which is relatively easy to obtain by analyzing the process flow diagram. Each process variable is represented as a random variable node in the graphic structure. These nodes are connected by arcs that describe the casual dependence amongst the process variables. In addition, the statistical dependence among the process variables is quantified by the probabilistic measure of influence of one process variable on another. Two types of logical reasoning are incorporated with BN, namely deductive reasoning and abductive reasoning

28

. Deductive reasoning allows inference of the states of the

unobserved process variables given the state of an observed process variable. On the other hand, abductive reasoning, also known as the most probable explanation (MPE) 29, determines the most likely combination of states of various unobserved process variables that best explain the state of the observed process variable. Utilizing these reasoning mechanisms, BN is able to isolate the true root-cause from unobserved (not monitored) process variables. In this study, a combination of modified ICA and BN is applied for two-stage fault diagnosis technique of industrial processes. Here, the modified ICA identifies the faulty monitored variable and is used as the evidence node in BN for finding the true root-cause amongst the intermediate variables through deductive and abductive reasoning approaches. The proposed two-stage fault diagnosis technique is demonstrated with two illustrative examples. 2

Background

2.1

Independent Component Analysis

ICA is able to extract statistically independent components from highly-correlated and highdimensional data and has been widely applied for blind source separation and signal separation

30

. In

recent years, ICA has also been extensively applied to monitor complex industrial processes. ICA performs better than many conventional techniques due to its ability to extract non-Gaussian features which are considered to be dominating in modern processes 31-33. For process monitoring, the monitored process data can be considered as a linear combination of signals that are generated from a subset of independent sources or latent variables. Suppose a process data matrix X ∈ R d ×n having d process


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variables and n samples is generated from the normal operating condition of a process. The ICA decomposition of X is expressed as

X = AS + E Where

A

and

E

are

the

mixing

matrix

(1) and

the

residual

matrix

respectively.

S = [ s1n , s2n , s3n ,..., s np ] ∈ R p×n is the independent component matrix consisting of p (p

Modified Independent Component Analysis and Bayesian Network

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