Nonlinear and Non-Gaussian Process Monitoring Based on Simplified

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Process Systems Engineering

Nonlinear and Non-Gaussian Process Monitoring Based on Simplified R-vine Copula Nan Zhou, and Shaojun Li Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00701 • Publication Date (Web): 16 May 2018 Downloaded from http://pubs.acs.org on May 20, 2018

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Industrial & Engineering Chemistry Research

Nonlinear and Non-Gaussian Process Monitoring Based on Simplified R-vine Copula Nan Zhou, Shaojun Li* Key Laboratory of Advanced Control and Optimization for Chemical Processes, East China University of Science and Technology, Ministry of Education, Shanghai 200237, China Abstract: In the field of chemical process monitoring, the vine copula model provides a new idea for describing

the interdependence between high-dimensional complex variables, and directly characterizes the correlation

without dimensional reduction. However, in actual industrial processes, the number of pair copulas to be optimized

and the parameters to be estimated increase rapidly when the dimensionality of the variables is large. This greatly

increases the computational load and reduces the detection efficiency. In this paper, a fault diagnosis method based

on a Simplified R-vine (SRV) model is proposed. Without reducing the precision of the model significantly, the

simplified level is set to reduce the complexity of the workload and calculations. The simplified level of an R-vine

model is obtained by a Vuong test. Then, the generalized local probability (GLP) of the non-Gaussian state is

constructed by using the theory of highest density region (HDR) and a density quantile table. The monitoring

results of the Tennessee Eastman (TE) process and a real acetic acid dehydration distillation system show that the

proposed SRV approach achieves good performance in monitoring results and computational load for chemical

process fault monitoring. Keywords: process monitoring; nonlinear and non-Gaussian; Vuong test; SRV copula; highest-density region

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1 Introduction In modern chemical process, process safety and product quality are two important issues of interest. In order

to improve product quality and efficiency and to ensure the safety of production, it is crucial to establish an

effective real-time process monitoring method. Generally, process monitoring can be divided into three categories: analytic model-based methods, knowledge-based methods, and data-based methods.(1)-(4) Analytic model-based

methods need to establish an accurate mathematical model of the process, which is unrealistic for complex

chemical processes. Knowledge-based methods have poor generality and rely on detailed operational experience

and prior knowledge. However, data-driven methods take the process measurement data as the research object,

rather than an exact model of the process and abundant prior knowledge.

In recent years, the DCS system and data acquisition system, which have significantly increased the popularity of data-driven process monitoring methods, have been widely used in chemical plants.(5),(6) Most of the multivariate statistical process monitoring (MSPN)(7),(8) methods use multivariate projection techniques to map

high-dimensional data to low-dimensional space to represent the overall information of the data with as few variables as possible. These methods, including principal component analysis (PCA)(9) and Partial Least Square (PLS),(10) usually assume that the process variables are linearly correlated or subject to a Gaussian distribution.

However, the practical processes often show nonlinear or non-Gaussian characteristics.

To deal with these problems, some improved methods for traditional PCA and PLS, and some new methods, have been proposed. For example, a nonlinear process method based on kernel KPCA,(11) the improved KPCA-based the technique of local approach,(12) a nonlinear PCA method based on neural networks,(13) the kernel partial least squares (KPLS)(14) method, etc., perform well in tackling nonlinear problems. Kano et al.(15) proposed the independent component analysis (ICA) method to solve the non-Gaussian problem. Based on this, Lee et al.(16)

defined a monitoring index of the ICA model. A two-step ICA-PCA based monitoring approach has been proposed


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both fault detection and identification by Ge et al.,(17) and then was adopted for time-varying and batch process monitoring.(18) Jie et al.(19) used a limited number of Gaussian elements to characterize the distribution model of

data, and solved the problem of describing non-Gaussian problems. In order to reduce the dimensionality of the process data, the GMM model is sometimes combined with the traditional PCA model.(4) All of the methods

mentioned above tend to adopt the idea of dimensionality reduction or Gaussian assumption, which may result in

information loss. Therefore, it is of great value to provide an efficient process-monitoring strategy by describing

the intricate dependencies between variables directly.

As an efficient statistical tool in dependence modeling, a copula combining the joint distribution function with

the marginal distribution function and its correlation structure has become increasingly popular in the fields of economics,(20) finance,(21) and meteorology.(22) Recently, the copula has received much attention in chemical process systems engineering. Anjana et al.(23) proposed a dynamic failure assessment method based on Bayesian

theory, which used a copula function to estimate the failure probability of the safety system in the chemical process. In the safety, quality, and operability systems (SQOSs) of chemical processes, Ankur Pariyani et al.(24) used a

Cuadras-Auges copula and multivariate Gaussian copula to characterize the nonlinear correlation of process variables, and considered the impact of operator behavior factors. Yu et al.(25) used the Rolling Pin method and a

Gaussian copula to establish a nonlinear dependence model between variables. However, the copula theory is limited by the cumbersome and inefficient optimization of the traditional multivariate copula.(26) To address this issue, Joe et al.(27) proposed the vine copula method to analyze various dependence structures.

The vine copula is actually a hierarchical model with bivariate copulas as simple building blocks. By using a vine

copula, multivariate dependence problems are simplified to a series of optimization problems of bivariate copulas

problems, which significantly promotes the application of copula theory in chemical process systems modeling. Ren et al.(28) proposed a novel vine-copula-based dependence description (VCDD) monitoring approach using a



C-vine copula to establish a correlation model among process variables, and constructed a generalized Bayesian

inference-based probability (GBIP) index under a given control limit, to achieve real-time fault detection of chemical processes. Zheng et al.(29) proposed a mixture of a vine copula by incorporating a D-vine copula into a

finite mixture (MDVC) model, which can reveal and fully extract the complex and hidden dependence patterns in multivariate data. However, the two vine-copula (C/D-vine) models(30)(31) have their own application scope.

Strongly correlated variables are suitable for a C-vine model with a star structure. By contrast, if the correlation between variables is weak, the linear D-vine decomposition structure is more effective.(32) Generally, a practical

chemical process often shows extreme complexity, and it is difficult to distinguish the difference of the correlation

relationship between variables. Therefore, providing an efficient and flexible vine copula model to describe the

complex dependencies between variables is necessary. Bedford-Cooke(33) proposed a highly effective R-vine decomposition model with more widespread

application. Compared with C/D-vine, the R-vine model is more flexible and accurate in describing the actual

behavior of the variables, and can fully explore the correlation information among random variables. Since it is not

a prespecified structure like C/D-vine, the construction process for an R-vine structure can take a considerable amount of time to complete. To overcome this problem, Kurowicka et al.(34) proposed a truncated R-vine (TRV)

copula, which identifies an appropriate truncation level, that is, the number of vine trees to be used in the model.

According to a truncated regular vine at level K, all R-vine bivariate copulas with conditioning set equal to or

larger than K are replaced by independent copulas. Such a PCA-like approach will inevitably result in the loss of

some information in trees larger than K. Later, an improved TRV copula was studied by Brechmann et al.(35),(36) This copula replaced independent

copulas with a kind of bivariate copula at level K. This is called a simplified R-vine (SRV) copula. As the number

of trees increases, the bivariate copula gradually becomes independent because the high-level trees contain less


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information and weaker relevance. Since bivariate Gaussian copulas are easy to specify and are sufficient to

describe this progressive independence, SRV usually replaces all pair copulas in high-order trees with bivariate

Gaussian copulas. Through simplification, it sharply reduces the computational load since the optimal selection

and parameter estimation of bivariate copulas are only for trees below K.

In this study, a process monitoring approach based on the simplified R-vine (SRV) copula is proposed. The

establishment of an SRV probabilistic model for complex industrial processes can fully describe the non-Gaussian

and nonlinear behavior of process variables, and can also significantly improve the monitoring performance.

Taking the strongest dependency as the criterion, the tree structure is constructed by using the maximum spanning

tree algorithm under different simplified levels. By selecting fixed bivariate Gaussian copulas for trees equal to or

larger than K and optimizing bivariate copulas only for the others, the most appropriate SRV model can be

obtained by using a Vuong test. Furthermore, the highest-density region (HDR) and an efficient mathematical

technique called the density quantile approach (DQA) are employed to construct a generalized local probability

(GLP) index, which realizes real-time chemical process monitoring. In this study, KPCA and FGMM methods are

selected for performance comparisons since they are typical process monitoring methods to deal with nonlinear

and non-Gaussian problems. Applications in the TE process and a real acetic acid dehydration distillation system

demonstrate that the proposed method can achieve good performance in both monitoring results and computational

load.

The rest of this paper is organized as follows. In Section 2, we provide necessary background on SRV,

including pair copula construction, conditional distribution calculation, and identification of the simplified level. In

Section 3, a type of SRV monitoring approach that includes a brief description of DQA and the pair copula selection process is further developed. Compared with the KPCA(11) method, finite Gaussian mixture model (FGMM)(19) approach, and VCDD(28) method, the validity and effectiveness of the proposed approach are



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illustrated in a TE process and a real acetic acid dehydration distillation system in Section 4. Section 5 contains

some concluding remarks.

2 Preliminaries A copula is a multivariate joint distribution function with a uniformly distributed marginal. It provides a

dependence structure to combine the corresponding multivariate cumulative distribution function (CDF) with the univariate cumulative marginal distributions.(37) According to the Sklar theorem,(38) a d-dimensional joint

distribution can be decomposed into d marginal distributions and a d-dimensional copula function. Let F be a joint CDF of d-dimensional random vector X = ( X1 , X 2 ,L X d ) . There exists a copula function C : [0, 1]d→[0, 1] such

that F ( x1 , L , x d ) = C ( F1 , F2 , L , Fd )

(1)

where Fi (xi) denotes the marginal CDF of random vector Xi, which satisfies

Fi ( x i ) =

∫

xi 0

fi ( xi ) d xi ,

Fi ( x i ) ∈ [0 ,1]

(2)

where fi (xi) represents the probability density function (PDF). By taking derivative on both sides of Eq. (2), the joint PDF f ( x ) can be expressed as the product of f i ( xi ) and the copula density function c: d

f ( x ) = c ( F1 , F2 , L , Fd )∏ f i ( xi )

(3)

i =1

By using the maximum likelihood method to estimate the parameters in Eq. (3), the joint PDF of

multidimensional data can be obtained. Obviously, it will become more difficult to fit the copula density function

with the increase of variable dimension, thus causing a curse of dimensionality. Fortunately, the vine copula

method proposed by Joe et al. presents a solution to the optimization problems of multivariate copula.

2.1 R-vine copula A multivariate joint distribution function can be transformed into a combination of bivariate copulas as simple

building blocks and marginal distributions by using a vine copula. C-vine, D-vine, and R-vine are the three most

popular cases of vines, and are shown graphically in Figure 1. Compared with the C/D-vine, the structure of the


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R-vine in Figure 1 (c) is more complicated. The R-vine has no specific structure because it not only contains the

star structure of a C-vine but also the linear structure of a D-vine. In fact, the C-vine and D-vine are just two

special structures of the R-vine.

(a)

C-vine with five

variables

(b)

D-vine

with five variables

(c)

R-vine with five variables

Figure 1. Illustration of five-dimensional trees with edge indices for C -vine, D-vine, and R-vine

For the sake of convenience, we define the node set E = { E 1 , E 2 , L , E d − 1 } , and each edge e j = j (e), k (e) | D (e)

N = { N 1 , N 2 ,L , N d − 1 }

and the edge set

in E j is associated with a bivariate copula density

c j ( e), k ( e)| D (e ) , where j ( e) and k (e) are nodes under condition set D ( e) . For a given d-dimensional random vector X = ( X1 , X 2 ,L X d ) , the R-vine copula specification with density is uniquely determined as follows: d

f ( x1 , x2 ,L , xd ) = c[ F1 ( x1 ), F2 ( x2 ),L , Fd ( xd ),θ ]∏ f i ( xi ) i

where c is defined as


(4)


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d -1

c ( F1 ( x1 ), F2 ( x 2 ),L , Fd ( x d )) = ∏ ∏ c j ( e ), k ( e )| D ( e ) [ F ( x j ( e ) | x D ( e ) , F ( x k ( e ) | x D ( e ) ),θ D ( e ) ]

(5)

i =1 e∈ Ei

xD(e) is the subvector of x determined by the indices contained in condition set D ( e) , θ(e) denotes the parameter(s) of the copula of c j ( e ), k ( e )| D (e ) , and F ( x j ( e) | xD ( e) ) is the conditional distribution function of x j ( e ) given xD ( e ) . Here, we use the partial differentiation of a bivariate copula proposed by Aas et al.(39) to calculate

these conditional distribution functions in Eq. 5. This is given as Eq. 7: Fx j | D ( x j | xD ) =

∂Cx j ,v | D− v ( Fx j | D− v ( x j | xD− v ), Fxv | D− v ( xv | xD− v );θ x j ,v|D ) −v

∂Fxv | D− v ( xv | xD− v )

(7)

where x j and xv are random scalars of x D (here, xv ∈ xD and xD− v is a column vector with xv removed), Fx j | D− v ( x j | x D− v ) denotes the conditional distribution function of x j given xD− v , and Cx j ,v |D− v is the copula

function corresponding to a bivariate copula density cx j ,v |D− v with the related parameter θ x j ,v|D . −v

For the convenience of recording the indices of bivariate copulas required in the joint PDF in Eq. 4 and condition set D ( e) , Dißmann et al.(20),(34) proposed a structure matrix method that involved the specification of a lower triangular matrix M = (mij | i, j = 1,2,L, d ) where M ∈ R d × d and mij ∈ {1, 2,L , d } . From the bottom up,

each row represents a tree identified by the corresponding column entry of the row and the diagonal entries of M, in which the diagonal entries are a sequence of the numbers 1, 2, L , d , and the condition set D ( e) consists of

the column elements below this row. In this way, we can construct matrices corresponding to M to record copula

types and parameters conveniently.

2.2 Selection and parameter estimation for bivariate copulas under the specified R-vine copula tree structure For low-dimensional data, we can traverse all of the possible R-vine structures to select the most suitable one.

However, the number of different possible R-vine structures will be very large with an increase in dimension. In order to build an effective structure quickly, Aas-Brechmann(35) used the maximal spanning tree (MST) method


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based on the strongest dependence to select reasonable R-vine copula trees. The specific optimization problem is

as follows:

∑

τ = max

| τ jk | D |

(8)

e ={ j , k | D } in Ti

where τ jk | D represents the Kendall τ

rank correlation coefficient of random variables X j and X k under the

condition set D. In particular, the condition set is empty for tree T1 . We construct the first tree that maximizes the sum of τ jk for all possible variable pairs

{ j , k} (1 ≤

j ≤ k ≤ d ) , and then select a bivariate copula family and

estimate the corresponding parameter(s). The conditional distribution functions

Fx j | D1 ( e ) ( x j | xD1 ( e ) ) and

Fxk | D1 ( e) ( xk | xD1 ( e) ) based on the condition set D1 determined by tree T1 can be calculated by Eq. 7, termed as

{ j , k | D1 (e)} (1 ≤

j ≤ k ≤ d − 1) . Therefore, the second tree T2 is obtained by maximizing the sum of τ jk |D1 (e ) for

all possible variable pairs

{ j , k | D1 (e)} (1 ≤

j ≤ k ≤ d − 1) . After that, the selection of a bivariate copula family and

estimation of the corresponding parameters proceed for each edge. The optimization process continues until the

last tree is fitted. Here, we use the maximum pseudo likelihood-based Akaike information criterion(40) to select the bivariate

copula family and estimate the corresponding parameters. This is described as follows: N  (θˆx j | D , λˆx j | D ) = arg max  ∑ log  c ( Fxkj | D− v ( x j | xD− v ), Fvk| D− v ( xv | xD− v );θ x j|D , λx j|D )  − γ    θ x j |D  k =1  λ

(8)

x j |D

where N is the number of data points, Fxk | D ( x j | x D ) represents the kth sample value of the conditional j

−v

−v

distribution function F ( x j ( e) | xD ( e) ) , and θ x j | D and λx j |D denote the parameter and family order of the corresponding pair copula, respectively. γ represents the number of parameters of a specific bivariate copula. In general, γ = 1, 2 .



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2.3 Simplification of R-vine When the dimensionality of the variables increases, the high-level trees contain less information and weaker

correlation. This means that most of the information in the R-vine model is captured in the first several trees.

Hence, by using simple bivariate copulas to fit the high-level trees, we can get the simplified R-vine model.

Here, we use bivariate Gaussian copulas to replace all bivariate copulas with conditioning sets equal to or

larger than K, where K is a level that determines which trees should be simplified. There are two reasons for

choosing bivariate Gaussian copulas. First, the bivariate Gaussian copula has a simple form that is very easy to

specify. The second reason is that the tail property of the bivariate Gaussian copula is progressive independence,

which is suitable for describing the dependence of the high-order trees. In particular, the bivariate Gaussian copula

becomes a bivariate independence copula when the corresponding parameter is equal to 0. In order to facilitate the description, we denote a simplified R-vine at level K by SRV(K) and let θSRV (K )

represent the corresponding parameters. Then, the density of an R-vine at simplified level K for a d-dimensional

random vector can be given as follows:  K   d −1  K cSRV (u | θ SRV ( K )) = ∏∏ c j ( e ), k ( e )| D ( e )  ×  ∏ ∏ c ρj ( e ), k ( e )| D ( e )   i =1 e∈Ei   i = K +1 e∈Ei 

(8)

K where the first term on the right side of the equation denotes the joint density cSRV of the first K trees of an

ρ R-vine, and the second term denotes the joint density cSRV of the simplified trees fitted by bivariate Gaussian

copulas with correlation parameter ρ j ( e ), k ( e)| D ( e) . The parameter set θ SRV ( K ) of SRV(K) contains two parts: one is K ρ for cSRV , and the other is for cSRV . This is rewritten as

θ SRV ( K ) = {θ j , j + i|1,L, j −1 : j = 1,L, K , i = 1,L, d − j} U { ρij |1,L, K : i, j = K + 1,L, d − 1, i ≠ j}

(9)

Usually, AIC and BIC tests(41) and the Vuong test(42) are used to evaluate this information gain from SRV(K)

to SRV(K + 1). According to Eq. 8, the log likelihood for SRV(K) is given by


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N

K

lSRV ( K ) (u | θ SRV ( K )) = ∑∑ ∑ ln c j ( e ), k ( e )| D (e ) [ F ( xt , j (e ) | xt , D( e ) , F ( xt , k ( e ) | xt , D( e ) )] t =1 i =1 e∈Ei N

+∑

d −1

∑ ∑ ln c ρ

j ( e ), k ( e )| D ( e )

t =1 i = K +1 e∈Ei

(10) [ F ( xt , j (e ) | xt , D( e ) , F ( xt , k ( e ) | xt , D( e ) )]

The AIC/BIC values of the given SRV(K) are defined as AIC ( SRV ( K )) = −2lSRV ( K ) (θˆSRV ( K ) | u ) + 2nSRV ( k )

(11)

BIC ( SRV ( K )) = −2lSRV ( K ) (θˆSRV ( K ) | u ) + nSRV ( k ) ln( N )

where nSRV (k ) and N are the number of the parameters θ SRV ( K ) and observations, respectively. We choose the

simplified model with level K if the value of SRV(K) is smaller when compared with SRV(K + 1). Note that AIC

and BIC are only used when comparing nested models where the “reduced” model is a special case of this “full” model and is parametrized in a subset of these parameters.(35) However, the simplified models SRV(K) and SRV(K + 1) are not nested. This means that SRV(K) ⊄ SRV(K + 1), and the AIC/BIC tests are not available in this case.

Here, the Vuong test is used to compare the simplified models. A Vuong test based on a likelihood ratio is proposed to compare two models f1 and f 2 with estimated parameters θˆ1 and θˆ2 , respectively. For each observation, the log difference of likelihoods between these two  f ( x | θˆ )  models is defined as mi = ln  1 i 1  . Then, we can compute the standardized sum v as Eq. 12:  f 2 ( xi | θˆ2 ) 

(1 N ) ∑ i =1 mi N

v=

∑

(12)

N

(mi − m)2 i =1

where m is the mean of { m i : i = 1, 2, L , N } . Model f1 is superior to model f 2 if v > Φ −1 (1 −

v < −Φ −1 (1 −

v ≤ Φ −1 (1 −

α 2

α 2

α 2

) , else if

) , the model f 2 is better. However, there is no decision on which model is suitable if

) . Here, Φ −1 denotes the inverse of the standard normal distribution function.

Similarly, for the models SRV(K) and SRV(K + 1), if v ≥ −Φ −1 (1 −

α 2

) , we can get the optimal simplified

R-vine at level K because SRV(K) is preferred to, or is indistinguishable from, SRV(K + 1) at significance level

α .(35) Now, detailed procedures to determine the simplified level are given as follows: Step 1. Collect a set of observations and determine the significance level α . Set i = 1.



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Step 2. Construct SRV(i) and SRV(i + 1) fitted by appropriate bivariate copulas.

Step 3. Execute a Vuong test between SRV(i) and SRV(i + 1). Evaluate the statistic v defined in Eq. (12). Step 4. If v ≥ −Φ −1 (1 −

α 2

) , then exit the loop with SRV(K) and simplify the R-vine model at level K = i.

Otherwise, perform i = i + 1 and go back to Step 2 until the satisfied simplified level is obtained.

3 Process fault monitoring based on simplified R-vine copula The R-vine copula is an efficient statistical tool that describes nonlinear and non-Gaussian dependencies and

can build a dependence structure between variables more flexibly. In this section, we apply a simplified R-vine

copula approach to the multivariate process monitoring field. Using the density quantile approach (DAQ) and the

highest density region (HDR), we construct a generalized local probability (GLP) index and realize real-time

chemical process monitoring.

3.1 Probability Calculation Using HDR and DAQ, and Construction of GLP index The highest-density region is a valid probabilistic distance measurement for arbitrary distribution that can

cover the sample space with the smallest possible volume for a given probability 1 - α. Every point inside the

region has a probability density at least as large as every point outside the region. Now, we give the specific definition from Hyndman.(43) Let f ( x ) be the joint PDF of random variable X. Then, the highest-density region with probability 1 - α is a subset of the given sample space R ( f δ ) of X such that R ( fδ ) = { x | f ( x ) ≥ fδ }

(13)

where fδ is the largest constant where P [ X ∈ R ( f δ )] ≥ 1 − δ , and 1 − δ represents the confidence level. For a given distribution function, the numerical integration approach (NIA)(44) and the density quantile approach (DQA) calculate the joint PDF in a specific HDR. According to Ren et al.,(28) NIA has a great deal of

computational complexity for high-dimensional data and requires a complete quantile method of the integral

region, which causes great trouble in the practical estimation process. The DQA method can avoid these


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disadvantages since it is focused on obtaining the information on the probability coverage of a given region of the

sample space. In fact, the probability of the HDR is equal to its confidence level. Therefore, the issue of

probability estimation can be transformed into finding the corresponding quantile. For a given HDR R ( f δ ) = [α , β ] where

f (α ) = f ( β ) , let random variable Y = f ( X ) . By the definition in Eq. (13), fδ satisfies

the following: P (α ≤ X ≤ β ) = P ( f ( X ) ≥ f δ ) = 1 − δ

(14)

where f δ is the δ quantile of random variable Y, which can be estimated from the sample data Yi = f ( X i ) ; th i = 1, 2,L , n ; n is the number of samples; and Xi denotes the i observations sampled from the joint PDF f ( x ) . In

fact, we just use the δˆ sample quantile of Y to estimate the δ

quantile of random variable Y.

For a non-Gaussian process, the probability index based on the Mahalanobis distance is not available.

Therefore, using the HDR and its probability estimation approach DQA, we can construct the generalized local

probability (GLP) index for non-Gaussian distribution as follows:

GLP( xtmonitor ) = P( f ( X ) ≥ f ( xtmonitor ))

(15)

where X is a d-dimension random variable, and f ( X ) is the corresponding joint PDF, which can be determined by the SRV method described in Section 2.3. For a given sample set with n observations x = ( x1 , x 2 ,L , x n ) , the corresponding joint PDF is y = f (x). For the current monitoring data xtmonitor , if there is qδyˆ = f ( x tmonitor )

(16)

GLP ( xtmonitor ) ≅ 1 − δˆ

(17)

then

where δˆ represents the estimated confidence level corresponding to the quantile value qδyˆ of the joint PDF y. Hence, the construction problem of the GLP index is equivalent to finding the corresponding confidence level of

the density quantile.



Page 14 of 37

To reduce the computational complexity, we design a density quantile table to estimate the GLP index defined

in Eq. (17). By discretizing the joint PDF y with the discretized step size l, we can obtain a brief table with l

intervals, including l + 1 confidence levels and the corresponding l + 1 density quantile values. For a given control limit CL ∈ (0,1) , the step size for discretization is defined as [

1 ] , where [ ] means to round up. If l is large 1 − CL

enough, the value of the GLP index can be set as the mean of the upper and lower bounds of the corresponding probability interval. Under a specific control limit, if GLP ( xtmonitor ) < CL, then the monitoring data xtmonitor is in normal operating condition; otherwise, the process is determined within an abnormal process condition.

3.2 Operation procedure of fault detection based on SRV copula According to the analysis described previously, the fault detection procedure using an SRV copula includes

two stages: offline modeling and online monitoring. An algorithm flowchart of the proposed approach is shown in

Figure 2, and the formulation is presented as follows:

Offline modeling: (1) Collect a set of historical training data, and then determine the Kendall τ

rank correlation coefficient

among the variables using the empirical cumulative distribution function (ECDF). Set i = 1. (2) Specify model SRV (i) by constructing high order trees Ti +1 , Ti + 2 ,L , Td −1 with bivariate Gaussian

copulas. (3) Specify model SRV(i + 1) by constructing the additional tree Ti +1 and specifying high-order trees

Ti + 2 , Ti + 3 ,L , Td −1 fitted by bivariate Gaussian copulas.

(4) Perform a Vuong test between model SRV(i) and model SRV(i + 1), and evaluate the statistic v defined in

Eq. (12). (5) If v ≥ −Φ −1 (1 −

α 2

) , then simplify the R-vine model at level K = i. Otherwise, perform i = i + 1 and go

back to step (2).


Page 15 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(6) Determine the step size l for discretization by specifying a control limit CL. Use the SRV(K) copula

model to calculate the joint PDF of the training data, and then create a static density quantile table with

different confidence levels.

Online monitoring: (1) Evaluate the joint PDF of the current monitoring data xtmonitor based on the SRV model established previously. Then, determine the GLP index by searching the density quantile table.

(2) According to the given control limit CL in the GLP control chart, the process is in abnormal operating

condition if GLP > CL.



Figure 2. Algorithm flowchart of SRV copula monitoring approach

4 Applications In this section, the TE process and a real acetic acid dehydration distillation process are analyzed to validate the efficiency of the proposed SRV approach in handling non-Gaussian and nonlinear processes. The fault detection results show the superior performance of the SRV approach by comparing it with the FGMM, KPCA, and VCDD (C-vine) methods.


Page 16 of 37

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4.1 Application to the TE process The TE benchmark process is a test simulation platform developed by the American Eastman Company based

on the actual chemical reaction process. This process includes five major units: a reactor, a product condenser, a vapor-liquid separator, a recycle compressor, and a product stripper.(45) As a typical complex process case, the TE

process is widely used in the field of fault detection and diagnosis. The process has 12 manipulated variables, 22

continuous process measurements, 19 compositions, and 21 kinds of process faults. In this study, we adopt 22

continuous process variables to build a process model and monitor 21 faults in real time. The details of the TE

process system, 22 process variables, and the 21 kinds of process faults are shown in Figure 3, Table S1, and Table

S2. The training data set and test data set both contain 960 pieces of data sampled at an interval of 3 min. All faults in the test data set are introduced from the 161st sample. The TE process data can be downloaded at

http://web.mit.edu/braatzgroup/links.html.

FI XC FI

FC

CMS

7 1

CMR

SC FI

Condensor

FC

FC

FC

PH

Comp LI

PI

5

2 D

TI XC

LC PI

FC

FI

FI 10

LI 3

CMS

E

XA XB XC XD XE XF FI

A N A L Y Z E R FC

PI

TI TC XC

TI

LI

FI

FC CMR

12

Reactor TI

FI

TC

Stm

Cond FI

Purge XA

A N A L Y Z E R

XB XC XD XE XF XG XH

Vap/Liq Separator

Stripper

TC

6

XC 9

JI

13

XC

FI

8

TI

A

FC

A N A L Y Z E R

XD XE XF XG XH Prod

LC

11

4

LG

A/B/C



Figure 3. Flowchart of TE process.

For the R-vine approach, 21 trees for 22 variables need to be specified, and 231 bivariate copulas need to be

optimized. The R-vine matrix M for 22 continuous process measurements of the TE process is shown in Table 1.

As described in Section 2.1, the bottom row of M corresponds to T1, the second row from the bottom corresponds to T2, and so on. The edges in T1 can be determined by using the numbers in the bottom row and the diagonal elements in the corresponding columns. For example, the edges are (5,17), (17,21), (4,20), and so on. For the

second tree T2, in addition to the numbers in the last second row and the diagonal elements in the corresponding columns, the elements below the last second row are used to determine the edges. For example, the giving edges

are (5,21|17), (17,2|21), (4,21|20), and so on, where {17}, {21}, and {20} are the conditioned set. Table 2 lists the

chosen bivariate copulas for the R-vine copula model without simplification. The numbers in this table represent

the family members corresponding to the edges in M, where the number 0 indicates the independence copula, and

the highlighted nonzero numbers represent different bivariate copulas optimized. For example, in tree T1, the first copula is chosen as the Frank copula ( λ = 5) corresponding to the edge (5,17), and the third pair copula is chosen as a Gaussian copula ( λ = 1) corresponding to the edge (4,20). In the tree T2, the first three pairs of copulas are chosen independent copula, the fourth pair of copula is chosen as a Gaussian copula corresponding to the edge (1,21|20), the 15th and 20th pair copulas are chosen as a Frank copula and a rotated Clayton copula with 270º ( λ =

33) corresponding to the edges (11,16|7) and (2,20|21), respectively, and so on.

As shown in Table 2, among the 231 pairs of copulas, we only need to optimize 72 bivariate copulas, while

the rest of the paired copulas are fitted as independent ones whose copula density is c = 1. Obviously, most of the

non-independent bivariate copulas are located in the low-order trees, while the high-order trees contain more

independent bivariate copulas. Among the 72 non-independent bivariate copulas, there are 47 in the first 5 trees

that account for half the total copulas, and the number increases to 63 in the first 8 trees. This means almost all


Page 18 of 37

Page 19 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dependence information is captured in the first few trees, while the dependencies among the nodes will become

weaker and tend to be independent as the order increases. Therefore, through fitting the high-order trees containing

less independence information with bivariate Gaussian copulas, the R-vine model can be simplified. There are 33

bivariate Gaussian copulas in this table, which account for almost half the total. This indicates the choice is

reasonable. Since all of the paired copulas are independent from tree T15 with a fixed copula density c = 1, the level of simplification is at least 7, which corresponds to a full R-vine model without simplifying.

Table 1. R-vine matrix M of TE process

M 5

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T21 10 17

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T20 15 10

4

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T19 14 15

10

1

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T18 12 14

15

10

6

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T17 8

12

14

15

10

9

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T16 22

8

12

14

15

10

15

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T15 19 22

8

12

14

15

10

14

\

\

\

\

\

\

\

\

\

\

\

\

\

\

T14 18 19

22

8

12

14

3

10

12

\

\

\

\

\

\

\

\

\

\

\

\

\

T13 11 18

19

22

8

12

2

3

10

10

\

\

\

\

\

\

\

\

\

\

\

\

T12 16 11

18

19

22

8

21

2

3

3

8

\

\

\

\

\

\

\

\

\

\

\

T11 3

16

11

18

19

22

20

21

2

8

3

22

\

\

\

\

\

\

\

\

\

\

T10 7

3

16

11

18

19

19

20

21

22

2

3

19

\

\

\

\

\

\

\

\

\

T9 13

7

3

16

11

18

18

19

20

19

21

2

3

18

\

\

\

\

\

\

\

\

T8

6

13

7

3

16

11

13

18

19

2

20 21

2

3

11

\

\

\

\

\

\

\

T7

4

6

13

7

3

16

16

13

18

21

19 20

21

2

3

3

\

\

\

\

\

\

T6

1

4

6

13

7

3

12

16

13

18

18 19

20

21

2

16

16

\

\

\

\

\

T5 20

1

9

6

13

7

8

12

16

11

13 18

13

20

21

7

2

7

\

\

\

\

T4

9

20

1

9

20

13

7

8

7

16

16 13

11

13

20

2

21

2

13

\

\

\

T3

2

9

2

2

9

20

11

7

22

7

7

16

7

11

13

21

20

21

2

2

\

\

T2 21

2

21

21

2

21

22

11

8

20

22

7

16

7

16

20

13

20

21

20

20

\

T1 17 21

20

20

21

2

14

22

11

13

11 11

18

16

7

13

7

13

20

21

21

21



Page 20 of 37

Table 2. Entire optimized bivariate copulas in R-vine Model for TE process 21

0

\

\

\

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20

0

0

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\

\

\

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\

\

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\

\

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\

\

\

\

\

\

19

0

0

0

\

\

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\

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\

\

\

\

\

\

\

\

\

\

\

\

\

\

18

0

0

0

0

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

17

0

0

0

0

0

\

\

\

\

\

\

\

\

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\

\

\

\

\

\

\

16

0

0

0

0

0

0

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

\

15

0

0

0

0

0

0

0

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\

\

\

\

\

\

\

14

a

0

5

0

0

0

0

0

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\

13

0

0

1

0

0

0

0

0

0

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\

12

0

0

0

13

0

0

0

0

0

0

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11

0

0

5

3

0

14

0

0

0

0

0

\

\

\

\

\

\

\

\

\

\

\

10

0

0

0

0

0

0

0

0

0

1

0

0

\

\

\

\

\

\

\

\

\

\

9

0

0

1

5

0

0

0

0

0

0

0

0

0

\

\

\

\

\

\

\

\

\

8

0

0

0

1

4

1

0

0

0

14

0

0

0

3

\

\

\

\

\

\

\

\

7

0

0

0

4

0

0

0

0

0

0

0

4

5

1

0

\

\

\

\

\

\

\

6

0

0

0

0

0

0

0

0

0

13

33

1

1

23

1

13

\

\

\

\

\

\

5

0

0

0

0

33

0

0

0

0

0

0

24

0

1

1

0

1

\

\

\

\

\

4

0

0

0

0

0

1

0

0

0

0

14 14

1

5

1

0

1

0

\

\

\

\

3

0

0

0

5

0

0

0

0

0

0

0

1

5

14

1

0

34

13

0

\

\

\

2

0

0

0

1

0

1

0

0

0

1

0

1

1

5

5

0

1

1

0

33

\

\

1

5

0

1

29

5

1

0

23

0

1

1

1

1

40

40

5

4

1

20

1

5

\

The resulting simplified scores based on AIC/BIC and the Vuong test are shown in Table 3. Each row

corresponds to a different simplification procedure. The second and third columns represent the AIC/BIC values of

R-vine models with different simplified levels, and the last column represents the statistics of a Vuong test between two different simplified models SRV(K) and SRV (K + 1). Here, the significance level α is set at 0.05, so Φ-1(1-

α/2) = 1.96. For the Vuong test criteria mentioned above, the statistic between SRV(0) and SRV(1) is -4.1121,

which is significantly smaller than -1.96 and indicates that the SRV(1) model is better than the SRV(0) model.

However, the statistic v = -1.8834 between SRV(1) and SRV(2) is higher than -1.96, which meets the terminating

condition of iteration in the Vuong test. For the AIC/BIC criteria, we cannot obtain a terminating model because

the AIC/BIC value of SRV(K + 1) is always larger than or equal to that of SRV(K). Thus, we conclude that most of


Page 21 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the dependencies have been captured in the first tree of the R-vine model, and that SRV(1) is the best fitting model.

Actually, we can obtain the terminating condition of iteration by calculating the Vuong test statistics of the first

three simplified models [SRV(0), SRV(1), and SRV(2)]. Here, for the convenience of readers, we give the statistics

of the first 14 simplified models. Table 3. Simplification scores for AIC/BIC and the Vuong criteria Simplified Level

AIC

BIC

Statistic of Vuong test

SRV(0)

-14333.9

-13983.5

-4.112143

SRV(1)

-14562.3

-14192.4

-1.883414*

SRV(2)

-14580.9

-14211.1

0.219676*

SRV(3)

-14575.1

-14205.2

-0.487681*

SRV(4)

-14578.7

-14208.8

-0.043278*

SRV(5)

-14578.9

-14209.3

-1.532768*

SRV(6)

-14591.6

-14221.7

-1.443577*

SRV(7)

-14601.4

-14231.6

-0.878382*

SRV(8)

-14607.2

-14237.3

-1.231191*

SRV(9)

-14611.3

-14241.5

0*

SRV(10)

-14611.3

-14241.5

-0.986312*

SRV(11)

-14618.4

-14248.5

-0.963486*

SRV(12)

-14625.6

-14255.7

0*

SRV(13)

-14625.6

-14255.7

-0.893892*

SRV(14)

-14627.1

-14257.3

\

Table 4 presents the fault detection rates (FDRs) of the two traditional multivariate statistical methods KPCA

and FGMM and three copula methods: VCDD (C-vine), VCDD (R-vine), and SRV. In order to ensure the validity

and fairness of the comparison, the control limit of the five methods is determined to be 0.99, and the KPCA, FGMM, and VCDD (C-vine) methods are taken directly from the corresponding literature.(11),(19),(46) The optimal

detection value for each fault is highlighted in bold in Table 4. The FDRs of the VCDD (R-vine) method perform

better than those of VCDD (C-vine) except for fault 8. The FDRs of the VCDD (R-vine) method are better than

those of KPCA and FGMM for all 21 faults. After simplification at level 1, SRV (1) exhibits a performance similar

to the full VCDD (R-vine) model and performs better than the other three methods in all faults except 12, 15, and

18.



Page 22 of 37

Table 4. Fault detection rates for 21 faults using different approaches (0.99 confidence level)

KPCA

FGMM

Fault NO.

VCDD

VCDD

(C-Vine)

(R-Vine)

SRV(1)

T2

SPE

GLP

GLP

GLP

GLP

1

99.75

99.75

99.88

99.75

99.88

99.88

2

98.13

98.25

98.43

98.25

98.63

98.75

3

4.38

5.00

1.76

3.25

6.13

6.50

4

2.00

2.25

2.23

2.25

3.25

3.38

5

27.00

27.00

24.50

26.38

28.88

29.50

6

100

100

100

100

100

100

7

42.38

42.63

35.78

40.00

45.25

44.50

8

97.38

97.75

97.62

98.50

98.25

98.50

9

3.38

4.88

3.71

3.13

7.25

6.38

10

45.00

60.00

70.87

66.13

84.75

82.00

11

34.50

40.88

19.75

36.50

47.63

47.25

12

99.50

99.13

98.50

99.25

99.63

99.38

13

94.75

94.63

94.63

94.75

95.00

95.00

14

99.88

99.88

99.88

99.88

100

100

15

10.13

7.13

8.30

6.75

9.63

9.63

16

32.38

35.38

21.75

31.13

53.00

48.88

17

95.38

94.63

93.37

96.38

97.00

97.13

18

89.88

89.88

89.75

90.38

90.50

90.25

19

4.13

16.63

10.88

24.13

46.50

45.38

20

45.00

50.63

70.25

74.50

83.50

83.38

21

44.63

49.75

38.25

46.13

57.50

55.50

The performance of the proposed approach is further analyzed through several typical faults. The FDR charts

of the TE process for step Fault 6 caused by a step change in the A feed is presented in Figure 4. Obviously, all

approaches can detect the faults with a 100% detection rate. This means the SRV model, like other methods, can

accurately characterize the dependencies between such variables, and is able to set apart the normal data from the

faulty data. The scatter plots of Fault 6 are depicted in Figure 5, where black points represent the training data, and

red points represent the testing data. It can be found that there is a very obvious difference in the distribution of the


Page 23 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


fault data and the abnormal data. The SRV approach can accurately depict this behavior and extract the faulty data

that is causing a large deviation.



Figure 4. Fault detection charts of TE process for Fault 6

Figure 5. Scatter plots of TE process for Fault 6

For random faults 8, 9, 10, and 11, the FDRs of VCDD (R-vine) and SRV are 98.25%/98.50%, 7.25%/6.38%,

84.75%/82.00%, and 47.63%/47.25%, which are all better than those of the other three methods. Figure 6 shows

the monitoring results of Fault 10 based on the five methods. Compared with Fault 6 caused by a step change in

the A feed, Fault 10 causes a variation in the stripper pressure owing to a random temperature change in stream 4.

According to Table 4, KPCA obtains only a 60.00% detection rate. Although the FDR of VCDD (C-vine) for Fault


Page 24 of 37

Page 25 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 was slightly improved, it is still lower than the result of the FGMM method (70.87%). VCDD (R-vine) achieves

a higher detection rate (84.75%) than the other three methods [KPCA, FGMM, and VCDD (C-vine)], and the

proposed SRV approach also achieves a similar monitoring performance (82.00%). Figure 7 shows a scatter plot of

some of the process variables for Fault 10, where black points represent the training data, and red points represent

the testing data. Compared with Fault 6, the difference between the fault and normal data of Fault 10 is very small

and difficult to distinguish. It is difficult to detect the faulty data near the normal data by KPCA, FGMM, or

VCDD (C-vine). The better monitoring performance of SRV probably results from its ability to accurately depict

the complex dependence behavior in complex process variables.



Figure 6. Fault detection charts of TE process for Fault 10

Figure 7. Scatter plots of TE process for Fault 10

For Faults 16, 17, 19, and 20, which are caused by unknown factors, the FDRs of VCDD (R-vine) and SRV

are 53.00%/48.88%, 97.00%/97.13%, 46.50%/45.38%, and 83.50%/83.38%, respectively. These achieve

significant performance compared with the other three methods, and indicate the better performance of SRV in handling these unknown faults. The fault detection charts of the TE process for Fault 20 are presented in Figure 8.

According to Figure 8 and Table 4, the proposed SRV method can detect faults efficiently and outperforms the


Page 26 of 37

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KPCA, FGMM, and VCDD (R-vine) methods. For Fault 19, the FDRs of all approaches are not satisfactory

because the deviation between the normal and faulty data is slight, and Fault 19 is quite difficult to detect.

However, the FDR of SRV is 45.38%, which is higher than that of KPCA (16.13%), FGMM (10.88%), and VCDD

(C-vine) (24.13%).



Figure 8 Fault detection charts of TE process for Fault 20

The FGMM approach, based on Mahalanobis distance or the corresponding local probability index, cannot

handle the non-Gaussian distribution of the TE process and cannot address the problems of process noise and nonlinearity.(29) Although the KPCA method enhances the ability of nonlinear processing, it still takes the idea of

dimensionality reduction, which inevitably leads to a partial loss of information in the process of data

transformation and feature extraction. The VCDD (C-vine) method has strong flexibility in characterizing

nonlinear and non-Gaussian processes, but owing to the fixed simple star structure, it is difficult to accurately

capture the dependence information between process variables. VCDD (R-vine) uses the variables’ correlations to

determine the dependence structure, as opposed to the VCDD (C-vine) method, which can fully explore the

inherent dependence structure between variables. Figure 9 and Figure 10 show the first trees of the R-vine and

C-vine models, respectively. Evidently, compared with the star structure in which all nodes are connected to node 7,

the first tree structure of VCCD (R-vine) is closer to the actual dependence behavior of the variable. To retain the

structure of VCCD (R-vine), we can obtain a simplified R-vine model by using the Vuong test, which can greatly

improve the efficiency of fault detection without significantly reducing the accuracy of the model. Table 5 shows


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the CPU time cost of VCDD (C-vine), VCDD (R-vine), and SRV in offline modeling, PDF calculation, and online

monitoring. The SRV method has the least time cost in online monitoring and has a similar time cost in offline

modeling and PDF calculation to VCDD (C-vine), indicating that the SRV method exhibits better performance in

process monitoring.

Figure 9. First tree of R-vine model for TE process

Figure 10. First tree of C-vine model for TE process

Table 5. CPU time cost of VCDD (C-vine), VCDD (R-vine), and SRV (Offline modeling and joint PDF for 960 training samples, online modeling per sample)

Offline modeling Joint PDF Online modeling

VCDD(C-Vine) 3.1281m 1.5955m 0.107s

VCDD(R-Vine) 3.4952 m 1.8981m 0.121s

SRV(1) 3.2232m 1.6350m 0.102s

4.2 Application to a real acetic acid dehydration process Purified terephthalic acid is an important raw material for polyester production widely used in the textile and

packaging industries. This acid is produced by the catalytic oxidation of paraxylene with subsequent



hydrogenation, crystallization, separation, and drying processes. In the oxidation process, acetic acid is an

important solvent in the reactor. Usually, part of the acetic acid will leave the top of the reactor together with water

during the reaction process. In order to reduce the loss of acetic acid, a solvent dehydration column is used to

recover acetic acid from the wastewater. A plant in China uses an ordinary distillation method to separate acetic

acid from water. Under normal circumstances, the top and the bottom products of the distillate column are water

and acetic acid, respectively. Although acetic acid and water do not form an azeotrope at atmospheric pressure,

using simple distillation to separate these two components requires many equilibrium stages and a biggish reflux

ratio since the system has a tangent pinch on the pure-water end. Usually, the top product contains acetic acid of

less than 1.15%, and the acetic acid product contains water from 5% to 8%, after the purification of acetic acid

returns the oxidation reactor.

Figure 11 shows the control system for the acetic acid dehydration process, which has 90 stages and 4 feeds,

while the mixture of acetic acid and water comes from the oxidation reaction. In this study, we choose 21

continuous process variables including temperature, pressure, flow, and conductivity to create a process model and

monitor the top acetic acid distillation in real time. A total of 500 sets of training data and 300 sets of testing data

came from DCS.


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Figure 11. Control system of dehydration process

In offline modeling, we obtain the simplified R-vine model at level 3 by using the Vuong test. This is called

SRV (3). Here, the 300 testing samples include normal data in the first 100 samples, the faulty data of the acetic

acid content of the top product rising from 1.1% to 1.2% in the second hundred samples, and normal data in the

third hundred samples. Table 6 shows the FDRs and false alarm rates (FARs) of the four methods of KPCA,

FGMM, VCDD (C-vine), and SRV under a control limit of 0.97. The fault detection charts for the monitoring

results using the four different methods are shown in Figure 12. Notice that the SRV method gets the best results

for both FDRs and FARs, indicating that the proposed method is a powerful tool for analyzing fault behaviors in

the acetic acid dehydration process.

Table 6. Fault detection rates and false alarm rates of acetic acid dehydration distillation system (CL = 0.97)

KPCA

FGMM

VCDD(C-Vine)

SRV(K=3)

T2

SPE

GLP

GLP

GLP

FDR

1

0.99

1

0.99

1

FAR

0.11

0.045

0.18

0.025

0.02



Figure 12. Monitoring results of acetic acid dehydration process

5 Conclusions In this paper, a process monitoring method based on the SRV model is proposed using the Vuong test method to determine the simplified level. The SRV approach simplifies the structure of the R-vine-based model, which fully captures the dependence behaviors between variables, greatly reducing the complexity of optimization problems and significantly increasing the efficiency of process monitoring. This method abandons the idea of dimensionality reduction and the Gaussian hypothesis of data processing to directly characterize the dependencies between variables. At the same time, this method uses the actual dependency behavior of data variables to


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construct a dependence structure. This is different from the method of VCDD (C-vine), which fixes the tree as a star structure. Therefore, this method can obtain a more accurate distribution model and explore the inherent dependence information. Application of the TE process and the acetic acid dehydration process shows that the SRV monitoring method has better monitoring results than the traditional KPCA, FGMM, and VCDD (C-vine) methods, while the SRV method has better detection efficiency without significantly reducing the model accuracy as compared with the VCDD (R-vine) method. As a result, the proposed SRV monitoring method in this paper is effective and practical in the field of process monitoring.

AUTHOR INFORMATION Corresponding Author *Tel.: +86-21-64253820. E-mail: [email protected].

ORCID Shaojun Li: 0000-0002-2891-2330

Notes The authors declare no competing financial interest.

Acknowledgments The authors of this paper appreciate the National Natural Science Foundation of China (under Project No. 21676086 and No. 21406064).

Reference (1)

Venkatasubramanian, V.; Rengaswamy, R.; Yin, K.; Kavuri, S. N. A review of process fault detection and diagnosis Part I: Quantitative model-based methods. Comput. Chem. Eng. 2003, 27, 293-311.

(2)

Venkatasubramanian, V.; Rengaswamy, R.; Kavuri, S. N. A review of process fault detection and diagnosis Part II: Qualitative models and search strategies. Comput. Chem. Eng. 2003, 27, 313-326.

(3)

Venkatasubramanian, V.; Rengaswamy, R.; Yin, K.; Kavuri, S. N. A review of process fault detection and diagnosis Part III: Process history based methods. Comput. Chem. Eng. 2003, 27 , 327-346.

(4)

Ge, Z.; Song, Z.; Gao, F. Review of recent research on data-based process monitoring. Ind. Eng. Chem. Res.

2013, 52, 3543-3562. (5)

Qin, S. J. Survey on data-driven industrial process monitoring and diagnosis. Annu. Rev. Control. 2012, 36, 220-234.

(6)

Ge, Z. Review on data-driven modeling and monitoring for plant-wide industrial processes. Chemom. Intell.

Lab. Syst. 2017, 171, 16-25.



(7)

Kano, M.; Nagao, K.; Hasebe, H.; Hashimoto, I.; Ohno, H.; Strauss, R.; Bakshi, B. R. Comparison of multivariate statistical process monitoring methods with applications to the Eastman challenge problem.

Comput. Chem. Eng. 2002, 26, 161−174. (8)

Chiang, L. H.; Russell, E. L.; Braatz, R. D. Fault Detection and Diagnosis in Industrial Systems; Springer: London, 2001.

(9)

Wold, S.; Esbensen, K.; Geladi, P. Principal component analysis. Chemom. Intell. Lab. Syst. 1987, 2, 37-52.

(10) MacGregor, J. F.; Jaeckle, C.; Kiparissides, C.; Koutoudi, M. Process monitoring and diagnosis by multiblock PLS methods. AIChE J. 1994, 40, 826-838. (11) Lee, J. M.; Yoo, C. K.; Choi, S. W.; Vanrolleghem, P. A.; Lee, I. B. Nonlinear process monitoring using kernel principal component analysis. Chem. Eng. Sci. 2004, 59, 223-234. (12) Ge, Z.; Yang, C.; Song, Z. Improved kernel PCA-based monitoring approach for nonlinear processes. Chem.

Eng. Sci. 2009, 64, 2245-2255. (13) Kramer, M. A. Nonlinear Principal Component Analysis Using Autoassociative Neural Networks．AIChE J.

1991, 37, 233-243. (14) Zhang, Y.; Hu, Z. Multivariate process monitoring and analysis based on multi-scale KPLS. Chem. Eng. Res.

Des. 2011, 89, 2667-2678. (15) Kano, M.; Tanaka, S.; Hasebe, S. Monitoring independent components for fault detection. AIChE J. 2003, 49, 969-976. (16) Lee, J.; Yoo, C.; Lee, I. Statistical process monitoring with independent component analysis. J. Process

Control 2004, 14, 467-485. (17) Ge, Z. Q.; Song, Z. H. Process monitoring based on independent component analysis-principal component analysis (ICA-PCA) and similarity factors. Ind. Eng. Chem. Res. 2007, 46, 2054-2063. (18) Ge, Z.; Song, Z. Online monitoring of nonlinear multiple mode processes based on adaptive local model approach. Control Eng. Pract. 2008, 16, 1427-1437. (19) Yu, J.; Qin, S. J. Multimode process monitoring with Bayesian inference-based finite Gaussian mixture models. AIChE J. 2008, 54, 1811-1829. (20) Dißmann, J.; Brechmann, E. C.; Czado, C.; Kurowickab, D. Selecting and estimating regular vine copulae and application to financial returns. Comput. Stat. Data Anal. 2013, 59, 52-69. (21) Wei, Y.; Zhang, S. Dependence Analysis of Finance Markets: Copula-GARCH Model and Its Application.

Syst. Eng. 2004, 4, 7-12. (22) Madadgar, S.; Moradkhani, H. Drought analysis under climate change using copula. J. Hydrol. Eng. 2011, 18, 746-759. (23) Meel, A.; Seider, W. D. Plant-specific dynamic failure assessment using Bayesian theory. Chem. Eng. Sci.

2006, 61, 7036-7056.


Page 34 of 37

Page 35 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(24) Pariyani, A.; Seider, W. D.; Oktem, U. G.; Soroush, M. Dynamic risk analysis using alarm databases to improve process safety and product quality: Part II—Bayesian analysis. AIChE J. 2012, 58 (3), 826-841. (25) Yu, H. Y.; Khan, F.; Garaniya, V. A probabilistic multivariate method for fault diagnosis of industrial processes. Chem. Eng. Res. Des. 2015, 104, 306-318. (26) Nelson, R. B. An Introduction to Copulas, 2nd ed.; Springer: Berlin, 2006. (27) Joe, H. Families of m-variate Distributions with Given Margins and m(m-1)/2 Bivariate Dependence Parameters. In Distributions with Fixed Marginals and Related Topics; Ruschendorf, L., Schweizer, B.,Taylor, M. D., Eds.; Institute of Mathematical Statistic, Hayward, 1996; pp 120-141. (28) Ren, X.; Tian, Y.; Li, S. J. Vine copula-based dependence description for multivariate multimode process monitoring. Ind. Eng. Chem. Res. 2015, 54, 10001-10019. (29) Zheng, W.; Ren, X.; Zhou, N.; Li, S. J. Mixture of D-vine copulas for chemical process monitoring. Chemom.

Intell. Lab. Syst. 2017, 169, 19-34. (30) Erhardt, T. M.; Czado, C.; Schepsmeier, U. Spatial composite likelihood inference using local C-vines. J.

Multivariate Anal. 2015, 138, 74-88. (31) Kurowicka, D.; Cooke, R. M. Sampling algorithms for generating joint uniform distributions using the vine-copula method. Comput. Stat. Data Anal. 2007, 51, 2889-2906. (32) Czado, C.; Jeske, S.; Hofmann, M. Selection strategies for regular vine copulae. J. Soc. Fr. Stat. 2013, 154, 174-191. (33) Bedford, T.; Cooke, R. M. Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intel. 2001, 32, 245-268. (34) Kurowicka, D.; Joe, H. Dependence modeling vine Copula Handbook; World Scientific: Singapore, 2011. (35) Brechmann, E. C.; Czado, C.; Aas, K. Truncated regular vines in high dimensions with application to financial data. Can. J. Stat. 2012, 40, 68-85. (36) Brechmann, E. C.; Joe, H. Truncation of vine copulas using fit indices. J. Multivariate Anal. 2015, 138, 19-33. (37) Patton, A. J. A Review of Copula Models for Economic Time Series. J. Multivariate Anal. 2012, 110, 4-18. (38) Sklar, A. Fonctions dé Repartition á n Dimension et Leurs Marges. Publ. Inst. Stat. Univ. Paris 1959, 8, 229-231. (39) Aas, K.; Czado, C.; Frigessi, A.; Bakken, H. Pair-Copula constructions of multiple dependence. Insur. Math.

Econom. 2009, 44, 182-198. (40) Schwarz, G. Estimating the dimension of a model. Ann. Stat. 1978, 6, 461-464. (41) Burnham, K. P.; Anderson, D. R. Model Selection and Multimodel Inference: A Practical

Information-Theoretic Approach; Springer: New York, 2002.



(42) Vuong, Q. H. Likelihood ratio tests for model selection and non-nested hypotheses. J. Econom. Soc. 1989, 307-333. (43) Hyndman, R. J. Computing and graphing highest density regions. J. Am. Stat. Assoc. 1996, 50, 120-126. (44) Sagara, S.; Zhao, Z. Y. Numerical integration approach to on-line identification of continuous-time systems.

Automatica 1990, 26, 63-74. (45) Lyman, P. R.; Georgakis, C. Plant-wide control of the Tennessee Eastman problem. Comput. Chem. Eng.

1995, 19, 321-331. (46) Ren, X.; Zhu, K.; Cai, T.; Li, S. J. Fault Detection and Diagnosis for Nonlinear and Non-Gaussian Processes Based on Copula Subspace Division. Ind. Eng. Chem. Res. 2017, 56, 11545-11564.


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Nonlinear and Non-Gaussian Process Monitoring Based on Simplified

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