Data-based decomposition plant for decentralized monitoring schemes: A comparative study

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Journal of Process Control
journal homepage: www.elsevier.com/locate/jprocont
Data-based decomposition plant for decentralized monitoring schemes: A
comparative study
M.J. Fuente, M. Galende-Hernández, G.I. Sainz-Palmero ∗
Department of System Engineering and Automatic Control, Industrial Engineering School, Universidad de Valladolid, Paseo Prado de la Magdalena
3-5, 47011 Valladolid, Spain
ARTICLE INFO
Keywords:
Fault detection
Canonical Variate Analysis
Regression
Correlation
Mutual information
Clustering
Decentralized process monitoring
Bayesian Inference
ABSTRACT
The complexity of the industrial processes, large-scale plants and the massive use of distributed control
systems and sensors are challenges which open ways for alternative monitoring systems. The decentralized
monitoring methods are one option to deal with these complex challenges. These methods are based on process
decomposition, i.e., dividing the plant variables into blocks, and building statistical data models for every block
to perform local monitoring. After that, the local monitoring results are integrated through a decision fusion
algorithm for a global output concerning the process. However, decentralized process monitoring has to deal
with a critical issue: a proper process decomposition, or block division, using only available data. Knowledge of
the plant is rarely available, so data-driven approaches can help to manage this issue. Moreover, this is the first
and key step to developing decentralized monitoring models and several alternative approaches are available.
In this work a comparative study is carried out regarding decentralized fault monitoring methods, comparing
several alternative proposals for process decomposition based on data. These methods are based on information
theory, regression and clustering, and are compared in terms of their monitoring performance. When the blocks
are obtained, CVA (Canonical Variate Analysis) based local dynamic monitors are set up to characterize the
local process behavior, while also considering the dynamic nature of the industrial plants. Finally, the Bayesian
Inference Index (BII) is implemented, based on these local monitoring, to achieve a global outcome regarding
fault detection for the whole process. To further compare their performance from the application viewpoint, the
Tennessee Eastman (TE) process, a well-known industrial benchmark, is used to illustrate the efficiencies of all
the discussed methods. So, a systematically comparison have been carried out involving different data-driven
methods for process decomposition to implement a decentralized monitoring scheme. The results are focused
on providing a reference for practitioners as guidelines for successful decentralized monitoring strategies.
1. Introduction
Process monitoring for complex large-scale industrial plants is an
effective way to improve safety, prevent damage to equipment and
maintain normal production by detecting anomalies and diagnosing
their root cause. Nowadays, the application of distributed control sys-
tems and modern measurement techniques is widespread, which can
make large amount of data about the plant available. Due to all this,
data-driven process monitoring has become more popular and usual,
particularly Multivariate Statistical Process Monitoring (MSPM) which
has been intensively researched, obtaining impressive progress over the
last decades [1–4].
These techniques, including Principal Component Analysis (PCA),
Partial Least Squares (PLS) or Independent Component Analysis (ICA),
effectively extract the underlying characteristics from historical data
∗Corresponding author.
E-mail addresses: mariajesus.fuente@uva.es (M.J. Fuente), marta.galende@uva.es (M. Galende-Hernández), gregorioismael.sainz@uva.es
(G.I. Sainz-Palmero).
and drive normal operation models by accommodating acceptable vari-
ations, and detecting abnormal conditions using statistical metrics [5–
7]. However, it is accepted that these traditional methods may not be
the best for complex industrial plants with their dynamic and non-
linear nature. In order to manage these problems, complementary
approaches are also included, such as Canonical Variate Analysis (CVA)
and Dynamical PCA (DPCA) for dynamic process monitoring, or kernel
techniques (KPCA, KPLS and KICA) for non-linear processes [8–14].
In order to face the challenge regarding wide-plant and large-
scale process monitoring some proposals address the challenge through
decentralized strategies. These approaches introduce some advantages:
first of all, the process decomposition into blocks can reduce the com-
putational loadings since each block employs a smaller set of relevant
https://doi.org/10.1016/j.jprocont.2024.103178
Received 18 October 2023; Received in revised form 13 January 2024; Accepted 4 February 2024

Journal of Process Control 135 (2024) 103178
2
M.J. Fuente et al.
variables to process. Secondly, by dividing the plant into multiple
blocks, it is possible to detect fault events in a distributed manner,
and the most responsible faulty sections and faulty variables can be
isolated. Finally, through building multiple blocks the robustness of
the whole statistic model can be enhanced for maintenance process,
because regular maintenance and updating of some blocks has no
impact on other blocks [15].
A common key issue of these decentralized strategies is the decom-
position of the plant variables into blocks to be processed by a local
monitoring model and, finally, a central processor to collect all local
outcomes and fuse them for a global decision regarding the normal
or faulty state of the whole process. However, the main challenge of
this strategy is the proper process decomposition, i.e., to divide the
variables into overlapping or disjoint blocks, which is the first and key
step for almost all decentralized monitoring strategies.
The traditional decomposition methods are usually based on prior
knowledge or process topology [16–18]; however, this knowledge is
not usually available, especially for complex industrial plants. So,
other alternatives, such as data-driven approaches, are interesting for
dealing with this critical issue: such alternatives would include [19]
sparse PCA; or [20,21] using PCA to built the blocks in the different
principal component directions, or [22–25] using mutual information
and correlation between the variables, respectively; or [26] first divides
the variables into Gaussian and non-Gaussian subspaces by Jarque–
Bera detection method, and after that uses the mutual information (MI)
in both subspaces to determine quality-relevant and quality-irrelevant
variables to obtain the sub-blocks.
Other authors, as [27,28] use the minimal redundancy and maxi-
mal relevance (mRMR) method to carry out the decomposition; also
similarly the method proposed by [29] is based on relevance and
redundancy (RRVS) taking into account the strong dynamic relation
between the variables of the process. We have also proposed several
of our own techniques to decompose the plant by linear regression
(Sparse PLS) [30], LASSO, Elastic-net and non-linear regression by
neural networks [31], or based on clustering [32]. Recently, [33]
combine the mechanism knowledge and data analysis to decentralize
the plant, i.e., the plant wide process is preliminary map as an undi-
rected graph corresponding to the mechanism knowledge and process
structure which divides the plant into blocks and secondly the authors
use mutual information based Louvain algorithm (MI-Louvain) to fine
decompose the process into reasonable sub-blocks, considering the
correlation among variables and the graph structure.
Others authors use causal networks that holds a large amount of
information on how process variables relate each other. For example,
in [34] the process variables are divide into modules, defined as
communities, based on the analysis of this causal network through
a community detection algorithm that evaluates the network topol-
ogy and the density of associations between variables and then each
module is monitored by a causal method called Sensitivity Enhancing
Transformation (SET). Another idea based on networks is the cascaded
monitoring network called MoniNet [35], that takes into consideration
the local correlation between the different operation units in an indus-
trial plant. In this method a convolutional operation for each variable
is carried out to extract simultaneously temporal information. This
reveals dynamic correlation on process data and spatial information,
reflecting local characteristics within each operation unit. For each
feature obtained, a sub-model is developed, and all the sub-models
are integrated to generate a final monitoring model. The MoniNet
model can be expanded to capture deeper information by adding more
convolutional layers.
Also, [36] uses convolutional feature extraction as MoniNet, but in
this method is firstly identified the causal relationship within the in-
dustrial process to analyze the interaction between different variables,
and based on this graph structure, convolutional filters are designed to
extract feature from directly related variables and their corresponding
time lags. The obtained features of all process variables are combined
into feature matrices based on which the sub-models are developed,
using Autoencoders (AE) in this case.
Other perspective for the division of the variables of the wide-
plant is the control-oriented decomposition for identifying subsystems
within the plant each of which can be controlled effectively without
affecting the other subsystems significantly [37]. This problem related
to the classical problem of control structure design, i.e., the selection
and pairing of manipulated input and controlled output variables, can
be viewed from a network perspective, and corresponds to identify
communities whose members interact strongly among them, yet are
weakly coupled to the rest of the network members. So, this can be
solve from the point of view of the graph and network theory [37]. But
this kind of decomposition is out the scope of this paper.
All these proposals decompose the plant using only historical data
to manage the shortage of knowledge. However, there is a lack of com-
paratives regarding the performance of these decomposition methods
in terms of fault detection schemes in the bibliographic publications to
rank them and know the best cases.
So, the main objective and contribution of this paper is focused
on supplying a comparative performance study between different full
driven-data methods for process decomposition into blocks to imple-
ment a decentralized monitoring scheme. The methods involved for
this comparison are: based on linear regression, such as LASSO, Elastic
net [31] or Sparse Partial Least Squares (SPLS) [30]; and based on
non-linear regression using MLP neural networks [31]. The third kind
of methods are based on information theory such as Sparse Principal
Component Analysis (SPCA) [19], correlation analysis [23], mutual
information [22], Detrended Cross-Correlation analysis (DCCA) or min-
imal redundancy and maximal relevance (mRMR) [28]; and finally,
methods based on clustering [32]. All these methods are data driven
proposals to divide the plant variables into blocks of variables. How-
ever, the majority of the distributed approaches use PCA for motioning
each block, that only consider the static process variation, ignoring the
dynamic characteristics of the industrial processes, so in this work for
each block, a local dynamic multivariate statistical model, here based
on Canonical Variate Analysis (CVA), is used for local monitoring.
Finally, all the outcomes from the blocks are fused through the Bayesian
inference strategy to provide an operation status of the whole plant.
The performance comparison between the different plant decompo-
sition methods is based on the fault detection results using the following
indexes: fault alarm rate (𝐹 𝐴𝑅), missed detection rate (𝑀𝐷𝑅) and
detection delay (𝐷𝐷).
On the other hand, another important index to consider is the com-
plexity of the decomposition method, i.e., the number of blocks defined
for every method (𝑁𝐵 ): some are fully decentralized approaches, [23,
25,30], defining a block for each plant variable which can be un-
bearable for large scale plants. Therefore, to carry out this perfor-
mance comparative the well-known Tennessee Eastman Process (TEP)
benchmark is used.
In short, the main contribution of this work is to provide perfor-
mance guidelines for data driven decomposition methods in decen-
tralized monitoring strategies when facing large and complex process
plants. Some of the block decomposition approaches included in this
comparison have not been previously used for this goal in the known
bibliography, as the DCCA method.
The rest of the paper is organized as follows. Section 2provides
some background knowledge about CVA and explains the different
methods to perform the decentralization, while the probabilistic
Bayesian fusion technique is also detailed here. Section 3elaborates
the decentralized fault detection method used in the paper. The results
of the comparison between the different plant decomposition methods
tested over the Tennessee Eastman Plant are summarized in Section 4,
followed by the conclusions in Section 5.

Journal of Process Control 135 (2024) 103178
3
M.J. Fuente et al.
2. Materials and methods
The relevant methods for the decentralized monitoring scheme are
reviewed in this section. This is made up of three steps: first the meth-
ods used for the process decomposition based on data are described,
followed by a brief review of CVA, which serves as the basis of the local
fault detection method for each block, and finally the decision making
to fuse all the results provided by each block, the Bayesian inference
index (BII) is introduced.
2.1. Plant decomposition based on data
In this section, the different data-based techniques to decompose
a large plant are briefly introduced. These methods are based on
regression (linear: LASSO, Elastic net and SPLS and non linear: artificial
neural networks); on information concerning the variables (Sparse PCA,
correlation, mutual information, minimal redundancy and maximal rel-
evance (mRMR) and Detrended Cross-Correlation analysis); and finally
based on clustering.
LASSO method.Least Absolute Shrinkage and Selection Operator
(LASSO) method [38] is a linear regression method, which penalizes
the coefficients of the regression, making some of them zero, permitting
a model simpler and more interpretable. This technique can be used to
plant decomposition, using only the most relevant regressors, which
have non-zero coefficients [31].
A linear model with 𝑚predictors: 𝑋(𝑡)=(𝑥1(𝑡), 𝑥2(𝑡),…, 𝑥𝑚(𝑡)) and
one response variable 𝑦(𝑡)can be expressed as follows:
𝑌=𝛽𝐗+𝐸(1)
The LASSO method solves the problem as:

𝛽(𝜆) = argmin
𝛽∥𝑦−𝛽𝐗∥2
2
𝑛+𝜆∥𝛽∥1(2)
where 𝜆≥0. This hyper-parameter 𝜆controls the process: when its
value increases, more coefficients are forced to be zero.
Elastic net method. The elastic net regression was introduced by
[39] and can be seen as a compromise between Ridge and LASSO
regression, i.e., it selects variables such as LASSO and shrinks the
coefficients according to Ridge. So, the elastic net regression solves the
optimization problem:

𝛽(𝜆, 𝛿) = argmin
𝛽∥𝑦−𝛽𝐗∥2
2
𝑛+𝜆1 − 𝛿
2∥𝛽∥2
2+𝛿∥𝛽∥1(3)
The elastic net is more flexible, and for 𝛿= 1, it gives the LASSO
solution and for 𝛿= 0 the Ridge regression is obtained. A frequent
strategy is to assign a big value to the 𝐿1penalization in order to get
a lower number of predictors, i.e., putting a value of 𝛿near to 1, and
giving a little weight to the 𝐿2regularization so as to provide some
stability if some of the predictors are highly correlated. This technique
can be used to perform feature selection, for the regression model of
a variable 𝑥𝑖only the most relevant regressors are take into account,
i.e., the variables 𝑥𝑗with a coefficient 𝛽𝑗> 𝑙𝑒𝑛𝑖(see Eq. (1)) have to be
in the same block as 𝑥𝑖with 𝑙𝑒𝑛𝑖a certain threshold [31].
SPLS method. The Sparse Partial Least Squares (SPLS) method [40]
is also an improvement over the well known PLS method. This adjusts
a linear model using least squares over new discovered features which
are combinations of the original ones: 𝐗the predictor matrix (𝑛×𝑚)
and 𝐘the response matrix (𝑛×𝑝):
𝐗=𝐓𝐏𝐓+𝐄(4)
𝐘=𝐓𝐐𝐓+𝐅(5)
where 𝐓=𝐗𝐖 contains the first 𝑘terms of the latent variables or the
score vectors, 𝐖= (𝑤1, 𝑤2,…, 𝑤𝑘)is a matrix of direction vectors, 𝐏
and 𝐐, respectively, are the loading vectors of the data matrices 𝐗and
𝐘, and 𝐄and 𝐅are the residual terms of PLS. PLS has successively
to calculate the vectors 𝑤that maximize the covariance between the
explanatory variables 𝐗and the responses 𝐘, obtaining the regression
model, with 𝐁𝐏𝐋𝐒 =𝐖𝐐:
𝐘=𝐗𝐁𝐏𝐋𝐒 +𝐅(6)
Now the objective of Sparse Partial Least Squares (SPLS) [40] is to
obtain a number of coefficients equals to zero in vector 𝐁𝐏𝐋𝐒, i.e., to
assure sparsity on the vector 𝐁𝐏𝐋𝐒 and as consequence in the vector 𝑤.
This is done imposing an 𝐿1term in this 𝑤vector in the optimization
objective similar to the LASSO model. This means the method selects
the most relevant predictors in the regression, making easier the un-
derstanding of the model, allowing also to perform a variable selection
including only the variables with a regression coefficient different from
zero [30] .
MLP-ANN Method. Artificial Neural Networks [41], and in par-
ticular the Multilayer Perceptron (MLP), are used to model non-linear
systems. This is due to its ability to approximate any continuous
function as accurately as necessary. Artificial neural networks are
well-known algorithms applied in many fields with a large variety of
architectures [42]. The trained weights that connect each input with
the output through various neurons are used for the feature selection
task [31,43]. So, for the MLP regression model of a variable 𝑥𝑖only
the variables that have obtained a greatest score are take into account,
i.e., the variables 𝑥𝑗with a score 𝑅𝑖𝑗 > 𝑙𝑛𝑛𝑖have to be in the same
block as 𝑥𝑖with 𝑙𝑛𝑛𝑖a certain threshold. This score is calculated as the
product of the synaptic weights that connect each input with the output
through the neurons in the neural model, i.e.,
𝑅𝑖𝑗 =
𝐻

𝑘=1
𝑊𝑗𝑘 𝑊𝑘𝑖 (7)
where 𝑅𝑖𝑗 is the relative importance, or score, of the input variable
𝑥𝑗,𝑗= 1,…, 𝑚 and 𝑗≠𝑖with respect to the output neuron, i.e., for
the variable 𝑥𝑖that we are modeling, 𝐻is the number of neurons in
the hidden layer, 𝑊𝑗𝑘 is the synaptic connection weight between the
input neuron 𝑗and the hidden neuron 𝑘, and 𝑊𝑘𝑖 is the synaptic weight
between the hidden neuron 𝑘and the output neuron.
SPCA method. The Sparse Principal component analysis (SPCA)
method is also based on PCA, and, given a data matrix 𝐗and its
covariance 𝐒𝐱, the objective of this method is to decompose 𝐒𝐱into its
components [𝐯𝟏,𝐯𝟐,…,𝐯𝐝]while constraining the number of elements
of each vector 𝐯different from zero to 𝑟, so 𝑟is the cardinality of the
vector 𝐯. This is calculated maximizing the variance of 𝐯∈ℜ𝑑with the
constraint of the cardinality, i.e.,
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝐯𝐓𝐒𝐱𝐯
𝑠𝑢𝑏𝑗𝑒𝑐 𝑡 𝑡𝑜 𝐯2= 1
𝐶𝑎𝑟𝑑 (𝐯)≤𝑟
(8)
In general, in PCA technique only the first 𝑏components explain
the most of the data variance. Here, SPCA is applied to the matrix 𝐗to
obtain 𝐵sparse components, i.e., [𝐯𝟏,𝐯𝟐,…,𝐯𝐁]defining each compo-
nent a block, and the non-zero coefficients of each sparse component,
𝐯𝐢, 𝑖 = 1,…, 𝐵 define the variables for that block [19].
Correlation. The relationship between two variables 𝑥𝑖and 𝑥𝑗can
be calculated as the absolute value of the correlation coefficient :
ℜ𝑖(𝑥𝑖, 𝑥𝑗) = 𝑥𝑇
𝑖𝑥𝑗∕𝑥𝑖𝑥𝑗(9)
where 𝑖, 𝑗 = 1,…, 𝑚. This coefficient is a direct measure of the corre-
lation between variables and, after the calculation of vector ℜ𝐢for the
𝑖th variable, the features selection for defining a block is carried out by
selecting the variables with higher correlation values, i.e., the variables
𝑥𝑗with ℜ𝐢> 𝛿𝑖, are the selected variables to include for the block
associate to the variable 𝑥𝑖, with 𝛿𝑖being a cut off parameter for every
variable that depends on the correlation present in the system [23].

Journal of Process Control 135 (2024) 103178
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M.J. Fuente et al.
Mutual Information. The mutual information of two variables
measures their dependence considering the entropy [44]. In contrast
to cross-correlation, it takes into account the higher-order statistics
and is able to capture the non-Gaussianity of stochastic systems. So,
this quantification would not only consider the linear correlations, but
also takes into account the non-linear relations between variables. So,
this information can be used to decompose the plant, i.e., variables
with high mutual information values should be chosen to form the
corresponding block [22]. MI can be calculated as:
𝑙(𝑥1, 𝑥2) = ∫ ∫𝑥1𝑥2
𝑝(𝑥1, 𝑥2) log( 𝑝(𝑥1, 𝑥2)
𝑝(𝑥1)𝑝(𝑥2))𝑑𝑥1𝑑 𝑥2(10)
where 𝑝(𝑥1, 𝑥2)is the joint probability density function while 𝑝(𝑥1)
and 𝑝(𝑥2)are the marginal probability density functions of 𝑥1and 𝑥2,
respectively.
The features selection for defining a block, in this case, is carried
out by selecting the variables with higher mutual information values,
i.e., the variables 𝑥𝑗with 𝑙(𝑥𝑖, 𝑥𝑗)> 𝑙𝑚𝑖𝑖𝑗 , are the selected variables to
include for the block associate to the variable 𝑥𝑖, with 𝑙𝑚𝑖𝑖𝑗 being a
threshold to be defined.
Minimal redundancy maximal relevance. Maximal relevance max-
imizes the relevance of the variables in the set 𝑆with respect to the
target 𝑐as [45]:
𝑚𝑎𝑥𝐷(𝑆, 𝑐 ) => 𝐷 =1
𝑆
𝑥𝑖∈𝑆
𝑙(𝑥𝑖, 𝑐)(11)
where 𝑆represents the number of variables in 𝑆and 𝑙(𝑥𝑖, 𝑐)is the
mutual information between 𝑥𝑖and 𝑐. Note that the selected variables
with maximal relevance could have rich redundancy. Therefore, the
minimal redundancy rule must be added to select mutually exclusive
variables,
𝑚𝑖𝑛𝑅(𝑆) => 𝑅 =1
𝑆2
𝑥𝑖,𝑥𝑗∈𝑆
𝑙(𝑥𝑖, 𝑥𝑗)(12)
So, in order to optimize 𝐷and 𝑅simultaneously, the function 𝛷(𝐷, 𝑅) =
𝐷−𝑅is defined. Now, the variables that maximized 𝛷(𝐷, 𝑅)(Eq. (13))
would be in the same block as the target variable 𝑐.
𝑚𝑎𝑥𝛷(𝐷, 𝑅) =>max
𝑥𝑗∈𝑋−𝑆𝑚−1 
𝑙(𝑥𝑗, 𝑐) − 1
𝑚− 1 
𝑥𝑗∈𝑆𝑚−1
𝑙(𝑥𝑖, 𝑥𝑗)
(13)
Here, the minimal redundancy maximal relevance (mRMR) algorithm
not only considers the relevance but also reduces the redundancy
among the abundant variables in plant-wide processes [28,45].
DCCA method. Detrended Cross-Correlation Analysis (DCCA) is
a modification of the standard covariance analysis in which global
average is replaced by local trends [46,47]. The DCCA coefficient
for two variables 𝑥and 𝑦is calculated as the relation between the
covariance without trend between 𝑥and 𝑦(𝐹2
𝑥𝑦) and the covariance
without trend of 𝑥and 𝑦, i.e.,
𝜌𝐷𝐷𝐶𝐴(𝑥,𝑦)=
𝐹2
𝑥𝑦
𝐹𝑥𝐹𝑦
(14)
The range of this coefficient is [−1 1], taking a value of 0when there
is no correlation between the variables. Here, the DCCA coefficient is
calculated for each two variables in the process, and the variables with
a large value for this coefficient are grouped together in a block.
Clustering. Clustering is an unsupervised machine learning ap-
proach for detecting groups of elements (clusters) according to some
type of similarity or nearness, i.e., the main objective is to find clusters
that minimize the inter-cluster variability and maximize the intra-
cluster variability. The cluster algorithm used in this work is the
Density-Based Spatial Clustering of Applications with Noise (DBSCAN)
[48], that is a density-based clustering algorithm, which uses the
hyper-parameter 𝜖for checking the neighborhood, or density of points,
around each point, thus permitting clusters of arbitrary shape to be
discovered.
Table 1 offers a brief comparison among all the discussed data-
driven methods to divide the plant variables into blocks, briefly de-
scribed in this section, in which the basic nature of the method (linear
o nor lineal), the parameters to be defined and the technique used are
described.
2.2. Canonical Variate Analysis
Canonical Variate Analysis (CVA) is a well-known multivariate
method that maximizes the correlation between two sets of variables.
It has been proposed for multivariate statistical analysis and was also
developed for identifying state-space models [49]. Consider time series
output data 𝒚(𝑡) ∈ ℜ𝑚𝑦and input data 𝒖(𝑡) ∈ ℜ𝑚𝑢, in the instant
𝑡∈ (1,…, 𝑛), the past vector 𝒑(𝑡)containing past outputs and inputs
is defined as:
𝒑(𝑡)=[𝒚𝑇(𝑡− 1),𝒚𝑇(𝑡− 2),…,𝒚𝑇(𝑡−𝑙),𝒖𝑇(𝑡− 1),𝒖𝑇(𝑡− 2),…,𝒖𝑇(𝑡−𝑙)]𝑇
(15)
while the future vector 𝒇(𝑡), comprising the outputs in the present and
future, is:
𝒇(𝑡)=[𝒚𝑇(𝑡),𝒚𝑇(𝑡+ 1),…,𝒚𝑇(𝑡+ℎ)]𝑇(16)
For an assumed state order k, the CVA algorithm computes an
optimal matrix 𝑱𝑘that linearly relates the past vector 𝒑(𝑡)to the
reduced state vector 𝒙𝑘(𝑡)𝜖ℜ𝑘, via the singular value decomposition as:
𝜮𝒑𝒑−1∕2 𝜮𝒑𝒇 𝜮𝒇 𝒇 −1∕2 =𝑼 𝜮 𝑽 𝑻(17)
where 𝜮𝑝𝑝,𝜮𝑓 𝑓 and 𝜮𝑝𝑓 are the covariances of 𝒑(𝑡),𝒇(𝑡)and the
cross-covariance of 𝒑(𝑡)and 𝒇(𝑡), respectively. 𝜮is the diagonal matrix
of non-negative singular values with descending order, 𝑼and 𝑽are
matrices of the right and left singular vectors, and so the matrix 𝑱𝑘is
obtained by
𝑱𝒌=𝑼𝑻
𝒌𝜮𝒑𝒑−1∕2 (18)
where 𝑼𝑘contains the first 𝑘columns of 𝑼, so the state vector 𝒙𝑘(𝑡)is:
𝒙𝑘(𝑡) = 𝑱𝑘𝒑(𝑡) = 𝑼𝑇
𝑘(
𝜮𝑝𝑝)−1∕2 𝒑(𝑡)(19)
The values for 𝑙and ℎ, i.e., the lags to include in the input vectors,
and the state order 𝑘is not known a priory, however they can be cal-
culated by the lags and the order that minimize the Akaike information
criterion [50].
In the CVA method, two types of statistics are used to detect
the behavior of the system: 𝑇2
𝑠, which measures the variations in
the canonical subspace, and 𝑇2
𝑟, for the variations inside the residual
subspace [8,51]:
𝑇2
𝑠=𝒙𝑇
𝑘(𝑡)𝒙𝑘(𝑡)
𝑇2
𝑟=𝒙𝑇
𝑟(𝑡)𝒙𝑟(𝑡)(20)
where 𝒙𝑟(𝑡) = 𝑱𝑟𝒑(𝑡) = 𝑼𝑇
𝑟(
𝜮𝑝𝑝)−1∕2 𝒑(𝑡)and 𝑼𝑟are the remaining
𝑙(𝑚𝑢+𝑚𝑦)−𝑘columns of 𝑼after extracting 𝑼𝑘. The state of the process
is determined using the thresholds of these statistics [8]. Another
possibility to detect faults is using the residual vector:
𝒓(𝒕)= (𝑰−𝑱𝑇
𝑘𝑱𝑘)𝒑(𝑡)(21)
which allows the statistic 𝑄to be obtained:
𝑄(𝑡) = 𝒓𝑻𝒓(22)
2.3. Bayesian Inference (BI)
In this decentralized CVA method, each block returns its own fault
indexes, i.e., the three statistics defined in Eqs. (20)–(22) for each block.
Then, it is necessary to fuse these multiple monitoring outcomes to
obtain a global outperformed result. Various decision fusion strategies

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5
M.J. Fuente et al.
Table 1
Characteristics of the data-driven process decomposition methods.
Method Linear/Non-linear Parameters Technique
LASSO Linear 𝜆Regression
Elastic net Linear 𝜆,𝛿,𝑙𝑒𝑛𝑖: thresholds Regression
SPLS Linear 𝜆1: sparsity Regression
MLP Non-linear 𝐻: number of neurons in the hidden layer, 𝑙𝑛𝑛𝑖: thresholds Regression
SPCA Linear 𝑟: cardinality, 𝐿𝑉 : number of latent variables Information theory
Correlation (C1 and C2) Linear 𝛿𝑖,𝑙𝑐𝑜𝑖𝑗 : thresholds Information theory
Mutual Information (MI1 and MI2) Non-linear 𝑙𝑚𝑖𝑖𝑗 : thresholds Information theory
mRMR Non-linear 𝑙𝑟𝑚𝑟𝑚𝑐𝑖: thresholds Information theory
DCCA (DCCA1 and DCCA2) Linear 𝑙𝑑𝑐 𝑐𝑎1𝑖,𝑙𝑑𝑐 𝑐𝑎2𝑖𝑗 : thresholds Information theory
DBSCAN Non-linear 𝜖Clustering
can be used, though the Bayesian Inference (BI) is the most popular one
to fuse fault indexes [23,24,52,53], giving a single result for the whole
plant.
In this fusion strategy the fault probability of the statistic 𝑆𝑇 (𝑆 𝑇
can be 𝑇2
𝑠,𝑇2
𝑟or 𝑄) in each block 𝑖(𝑖= 1,2,…, 𝐵), 𝐵being the number
of blocks, can be calculated by (where 𝑁denotes the normal condition,
and 𝐹denotes the faulty condition)
𝑃𝑆𝑇 (𝐹∣𝑥𝑖) = 𝑃𝑆𝑇 (𝑥𝑖∣𝐹)𝑃𝑆 𝑇 (𝐹)
𝑃𝑆𝑇 (𝑥𝑖)(23)
where
𝑃𝑆𝑇 (𝑥𝑖) = 𝑃𝑆 𝑇 (𝑥𝑖∣𝑁)𝑃𝑆𝑇 (𝑁) + 𝑃𝑆𝑇 (𝑥𝑖∣𝐹)𝑃𝑆𝑇 (𝐹)(24)
𝑃𝑆𝑇 (𝑁)and 𝑃𝑆 𝑇 (𝐹)are the prior probabilities of the process being
normal and faulty, which can be simply assigned with confidence
level 𝛼and 1 − 𝛼respectively [24,52,53]. The conditional probabilities
𝑃𝑆𝑇 (𝑥𝑖∣𝑁)and 𝑃𝑆 𝑇 (𝑥𝑖∣𝐹)are calculated as:
𝑃𝑆𝑇 (𝑥𝑖∣𝑁) = 𝑒(−𝑆 𝑇𝑖∕𝑆𝑇𝑖,𝑙𝑖𝑚 ), 𝑃𝑆𝑇 (𝑥𝑖∣𝐹) = 𝑒(−𝑆𝑇𝑖,𝑙 𝑖𝑚∕𝑆 𝑇𝑖)(25)
where 𝑆𝑇𝑖represents the statistic 𝑆 𝑇 of the current sample in the 𝑖th
block and 𝑆𝑇𝑖,𝑙𝑖𝑚 is the corresponding threshold for the statistic 𝑆 𝑇 in
the block 𝑖.
The 𝐵𝐼 𝐼 index for 𝑆𝑇 in the whole plant is obtained by fusing all
local results:
𝐵𝐼 𝐼𝑆𝑇 =
𝐵

𝑖=1 𝑃𝑆𝑇 (𝑥𝑖∣𝐹)𝑃𝑆 𝑇 (𝐹∣𝑥𝑖)
𝐵
𝑖=1 𝑃𝑆𝑇 (𝑥𝑖∣𝐹)(26)
If the 𝐵𝐼 𝐼 value for the statistic 𝑆𝑇 is over (1 −𝛼), a fault is detected
with this statistic.
3. Comparative study
In this section a very summarized description is carried out about
the decentralized monitoring scheme for testing the alternative decom-
position approaches. This is made up of three steps: data driven process
decomposition into blocks, local fault detection based on the CVA
method for each block, and decision making based on all local results
provided. This overall decentralized monitoring framework is show in
Fig. 1 and the implementation is explained in the following subsection.
The performance comparison is carried out over the Tennessee Eastman
process (TEP). The TEP model is a well-known realistic model of a
chemical plant and benchmark for control and monitoring studies [54].
3.1. Decentralized monitoring scheme
(A) Off-line procedure.
•Step 1.- Collect the training data under normal operation condi-
tions, scale it to zero mean and unit variance, and construct the
input matrix 𝑿.
•Step 2.- Decompose the input data matrix 𝑿into 𝐵blocks us-
ing one of the methods defined in Section 2: such as LASSO
regression, Elastic net regression, SPLS regression, ANN-MLP,
SPCA, Correlation, Mutual Information, Maximal relevance min-
imal redundancy (mRMR), Detrended Cross-Correlation Analysis
or DBSCAN clustering.
•Step 3.- Use a local monitoring scheme in each block. In this
case, a CVA dynamic monitoring method is carried out, taking
into account the auto and cross-correlated data of the industrial
plants. Calculate the threshold for each statistical index and for
each block.
(B) On-line Global monitoring procedure.
•Step 1.- Collect test data from the plant, normalize the data and
build the input test matrix 𝑿𝒕.
•Step 2.- Divide the matrix 𝑿𝒕into 𝐵blocks by the decomposition
method chosen in the off-line procedure.
•Step 3.- Apply the CVA model in each block, calculating the 𝑇2
𝑟𝑖,𝑇2
𝑠𝑖
and 𝑄𝑖statistics in each block 𝑖= 1,…, 𝐵. Compare the statistics
with their corresponding thresholds.
•Step 4.- Combine the monitoring results of each block using
the Bayesian inference index to obtain global 𝐵𝐼 𝐼𝑇2
𝑟,𝐵𝐼 𝐼𝑇2
𝑠and
𝐵𝐼 𝐼𝑄, which are used to detect faults in the whole plant.
•Step 5.- If some of the BII indexes overpass the threshold (1 − 𝛼),
a fault is detected.
3.2. Indexes for comparison the process decomposition methods
•False Alarms Rate (𝐹 𝐴𝑅)takes into account the robustness of each
statistic. It is the percentage of non-faulty samples classified as
faulty. It can be calculated as:
𝐹 𝐴𝑅 = 100 𝑁𝑁 ,𝐹
𝑁𝑁
%(27)
where 𝑁𝑁,𝐹 is the faultless samples identified as faults, and 𝑁𝑁
is the number of faultless samples.
•Missed Detection Rate (MDR), quantifying the sensitivity to pos-
sible faults. It denotes what percentage of faulty measures are
classified as faultless samples, and can be calculated as:
𝑀𝐷𝑅 = 100𝑁𝐹 ,𝑁
𝑁𝐹
%(28)
where 𝑁𝐹 ,𝑁 is the number of fault samples identified as normal,
and 𝑁𝐹is the number of fault samples.
•Fault Detection Delay (𝐹 𝐷𝐷). This measures how many samples
are needed to detect a fault after its occurrence.
•Number of Faults Detected (𝑁𝐹 𝐷 ).
•Number of blocks (𝑁𝐵 )of the corresponding decomposition
method.

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M.J. Fuente et al.
Fig. 1. The CVA-based decentralized monitoring framework.
3.3. Tennessee Eastman process benchmark
The Tennessee Eastman Process is a well-known benchmark for
monitoring and control [54,55]. Fig. 2 shows the flow diagram of
the process with five major units, i.e., reactor, condenser, compressor,
separator and stripper.
The process has two products from four reactants; an inert product
and a by-product are also present, making, a total of 8 components
denoted as A, B, C, D, E, F, G and H. The process allows 52 measure-
ment, of which 41 are process variables and 11 are manipulated vari-
ables [55]. The data sets given in [50] are widely accepted for process
monitoring studies. These available data, which can be downloaded
from http://web.mit.edu/braatzgroup, are formed by 22 training sets
(including normal and 21 fault operation conditions) collected to record
the process measurements for 24 operation hours; and the correspond-
ing 22 test data sets including, 48 h of the operation plant, in which the
faults were generated after 8simulation hours. The cited 21 faults are
included in Table 2.
4. Results & Analysis
4.1. Experimental setup
First, the methods involved in this comparison (see Section 3, A-Step
2) have been tuned using the standard experimental methodologies
applicable for each one, such as a cross validation and minimizing the
root-mean-square-error (rMSE). Also, the thresholds defined for each
method (𝑙𝑒𝑛𝑖, 𝑙𝑛𝑛𝑖, 𝑙𝑚𝑖𝑖,𝑗 , 𝑒𝑡𝑐 .)are calculated in a grid search looking for
the best results in terms of fault detection, i.e., the best results in terms
of FAR, MDR and FDD for each method.
Notice that, as a CVA method is used, a state space model is
calculated for every block to monitor the process; so, in this case, the
variables chosen to perform every block must be the least correlated
variables. On the other hand, the CVA hyper-parameters, 𝑙and ℎ, the
lags in the past and future vectors and the 𝑘canonical variables (CV)
to be retained to form the state 𝑥𝑘(Eq. (19)) for each block have been
tuned to maximize fault success and minimize fault dismissal. The local
thresholds used for the statistics for each block were adjusted to obtain
a significance level of 99%.
Finally, the threshold for the global decision index (BII index) and
the number of consecutive alarms (𝑁) necessary to detect a fault
were adjusted for each method through a grid search for the best
performance in terms of fault detection, and particularly 𝑁is tuned
to obtain zero real false alarm rate. The threshold for all the methods
is 𝛼= 0.9, and 𝑁is defined for every method as explained below.
•LASSO [31]. A model is carried out for every system variable.
Its hyperparameter 𝜆from Eq. (2) is adjusted for every case to
minimize 𝑟𝑀𝑆 𝐸 using a 3-cross validation procedure. Then, a
block is generated for every variable 𝑥𝑖(𝑖= 1,…, 𝑚): including
its own variable and those others with a zero coefficient in the
respective LASSO regression model (i.e., the least relevant vari-
ables). The CVA parameters were 𝑙= 4 and ℎ= 4 for most cases,
and 𝑁= 5 consecutive anomalous observations are necessary to
detect a fault.
•Elastic net (EN) [31]: An Elastic net based model was tuned for
every variable of the system, using 3-cross validation and looking
for the minimum average rMSE. The parameters 𝛿and 𝜆of Eq. (3)
were individually tuned for every model in a grid search of both
parameters. The block for a certain variable was made up of that
variable and those others with coefficients from its respective
model below a certain threshold, 𝑙𝑒𝑛𝑖, calculated as the mean
value of the coefficients in each model. The order for the CVA
models in each block, i.e., the parameters 𝑙and ℎfor Eqs. (15)
and (16) were such that 2, and 𝑁= 6 consecutive anomalous
observations are necessary to detect a fault.
•SPLS [30]. An SPLS based model for each system variable is
generated, taking as output 𝑌=𝑥𝑖(𝑖= 1...𝑚 ) and 𝑋with-
out 𝑥𝑖as predictor. Different values of sparsity were checked,
i.e., how many coefficients in 𝐵𝑖𝑃 𝐿𝑆 are considered zero, in order
to minimize 𝑟𝑀𝑆 𝐸 using a 3-cross validation procedure. A block
is created for each 𝑥𝑖(𝑖= 1,…, 𝑚) including its own variable
and those others with zero coefficient in its corresponding SPLS
model. Here, the CVA parameters were 𝑙= 4 and ℎ= 4, and 𝑁= 5
consecutive anomalous observations are used to detect a fault.

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Fig. 2. Tennessee Eastman process diagram.
Table 2
Tennessee Eastman process faults.
Fault Description Type
1 A/C feed ratio, B composition constant (Stream 4) Step
2 B composition, A/C ratio constant (Stream 4) Step
3 D feed (Stream 2) Step
4 Reactor cooling water inlet temperature Step
5 Condenser cooling water inlet temperature Step
6 A feed loss (Stream 1) Step
7 C header pressure loss-reduced availability (Stream 4) Step
8 A, B and C compositions (Stream 4) Random variation
9 D feed temperature (Stream 2) Random variation
10 C feed temperature (Stream 4) Random variation
11 Reactor cooling water inlet temperature Random variation
12 Condenser cooling water inlet temperature Random variation
13 Reaction kinetics Slow drift
14 Reactor cooling water valve Sticking
15 Condenser cooling water valve Sticking
16 Unknown –
17 Unknown –
18 Unknown –
19 Unknown –
20 Unknown –
21 Stream 4 valve Sticking
•MLP [31]. A model for each system variable is built by an MLP-
ANN. The MLP hyperparameters were considered in a wide range,
as well as a 3-cross validation procedure to minimize 𝑟𝑀𝑆 𝐸.
The block for a variable is made up of that variable and those
that have obtain the smallest scores [31] by the model, i.e., the
variables with a score 𝑅𝑖𝑗 see (Eq. (7)) below a certain threshold,
𝑙𝑛𝑛𝑖. This threshold was calculated as the mean value of the
relevance values of the variables in each model. Here, the CVA
order for each block was 𝑙= 5 and ℎ= 5, and 𝑁= 5 consecutive
anomalous observations are used to detect a fault.
•SPCA[19]. An SPCA model is built using all the system vari-
ables. The cardinality of the non-zero eigenvector elements (𝑟,
see Eq. (8)) was selected as 𝑟= 40, using a 3-cross validation
procedure to obtain the best model. The obtained 𝑟eigenvectors
were ranked and taking the first 𝐵= 26 latent variables that
explain 70% of the variability of the faultless behavior, defining
the resulting plant blocks. Zero indexes in every preserved eigen-
vector define each block structure, i.e., the variables considered
in that block. The CVA parameters in each block were 𝑙= 3 and
ℎ= 3, and 𝑁= 3 consecutive alarms are used to detect a fault.
•Correlation:
– C1 [23]. Each block 𝑖(𝑖= 1,…, 𝑚), being 𝑚the number
of variables, includes all the variables whose correlation
value with variable 𝑖does not overpass a certain threshold,
i.e., such that ℜ𝐢(𝑥𝑗, 𝑥𝑖)< 𝛿𝑖𝑗= 1,…, 𝑚 𝑗 ≠𝑖with
𝛿𝑖=𝑎⋅𝑚𝑒𝑑𝑖𝑎𝑛(ℜ𝐢). Here, 𝑎= 0.3to control the group size
and choose it in a trial and error procedure to obtain the
best results in the fault detection performance and ℜ𝐢is
the 𝑖th row of the correlation matrix. The CVA parameters
are 𝑙= 3 and ℎ= 3, and 𝑁= 3 consecutive alarms were
necessary to detect a fault.
– C2. In this case, the correlation ℜis calculated, and the
variables with minimal correlation regarding 𝑥𝑖go to the
𝑥𝑖block, i.e., if ℜ𝑖(𝑥𝑖,𝐱𝐣)< 𝑙𝑐𝑜𝑖,𝑗 , where 𝑙𝑐 𝑜𝑖,𝑗 = 0.3 ∗
𝑚𝑒𝑑𝑖𝑎𝑛(ℜ𝐢), then 𝑥𝑗is in the same block as 𝑥𝑖, giving disjoint
blocks and fewer blocks than variables. The CVA parameters
for each block are 𝑙= 3 and ℎ= 3, and 𝑁= 3 consecutive
alarms are used to detect a fault.
•Mutual Information.
– MI1 [22], 𝑀 𝐼𝑥𝑖,𝑥𝑗being the mutual information between
two variables 𝑥𝑖and 𝑥𝑗. If 𝑀𝐼𝑥𝑖,𝑥𝑗<=𝑙 𝑚𝑖𝑖𝑗 , with (𝑖=

Journal of Process Control 135 (2024) 103178
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M.J. Fuente et al.
1,…, 𝑚;𝑗= 1,…, 𝑚)then 𝑥𝑖and 𝑥𝑗are in the same block.
𝑙𝑚𝑖𝑖𝑗 is empirically tuned in a grid search to obtain the
best results from the fault detection performance, resulting
a value of 𝑙𝑚𝑖𝑖𝑗 = 1.3 ∗ 𝐼𝑖𝑚 where 𝐼𝑖𝑚 the median value of
𝑀𝐼𝑥𝑖,𝑥𝑗, 𝑗 = 1,…, 𝑚. The order for the CVA models are 𝑙= 3
and ℎ= 3 and 𝑁= 3 consecutive alarms are used to detect
a fault.
– MI2. In this case, a block is obtained for each variable 𝑖,
including all the variables whose mutual information values
regarding 𝑖does not overpass a certain threshold, i.e., each
block 𝑖(𝑖= 1,…, 𝑚) includes the variables 𝑥𝑗such that
𝑀𝐼𝑥𝑖,𝑥𝑗<0.05. This threshold is tuned by a trial and error
procedure to obtain the best results in the fault detection
performance. If a block is made up of fewer than 4vari-
ables, then this is removed and an extra block is created
containing all these variables. Similar to the MI1 case, the
CVA parameters are 𝑙= 3 and ℎ= 3, and 𝑁= 3 consecutive
alarms are used to detect a fault.
•Maximal relevance minimal redundancy (mRMR)[28]. Select-
ing arbitrarily a variable in X as a target variable 𝑐, for example
the first variable 𝑥1; if 𝑥𝑖,(𝑖= 2,…, 𝑚)satisfies 𝛷 > 𝑙𝑚𝑟𝑚𝑟𝑐,𝑖 then
𝑥𝑖must be in the same block as 𝑐. If 𝑘variables are selected to
be in this block, the rest of the variables are 𝑋−𝑋𝑘, with 𝑋𝑘
being the set with the 𝑘selected variables. Now, the variable
with the minimal 𝛷in the remain dataset (𝑋−𝑋𝑘) is selected
as the new target variable 𝑐, and the procedure is repeated until
all the variables are in a block. The threshold 𝑙𝑚𝑟𝑚𝑟𝑐,𝑖 is selected
experimentally looking for the best result in the fault detection
procedure, where 𝑙𝑚𝑟𝑚𝑟𝑐,𝑖 =𝐼𝑚𝑒𝑎𝑛 and 𝐼𝑚𝑒𝑎𝑛 is the mean value
of mRMR through the variables with the same target. The CVA
parameters in each block are 𝑙= 5 and ℎ= 5, and 𝑁= 6
consecutive alarms are used to detect a fault.
•Detrended Cross-Correlation:
– DCCA1: the 𝑖th row of the DCCA matrix is the DCCA co-
efficient of 𝑥𝑖regarding all other system variables. So, the
𝑖th block is composed of 𝑥𝑖and the variables in the 𝑖th row
lower than the threshold 𝑙𝑑𝑐 𝑐𝑎1 = 0.75 ∗ 𝑚𝑒𝑑 𝑖𝑎𝑛(𝝆𝐃𝐃𝐂𝐀𝐢),
tuned by a trial and error procedure to obtain the best
results in terms of the fault detection performance, i.e.,
minimum FAR and MDR. Here, CVA models are featured by
𝑙= 3 and ℎ= 3, and 𝑁= 3 consecutive alarms are necessary
to detect a fault.
– DCCA2, with fewer blocks than variables. The variables 𝑥𝑗
with 𝝆𝐃𝐃𝐂𝐀𝐢(𝐱𝐢,𝐱𝐣)< 𝑙𝑑𝑐 𝑐𝑎2𝑖,𝑗 are in the same block as 𝑥𝑖,
where 𝑙𝑑𝑐 𝑐𝑎2𝑖,𝑗 = 0.25 ∗ 𝑚𝑒𝑑 𝑖𝑎𝑛(𝝆𝐃𝐃𝐂𝐀𝐢). The CVA models
are parametrized by 𝑙= 5 and ℎ= 5, and 𝑁= 3 consecutive
alarms are needed to detect a fault.
•DBSCAN [32]. The kurtosis and skewness of each variable, the
mean and variance of the correlation, and the mutual informa-
tion of the variables are used by DBSCAN based clustering with
standard validation indexes. The hyperparameter neighbor ratio
(𝜖) is tuned to 𝜖= 0.2. The CVA models are parametrized by 𝑙= 5
and ℎ= 5, and 𝑁= 3 consecutive alarms are used to detect a
fault.
4.2. Results
Here the main results of the proposed comparison are shown and
analyzed: a decentralized monitoring, previously dividing the plant into
blocks, and a CVA-based fault detection method are implemented for
each block. Then, all the outcomes are fused by Bayesian Inference
Index. Simultaneously, the different decentralized methods are com-
pared regarding the centralized CVA method proposed by [8]. The
indexes for comparison described in Section 3are used here to check
the performance of every method. The results shown in the following
subsections have been obtained in this work, except the results of
the centralized CVA method which are shown as in [8], where the
hyperparameters of the CVA method have been tuned to obtain the best
performance.
4.2.1. Block decomposition: number of blocks (𝑁 𝐵)
This is a major challenge for most of the decentralized monitoring
approaches. Table 3 shows the number of blocks implemented by each
method. The decentralized methods based on regression (LASSO, Elas-
tic Net (EN), SPLS, neural networks (MLP)), Correlation (C1), Mutual
Information (MI2) and DCCA1 work with one block per variable, so
they all generate 52 blocks in this case study. This can be a serious
issue when the system has so many variables.
On the other hand, other methods have fewer blocks than variables,
such as SPCA (26 blocks), Mutual Information MI1 (10), correlation C2
(5), Maximal relevance minimal redundancy mRMR (10) and DCCA2
(4). So, these techniques are able to reduce the number of blocks
generated. Finally, the DBSCAN clustering method can deliver a de-
centralized approach with a reduced number of blocks as small as
3.
So, if the number of blocks is an important issue in the application,
it is best to choose a decomposition method with fewer blocks than
variables, as MI1 or mRMR. Otherwise, if there were enough computing
resources the method with best results should be chosen.
4.2.2. Fault detection and Fault Alarm Rate (𝐹 𝐴𝑅)
Table 3 shows that all the decentralized proposals detect more faults
than the centralized CVA method, particularly LASSO regression, SPCA,
C1 and MI1, which detect all 21 faults by the 𝑇2
𝑠statistic. DBSCAN was
also capable of detecting 20 faults by the 𝑇2
𝑟and 𝑄statistics, as central
CVA only gets to detect 18 faults in the best case.
The fault alarm rates (FAR) in %calculated as Eq. (27) for the test
data are also shown in Table 3, taking values from 0to 4.8 for all the
methods, clearly much lower than the central CVA for all the statistics.
Notice that methods such as LASSO regression, SPLS, Neural networks
(MLP), SPCA and DCCA2 show 𝐹 𝐴𝑅 = 0 for almost all the statistics.
4.2.3. Missed Detection Rate (𝑀 𝐷𝑅)
Tables 4–6 show the results of the Missed Detection Rate (MDR) for
all the methods, including the centralized CVA method, for statistics
𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄, respectively.
These tables show three groups of MDR results according to the
faults.
•Statistics 𝐵𝐼 𝐼𝑇2
𝑠,Table 4:
– Faults 1–2, 4–8, 12–14 and 17–18: all the methods offer
very low MDR, except the centralized CVA for faults 4and
7and SPCA and Correlation (C2) for fault 5. So, these faults
are very easy to detect, because the effects of the faults in
the measured variables are high, and all the methods have
good results.
– Faults 10–11, 16, 19–21: the methods based on LASSO
regression and MLP outperform the fault detection rate of
the remaining methods. However, both methods decompose
the plant into 52 blocks (as many blocks as variables), which
can be a serious issue in large-sized plants. Among the
methods with a lower number of blocks, the best is the
decentralized method based on maximal relevance and min-
imal redundancy (mRMR), followed by the method based on
mutual information (MI1).
– Faults 3, 9 and 15: all the methods provide high MDRs,
so they can not detect the faults, but even for this case, the
LASSO regression based method is the best.

Journal of Process Control 135 (2024) 103178
9
M.J. Fuente et al.
Table 3
Number of blocks (𝑁𝐵 ), False Alarm Rate in % (𝐹 𝐴𝑅) and Number of Faults Detected (𝑁𝐹 𝐷).
LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
NB 52 52 52 52 26 52 5 10 52 10 52 4 3 1
FAR 𝑇2
𝑠0 4.79 0 0 0.2 2.7 0.94 1.36 4.69 2.8 0.2 0 1.67 8.3
FAR 𝑇2
𝑟0 1.77 0 0 0 3 2.29 2.6 1.56 1.8 0 0 4.1 12.6
FAR 𝑄0.4 1.4 0.2 0 0 1.14 0.52 1.77 0.83 3.5 0.8 0 3.55 8.7
NFD 𝑇2
𝑠21 19 19 19 21 21 19 21 20 18 19 19 19 18
NFD 𝑇2
𝑟19 18 19 19 18 20 20 19 18 18 20 20 20 18
NFD 𝑄18 18 19 18 19 18 18 19 19 18 18 17 20 17
Table 4
Missed Detection Rate (𝑀𝐷𝑅) - 𝐵 𝐼𝐼𝑇2
𝑠.
Fault LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 0.4 0.4 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.5 0.5 0.4 0.4 0.1
2 1.5 1.6 1.4 1.7 2.1 1.6 1.6 1.4 1.6 1.7 1.7 1.9 1.9 1.1
4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4 4.4 5.4 68.8
5 0.1 0.1 0.1 0.1 67.1 4.4 74.1 0.1 0.1 0.1 0.1 0.25 0.1 0
6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0
7 0.1 0.1 0.1 0.1 0.1 0.1 0.4 0.1 0.1 0.1 0.1 0.1 0.1 38.6
8 2.1 1.8 2.5 1.6 2.2 2.1 2.4 2.6 2.5 2.1 2.2 2.4 2.5 2.1
12 0.2 0.2 0.2 0.2 0.6 0.4 0.7 0.2 0.2 0.4 0.4 0.4 0.2 0
13 4.4 4.6 4.5 4 4.9 4.6 4.9 4.6 4.9 5.1 4.5 5.5 5.1 4.7
14 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0
17 2.7 3.1 3.2 2.4 3.2 3 3.7 2.6 5.7 2.7 3 6.7 9.7 10.4
18 8.4 9.2 9.4 9.2 9.4 9.6 9.9 10 9.4 9.9 9.4 9.7 9.7 9.4
10 7.2 8.2 10 6.7 15.2 9.7 11 13 15.6 6.4 9.6 20.1 10.6 16.6
11 12.4 22.2 18.9 10.4 12.5 15.4 24.6 20.1 24.7 16.1 22.2 35.4 35.5 51.5
16 2.7 8.6 4.6 3.6 8.25 7.2 8.4 26.2 10.5 4.5 3.7 19.9 9.7 16.6
19 2.2 10.1 11.2 0.5 9.9 2.5 37.4 13.2 40.5 15.7 42.1 55.2 55.5 84.9
20 9.5 10.2 12.3 8.5 20.25 20 31.1 14.1 17.5 12.1 17.1 25.2 22.4 24.8
21 42.5 41.9 48.1 30.2 56.6 57 56 50.2 51.2 46.1 48.1 53.6 57.5 44
3 89.2 94.2 97.2 96.5 95 96.2 98.7 98.6 94.4 97.9 92.7 99.5 97.1 98.1
9 92.9 95.6 97 98.4 95.9 98.1 9.1 98.5 95.9 98 94.5 99.7 99.1 98.6
15 75.7 84.2 88.2 78.9 91.1 94.6 97.5 93.4 89.6 96.6 84.9 98.2 90.2 92.8
Mean 16.9 18.9 19.5 16.8 23.6 20.3 22.5 21.5 22 19.8 20.8 25.6 24.3 31.6
MMDR* 5.4 6.9 7.1 4.4 11.8 7.6 14.8 9 10.1 6.9 9.2 13.2 12.6 20.7
MMDR* =Mean MDR for 18 faults (without faults 3, 9 and 15).
Fig. 3. MDR (MMDR*) average for the 𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄statistics in each
method.
For a more intuitive comparison, the average missed alarm rate is
shown in Fig. 3 (the graph in blue) and in Table 4, specifically in
the last two rows of the table, for all the faults (21) and (MMDR*)
for 18 faults (excluding faults 3,9and 15). Considering 𝐵𝐼 𝐼𝑇2
𝑠,
the methods based on LASSO (5.4) and MLP-ANN (4.4) regression
obtain clearly better results, followed by the Elastic net regression
(6.9), mRMR (6.9), SPLS (7.1) and correlation (C1 with a value
of 7.6) based methods.
On the other hand, the centralized CVA method [8] is better for
some faults (5,6,12 and 14) with 𝑀𝐷𝑅 = 0 (although very similar
to other methods such as LASSO, SPLS, MLP-ANN, Elastic net,
mRMR or MI1 and MI2 with values 𝑀𝐷𝑅 = 0.1or 𝑀𝐷𝑅 = 0.2).
However considering the mean of all the faults, the centralized
CVA method is the worst with a large difference.
The best MMDR*s, considering methods with a lower number of
blocks are for the mRMR with a value of 6.9 followed by the
Mutual Information (MI1, with a value of 9) and SPCA (11.8)
methods.
•Statistics 𝐵𝐼 𝐼𝑇2
𝑟,Table 5:
– Faults 1–2, 4–8, 12–14 and 17–18: all the methods offer
very low MDR, except for the SPCA and C2 methods in
fault 5that have a very high value. Also the value for the
mRMR method in the same fault is high, 15.1, compared
with the rest of the methods. Note that the best results are
for the centralized CVA method, close to 𝑀𝐷𝑅 = 0, but this
𝑀𝐷𝑅 is very similar to the remaining methods (MDR =0.1
or MDR =0.2). This result is because the FAR for CVA is
8.12 [8], a very high value with regard to the other methods
(see Table 3), i.e., this statistic threshold is very low, and a
very high FAR and a very low MDR values are obtained.

Journal of Process Control 135 (2024) 103178
10
M.J. Fuente et al.
Table 5
Missed Detection Rate (𝑀𝐷𝑅) - 𝐵 𝐼𝐼𝑇2
𝑟.
Fault LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 0.2 0.2 0.2 0.2 0.4 0.2 0.4 0.2 0.4 0.5 0.2 0.4 0.2 0
2 1.1 1.7 1.2 1.4 2 1.9 1.7 1.5 1.9 1.4 2 1.6 1.6 1
4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0
5 0.1 0.1 0.1 0.1 71.2 0.2 70.2 0.1 0.1 15.1 0.1 0.2 0.1 0
6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0
7 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0
8 1.7 2.5 1.9 1.7 2.2 2 2.1 2.4 2 2.7 2.4 2.5 2.6 1.6
12 0.2 0.2 0.2 0.2 0.9 0.2 0.4 0.2 0.2 1.1 0.2 0.4 0.2 0
13 4.4 4.5 4.4 4.2 4.9 4.5 4.7 4.5 5 5.1 4.6 4.7 4.7 4
14 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0
17 2.5 2.7 2.5 2.4 3.7 2.6 2.4 2.5 5.1 3 2.5 2.9 3.5 2.4
18 9.4 9.7 9.5 9.4 9.7 9.2 9.2 9.4 9.7 9.4 9.4 9.4 9 9.2
10 7 13.4 8.2 7.5 18.1 10.1 9.9 17.9 15.5 10.2 9.6 57.7 7.6 9.9
11 10.7 18.7 12.2 10.6 10.9 10.2 13.6 8.1 19.2 5.2 7.6 11.5 11.5 19.5
16 3.6 11 5.1 4.5 10.5 5.7 6.9 17.5 14 6.6 3.2 8.5 5.2 8.4
19 0.6 4.1 1.6 0.4 1.6 0.6 1.9 1.2 8.6 5.9 1.1 0.7 1.1 1.9
20 8.6 18.6 9.1 8.5 22.8 18 22.2 11.1 16 18.4 10.5 14.2 8.2 8.7
21 35.1 58.6 36.6 40.7 59.7 55.7 59 53.6 58.6 57.2 55.2 49.2 54.5 34.2
3 97.5 98.4 97.6 96.9 98.5 95.6 95.4 96.1 98.1 98.5 97.7 98 93 98.6
9 98.9 99 98.6 98.5 98.4 96.7 96.1 98.4 99.1 98.9 96.7 98.7 96 99.3
15 82 93 86 82.1 98.1 95.5 95.9 88.5 85.2 99.5 94.9 98.1 79.2 90.3
Mean 17.3 20.82 17.9 17.6 24.5 19.5 23.5 19.7 21.41 20.95 18.9 21.9 18 18.5
MMDR* 4.8 8.2 5.2 5.1 12.2 6.8 11.42 7.2 8.7 7.9 6.1 9.1 6.1 5.6
MMDR* =Mean MDR for 18 faults (without faults 3, 9 and 15).
– Faults 10–11, 16, 19–21: the results are very similar for
all the methods but the best are those based on regres-
sion (LASSO, SPLS and MLP) and also those based on DB-
SCAN and Detrended Cros-correlation Analysis (DCCA1).
However, the CVA centralized method is not very behind.
– Faults 3, 9 and 15: the MDRs are very high and, really,
these faults are not detected by any of the methods.
Observing the average missing alarm rate (MMDR*) for all the
faults in Table 5 and Fig. 3 (the graph in red), the LASSO regres-
sion based method is the best one (4.8), followed by the methods
based on MLP-ANN (5.1), SPLS (5.2) and centralized CVA (5.6).
Taking into account the number of blocks, the best is the CVA
centralized method, which has only one block, followed by the
DBSCAN based decentralized method (6.1 and 3blocks) and the
mutual information based decentralized method (MI1, 7.2 but 10
blocks).
•Statistics 𝐵𝐼 𝐼𝑄,Table 6:
– Faults 1–2, 4–8, 12–14 and 17–18: low MDR values for
almost all cases except the centralized CVA for faults 4, 7
and 8; MLP-ANN, EN and DBSCAN for fault 4; SPCA for
faults 4 and 5; Correlation (C2) for faults 4, 5 and 7; and
Detrended cross-correlation (DCCA2) for nearly all these
faults. In this case, in general, the results with this statistic
are not so good as with 𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑇2
𝑠, as it is possible to
see in Figure Fig. 3 (the graph in grey), clearly higher than
the other statistics.
– Faults 10–11, 16, 19–21: the methods based on SPLS and
LASSO regression provide a better fault detection perfor-
mance followed by the method based on correlation (C1).
The method based on maximal relevance minimal redun-
dancy also gives good results for these faults, except for
faults 19 and 21, which are not detected by this method due
to the high MDR obtained for these faults. Notice that MLP-
ANN does not give as good results as for the other statistics,
maybe because MLP-ANN obtains a non-linear model for
each variable, extracting non-linear characteristics of the
plant to build the blocks. 𝐵𝐼 𝐼𝑇2
𝑠monitors the behavior of
the model and 𝐵𝐼 𝐼𝑄monitors the residual space, so if the
model is better, then it is expected that 𝐵𝐼 𝐼𝑇2
𝑠will be more
effective than 𝐵𝐼 𝐼𝑄.
– Faults 3, 9 and 15: MDR is very high, i.e., these faults are
not really detected by any of the methods, but even in this
case, the methods based on linear regression, i.e., LASSO
and SPLS, obtain better results.
Observing MMDR* for all the faults in Table 6 and Fig. 3 (the
graph in grey), it is possible to see that the methods based on
linear regression (SPLS and LASSO) are the best with values 6.8
and 7.5 respectively followed by the method based on correla-
tion (C1, with MDDR* =9.4). If the number of blocks was an
index to be taken into account, the best results would be for
the method based on mutual information (MI1), followed by the
mRMR method, which has a slightly different value, i.e., 13.1
and 13.2 respectively, followed by the method based on DBSCAN
(15.5).
Comparing the MDR results for the correlation methods (C1 and C2)
regarding the three statistics, it is possible to note significantly better
results always for the method C1, i.e., considering a block per variable
as in [23]. Nevertheless, if the MI1 and MI2 methods are considered,
the best results are always for the MI1 method, i.e., the method with
fewer blocks than variables (10 blocks), as proposed by [22] but the
differences in this case are not so much significant. It is also possible
to note that the DCCA method shows slightly better results than the
correlation based methods for statistics 𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑇2
𝑠, but they are
worse for the 𝐵𝐼 𝐼𝑄statistic, so no relevant improvement is achieved
by this complex method, at least for this plant.
Finally, if MI1 method is compared with the mRMR method, which
is an improvement over the mutual information method [27] , the
results are very similar. In the 𝐵𝐼 𝐼𝑇2
𝑠statistics, the mRMR method wins
with a median value of 6.9 compared to MI1, that has a value of 9, but
for the other two statistics, 𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄, the values are similar
(7.2 and 13.1 respectively for the MI1 method and 7.9 and 13.2 for the
mRMR method), so in this example nearly any improvement has been
obtained.
4.2.4. Fault Detection Delay (𝐹 𝐷𝐷)
The detection delay is required to be as low as possible, so the
monitoring scheme has to detect the faults as soon as possible. From

Journal of Process Control 135 (2024) 103178
11
M.J. Fuente et al.
Table 6
Missed Detection Rate (𝑀𝐷𝑅) - 𝐵 𝐼𝐼𝑄.
Fault LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 0.2 0.6 0.4 0.5 1 0.4 1.1 0.5 0.4 1.1 0.5 1.1 0.7 0.3
2 1.6 2.9 1.5 2.5 3.1 2.1 3.2 2 2 2.4 2.4 5.5 1.5 2.6
4 0.1 39.4 0.1 84.8 46.2 7.7 20.9 3.6 17.7 0.1 24 98.9 23.5 97.5
5 0.1 0.1 0.1 0.1 76.4 0.4 83.5 0.1 0.1 0.5 0.1 0.5 0.1 0
6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0
7 0.1 0.1 0.1 10.4 0.1 0.1 74 0.1 0.1 0.1 0.1 19.6 0.1 48.6
8 2.2 3.6 2 3.4 5.4 2.6 11.9 2.9 2.4 7.2 2.5 20.7 2.2 48.6
12 0.2 0.2 0.2 0.2 5.9 0.6 15.4 0.4 0.2 0.6 0.4 6.9 0.4 2.1
13 4.9 5.4 4.2 4.1 8.9 4.7 8.5 5.4 5.1 7.7 6.2 16.5 5.5 5.5
14 0.2 0.3 0.2 2.7 0.6 0.2 1 0.2 0.4 0.1 0.4 4.2 0.2 12.2
17 3.6 8.7 4 3.9 19.2 5.1 11 6.7 10.1 5 6.6 25.9 12.5 13.8
18 9.2 10.2 8.4 9.5 10.5 9.7 11.1 9.9 9.7 10.7 9.5 10.9 9.5 10.2
10 10 17.1 9.6 16.9 38.5 8.4 21.6 33.1 10.9 9.1 17.2 49.9 24.1 59.9
11 17.5 47.4 15.1 45.1 50.1 27.6 40 36.6 35.4 7.1 41 77.4 25.4 66.4
16 6.5 15.5 4.7 18.4 27.2 8 20.2 34.6 10.2 8.4 12 52.9 20.9 42.9
19 10.1 31.2 4.9 62.5 35 3 80.1 20.4 52.1 68.2 41.9 85.7 62.9 92.3
20 16.2 15.5 16.4 34.8 24.4 21.5 43.2 17.2 40.1 20.1 31.9 39.2 31 35.4
21 51.9 64.9 50.6 71.4 71.5 67 85.9 61.9 62.1 83.4 61.2 85.9 58.8 54.8
3 93.2 97.4 90.5 97.9 99.1 98.7 99.1 95.9 99 97.9 98.5 99 93.8 98.5
9 93.1 98 88.7 98.9 99.4 99 98.5 98.1 99.4 98.5 98.5 99 90.9 99.3
15 89.1 97.7 85.4 98.5 96.6 98.2 99.6 94.5 83.5 98 98.4 98.2 87.2 97.9
Mean 19.5 26.5 18.4 31.7 34.2 22.2 39.5 25 25.7 25.4 26.4 42.8 26.3 42.3
MMDR* 7.5 14.6 6.8 20.6 23.6 9.4 29.6 13.1 14.4 13.2 14.3 33.4 15.5 32.9
MMDR* =Mean MDR for 18 faults (without faults 3, 9 and 15).
an engineering point of view, a fault is usually indicated only when a
determined number of consecutive values of the statistics exceed their
thresholds [8]. In this work, this number of consecutive values, 𝑁,
for each of the methods is described in Section 4.1, and the detection
delay is recorded at the first time instance for which the control limit is
exceeded. Tables 7–9 and Fig. 4 show the detection delays for statistics
𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄, respectively.
•Statistics 𝐵𝐼 𝐼𝑇2
𝑠,Table 7:
Table 7 is also organized in three fault sets. Here however, every
method shows similar detection delay, except the centralized
CVA method for faults 4and 11, which shows very high values
regarding the remaining methods, and for fault 19, which is not
detected by CVA.
To look for the best method, we use the average detection de-
lay (MFDD*) for 18 faults excluding, faults 3, 9 and 15 (last
row of Table 7) and Fig. 4. Considering 𝐵𝐼 𝐼𝑇2
𝑠, the MLP-ANN
based regression is the best one, (26,5) with a slightly differ-
ence with regard to the LASSO based regression (with value 30)
and the DCCA2 based method (30,2). Also, the centralized CVA
method [8] obtains the worst results, showing a large MFDD*
with regard to the rest of methods, even considering that fault
19 is not detected; so only 17 faults are included in calculation
for MFDD* in this method (i.e., excluding faults 5, 9, 15 and 19).
Considering the MFDD* for the methods with few blocks, the De-
trended Cross-correlation Analysis (DCCA2) is the best followed
by the method based on mutual information (MI1), which is
slightly worse.
•Statistics 𝐵𝐼 𝐼𝑇2
𝑟,Table 8:
The Fault Detection Delay of each method, including the cen-
tralized CVA method, is very similar for every fault. The only
difference is the detection time in fault 21, where Mutual Infor-
mation (MI1) is clearly the best for that fault, followed by DCCA2.
The worst case is for the centralized CVA method.
All this can be much clearly observed using the average value,
MFDD* and Fig. 4: the best is the MI1 method, but no very
significant differences are observed, except for the DBSCAN, C2,
MI2, mRMR, EN, and centralized CVA methods, which are the
worst.
Fig. 4. FDD (MFDD*) average for the 𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄statistics in each method.
•Statistics 𝐵𝐼 𝐼𝑄,Table 9:
The results are very similar for all the methods, with a few
exceptions such as for fault 4by EN and MLP-ANN with a very
large 𝐹 𝐷𝐷, or DCCA2 method for fault 11 and the non-detection
of fault 4. The best MFDD*, see Fig. 4 and the last row of Table 9,
is for the SPCA method (28.1), followed by the methods based on
linear regression, i.e., SPLS (32.1) and LASSO (32.5). Note that
the CVA method only takes 17 faults for MFDD* (i.e., excluding
faults, 3, 9, 15 and 19), so this is not comparable with regard to
the other methods.
In any case, taking into account the eight indexes considered (MDR
and FDD for 𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and 𝐵𝐼 𝐼𝑄), the false alarms rate (FAR) and
the number of detected faults, the decentralized strategies achieved
better results for almost all of these indexes than the centralized CVA

Journal of Process Control 135 (2024) 103178
12
M.J. Fuente et al.
Table 7
Fault Detection Delay (𝐹 𝐷𝐷) - 𝐵𝐼 𝐼𝑇2
𝑠.
Fault LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 3 4 3 2 3 2 2 2 3 5 4 3 3 2
2 13 14 13 9 15 15 13 11 13 16 15 15 13 13
4 1 2 1 1 1 1 1 1 1 2 1 1 1 461
5 1 2 1 1 1 1 1 1 1 2 1 2 1 1
6 1 2 1 1 1 1 1 1 1 2 1 1 1 1
7 1 2 1 1 1 1 1 1 1 2 1 1 1 1
8 18 17 20 17 19 17 21 18 20 18 19 19 20 20
12 2 3 2 2 3 3 3 2 2 4 3 3 2 2
13 36 38 37 32 39 37 39 37 39 42 39 42 44 42
14 1 2 2 1 1 1 1 1 1 2 1 2 2 2
17 19 20 20 19 23 20 21 20 23 23 21 21 22 27
18 77 80 78 76 76 79 80 80 81 81 79 78 79 83
10 22 24 23 22 21 21 22 22 21 23 21 23 22 25
11 7 12 7 7 7 7 7 7 7 7 7 8 7 292
16 9 10 9 9 10 9 9 14 11 10 9 14 10 14
19 3 12 12 3 11 2 12 11 11 12 11 15 47
20 71 71 71 67 66 65 74 67 75 68 76 69 71 82
21 255 276 300 207 285 286 271 265 272 286 255 227 490 275
3 43 45 51 45 51 41
9 5 6 731 5
15 577 676 635 3 241 241 681 631 679 575 706 639 677
MFDD* 30 32.8 33.4 26.5 32.4 31.5 32.2 31 32.3 33.7 31.3 30.2 46.5 79
MFDD* =Mean value of fault detection delay for 18 faults (without faults 3, 9 and 15).
Table 8
Fault Detection Delay (𝐹 𝐷𝐷) - 𝐵𝐼 𝐼𝑇2
𝑟.
Fault LASSO EN SPLS MLP SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 2 3 2 2 2 2 3 2 3 5 2 3 2 3
2 11 15 11 12 16 15 15 12 15 12 16 13 13 15
4 1 2 1 1 1 1 1 1 1 2 1 1 1 1
5 1 2 1 1 2 2 1 1 1 3 1 2 1 1
6 1 2 1 1 1 1 1 1 1 2 1 1 1 1
7 1 2 1 1 1 1 1 1 1 2 1 1 1 1
8 17 21 18 18 20 16 19 16 19 23 19 20 19 20
12 2 3 2 2 3 2 2 2 2 3 2 3 2 2
13 35 39 35 34 39 36 39 36 41 45 38 38 39 39
14 1 2 1 2 1 1 1 1 1 2 1 2 1 1
17 20 20 20 19 23 20 20 20 23 21 20 21 20 20
18 76 80 77 76 80 76 79 76 79 77 80 76 76 79
10 22 24 22 22 25 21 24 24 24 24 23 36 20 23
11 7 12 7 7 6 6 6 6 7 8 7 7 6 11
16 9 9 9 10 9 8 9 11 10 10 8 10 9 9
19 2 12 3 3 2 2 2 2 11 13 11 2 2 11
20 66 66 67 66 65 65 65 65 72 66 67 66 65 66
21 257 490 226 287 286 286 484 174 479 474 286 211 456 511
3 484 54 586
9 731
15 643 643 3 741 391 746 132 659
MFDD* 29.5 44.7 28 31.3 32.3 31.2 42.9 25 43.9 44.1 32.4 28.5 40.8 45.2
MFDD* =Mean of DD for 18 faults (without faults 3, 9 and 15).
method. So, all this shows that the decentralized approaches are prefer-
able alternatives to centralized ones. In this comparison, the only index
with good results for the centralized CVA method is the MDR with the
𝑇2
𝑟statistic. However, if the division of the plant is not adequate the
results are not so good, such as for the DCCA2 method, which is the
worst decentralized method.
Comparing the decentralized methods, the LASSO regression based
method is the best, followed by the SPLS based method, i.e., in general,
the methods based on linear regression obtain the best results. So, these
methods are the best to carry out the division of the plant, followed by
the MLP-ANN regression based method with a very similar performance
regarding 𝐵𝐼 𝐼𝑇2
𝑠and 𝐵𝐼 𝐼𝑇2
𝑟, but not for 𝐵𝐼 𝐼𝑄and the C1 based method.
However, all these methods need a total decentralization of the plant,
meaning the same number of blocks as variables. If the number of
variables is very high, or it is necessary to reduce the number of blocks
due to computational issues, the Mutual Information (MI1) is the best
method with a lower number of blocks than variables.
5. Conclusions
In this paper, a comparative study of decentralized monitoring
schemes based on CVA has been carried out. These decentralized meth-
ods divide the plant-wide process variables into several blocks, then
statistical data models are built to perform local monitoring. All the
local monitoring from every block is integrated through a decision fu-
sion algorithm. In this work, different approaches for the plant variable
decomposition are studied: some based on regression, such as LASSO,
Elastic net, SPLS and MLP-ANN; others are based on information such
as Sparse Principal Component Analysis (SPCA), Correlation, Mutual
Information, Maximal Relevance and Minimal Redundancy (mRMR)
and Detrended Cross-correlation Analysis (DCCA); and finally others
are based on clustering such as DBSCAN method. Then, a Canonical
Variate Analysis (CVA) based local dynamic monitor is set up for each
of these established blocks, and finally the Bayesian Inference (BI) is
introduced to achieve a global decision of fault detection for the whole

Journal of Process Control 135 (2024) 103178
13
M.J. Fuente et al.
Table 9
Fault Detection Delay (𝐹 𝐷𝐷) - 𝐵𝐼 𝐼𝑄.
Fault LASSO EN SPLS ANN SPCA C1 C2 MI1 MI2 mRMR DCCA1 DCCA2 DBSCAN CVA [8]
1 2 6 3 4 8 3 10 4 3 10 4 10 6 2
2 14 31 14 19 25 19 27 15 19 22 19 46 13 25
4 1 79 1 249 1 1 5 1 1 2 11 1 0
5 1 2 1 1 4 3 7 1 1 2 1 4 1 0
6 1 2 1 1 1 1 1 1 1 2 1 1 1 0
7 1 2 1 1 1 1 1 1 1 2 1 3 1 0
8 18 23 20 21 26 22 26 22 20 26 23 30 22 21
12 2 3 2 2 10 4 27 2 2 6 3 4 2 0
13 41 48 35 35 47 39 47 45 43 49 50 58 49 43
14 2 2 2 10 3 2 3 2 3 2 3 2 2 1
17 21 24 20 21 28 22 24 22 25 24 23 30 28 23
18 80 80 78 79 85 79 89 82 80 87 83 88 82 84
10 24 25 22 25 29 22 25 27 22 23 26 36 25 44
11 8 12 6 9 9 6 7 9 11 8 7 99 11 27
16 10 10 9 10 14 9 10 12 9 11 9 20 14 11
19 12 13 12 19 11 2 13 13 12 12 11 15 17
20 71 65 68 74 64 64 76 65 77 68 76 78 73 72
21 276 561 283 489 140 480 639 288 299 712 492 648 485 302
3
9 43 239
15 676 118 677 46 524
MFDD* 32.5 55.1 32.1 59.4 28.1 43.3 57.6 34 35 59.5 46.8 65.1 46.3 38.5
MFDD* =Mean FDD for 18 faults (without faults 3, 9 and 15).
process based on the information coming from the local monitors. All
the discussed methods were tested over an industrial benchmark of TE
process to complete a detailed comparison study. As far as the results
are concerned, we would like to point out that:
•Taking into account all the indexes considered (MDR and Fault
Detection Delay (FDD) for the three statistics 𝐵𝐼 𝐼𝑇2
𝑠,𝐵𝐼 𝐼𝑇2
𝑟and
𝐵𝐼 𝐼𝑄, the False Alarms Rate (FAR) and the Number of Faults
Detected (NFD)), the decentralized methods, in general, are better
than the centralized CVA model. In this comparative, the only
index with good results for the CVA method is the MDR with the
𝑇2
𝑟statistic, but with a very high fault alarm rate, compared to
the rest of the methods.
•The decomposition methods based on regression, in general, give
lower MDRs in most of the faults than for the remaining methods
in the comparative, due to their ability to capture the strong
relationships between the variables. The methods based on linear
regression, such as LASSO and SPLS, provide good results for all
the indexes and all the statistics; while the method based on MLP-
ANN (a non-linear regression method) provides low values for
the statistics 𝐵𝐼 𝐼𝑇2
𝑠and 𝐵𝐼 𝐼𝑇2
𝑟, but worse results for the 𝐵𝐼 𝐼𝑄
statistic, for both MDR and Detection Delay indexes. The method
based on correlation (C1) also gives goods results, but they are
slightly worse than the methods based on regression. However,
all these methods decompose the plant into a very high number
of blocks.
•If the number of blocks is an important index to take into account,
the method based on Mutual Information (MI1) also gives low
values for the MDR and detection delay indexes in all the statis-
tics, and these results are better than the other method results,
which have a lower number of blocks than variables, i.e., SPCA,
C2, mRMR, DCCA2 and DBSCAN.
•Comparing the results and considering all the indexes for the
correlation methods (C1 and C2), it is worth noting that the
results are always better for the method C1, i.e., considering a
block per variable as proposed by [23], instead of using fewer
non-overlapping blocks with a clearly better perform. If a com-
parative between the Correlation methods and the Detrended
Cross-Correlation analysis is also carried out, it can be see that
the results are not very different, so no relevant improvement has
been reached with this more complex analysis, at least for this
case study.
•Comparing the results and considering all the indexes for the mu-
tual information methods (MI1 and MI2), it is worth nothing that
the results are always better for the method MI1, i.e., considering
a lower number of blocks than variables, as proposed by [22]
instead of using a block per variable as in the C1 method. If
a comparison between the Mutual Information methods and the
Maximal Relevance Minimal Redundancy (mRMR) is also carried
out, it can be see that results between the methods MI1 and
mRMR are not very different with the MDR indexes considering
the three statistics; both divide the plant variables into 10 blocks,
but the results for the fault detection delay (FDD) index and
the three statistics are clearly better for the MI1, so no notice-
able improvement has been obtained with this more complicated
analysis, at least for this case study.
CRediT authorship contribution statement
M.J. Fuente: Conceptualization, Funding acquisition, Investigation,
Methodology, Software, Supervision, Validation, Writing – original
draft, Writing – review & editing. M. Galende-Hernández: Concep-
tualization, Investigation, Methodology, Software, Supervision, Val-
idation, Writing – original draft, Writing – review & editing. G.I.
Sainz-Palmero: Conceptualization, Funding acquisition, Investigation,
Methodology, Software, Supervision, Validation, Writing – original
draft, Writing – review & editing.
Declaration of competing interest
None declared.
Data availability
Data will be made available on request.
Acknowledgments
This work was supported by the Spanish Government through the
Ministerio de Ciencia e Innovación (MICINN), Spain / Agencia Estatal
de Investigación (AEI), Spain under Grant (MICCIN/AEI/10.13039/
501100011033) PID2019-105434RB-C32.

Journal of Process Control 135 (2024) 103178
14
M.J. Fuente et al.
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