Fault Detection for Multimode Processes Using an Enhanced Gaussian Mixture Model and LC-KSVD Dictionary Learning

Abstract

Monitoring multimode industrial processes presents significant challenges due to varying operating conditions, nonlinear dynamics, and mode-dependent feature distributions. This paper proposes a novel process monitoring framework that integrates an enhanced Gaussian Mixture Model (GMM) for mode identification with Label Consistent K-SVD (LC-KSVD) for sparse dictionary learning. The improved GMM employs a parallelized Expectation–Maximization algorithm to achieve accurate and scalable mode partitioning in high-dimensional environments. Subsequently, the LC-KSVD then learns label-consistent, discriminative sparse representations, enabling effective monitoring across modes. The proposed method is evaluated through a simulation study and the widely used Continuous Stirred Tank Heater (CSTH) benchmark. Comparative results with traditional techniques such as LNS-PCA and FGMM demonstrate that the proposed method achieves superior fault detection rates (FDRs) and significantly lower false alarm rates (FARs), even under complex mode transitions and mild fault scenarios. Furthermore, the method also provides interpretable fault isolation through reconstruction-error-guided variable contribution analysis. These findings confirm that the proposed LC-KSVD-based scheme offers a reliable solution for fault detection and isolation in multimode process systems.

1. Introduction

To enhance operational safety and ensure product quality, process monitoring and fault isolation have become critical components in modern industrial systems [1]. Over the past few decades, various techniques have been developed to address these tasks, among which multivariate statistical process monitoring (MSPM) has emerged as a mainstream approach due to its effectiveness in handling high-dimensional process data and uncovering abnormal behavior based on statistical patterns [2,3,4].

Conventional MSPM techniques, such as principal component analysis (PCA) and its variants, typically assume that process data follow a time-invariant and unimodal Gaussian distribution [5,6,7]. However, practical industrial processes often operate under multiple distinct modes due to variations in raw material properties, production specifications, throughput demands, and control strategies [8,9,10]. Under such multimodal conditions, the performance of standard MSPM approaches degrades significantly, as they fail to capture the complex distributional shifts and mode-dependent dynamics.

In response to these limitations, a growing body of research has explored multimode process monitoring methods. For instance, Li et al. proposed a local neighborhood standardization strategy to preprocess multimodal data [11]. Yu et al. adopted a finite Gaussian Mixture Model (GMM) combined with Bayesian inference for mode-wise monitoring [12]. Jiang et al. further integrated GMM with optimal principal component selection to enhance diagnosis capability in multimode processes [13]. More recent studies have extended Bayesian or sparse approaches, such as mode identification with transitional modeling [14], sparse principal component selection coupled with Bayesian inference-based probability [15], and Bayesian reconstruction strategies for specific applications like coal mills [16]. Cui et al. additionally proposed dimensionality-reducing GMM-based reconstruction combined with deep learning models for nonlinear multimode monitoring [17]. These methods demonstrate the effectiveness of combining GMM partitioning with local models or probabilistic inference, yet they still mainly focus on clustering and detection, with limited exploration of discriminative dictionary learning or variable-level fault isolation. Further, Kodamana et al. developed mixtures of probabilistic PCA for modeling and monitoring multimode dynamic processes [18]. Zhang et al. proposed a modified PCA algorithm, designated PCA-EWC, for monitoring multimode processes to overcome catastrophic forgetting of PCA for successive modes [19]. Although these approaches improve upon traditional MSPM in handling multimodal characteristics, they are still fundamentally rooted in variance-preserving dimensionality reduction and assume smooth Gaussian latent structures.

In addition, real-world process data may exhibit sparsity and non-Gaussianity that are poorly modeled by PCA-type methods. Sparse representations not only enhance interpretability but also improve generalization and robustness by emphasizing the most informative features [20,21]. Recent studies have demonstrated that sparse coding techniques offer considerable advantages in high-dimensional and heterogeneous environments [22]. Nevertheless, most current multimode monitoring frameworks do not explicitly exploit sparse structures or address non-Gaussian feature distributions. Furthermore, despite increasing interest in fault detection, few studies have systematically tackled the more challenging problem of fault isolation in multimode systems, where overlapping features and operating conditions complicate variable-wise interpretation. Inspired by developments in computer vision and signal processing [23,24,25,26], where dictionary learning has proven effective in extracting sparse and discriminative representations, this paper proposes a novel process monitoring approach based on label-consistent K-SVD (LC-KSVD) dictionary learning [27,28]. Unlike traditional MSPM models, LC-KSVD embeds supervision directly into the dictionary learning process, enabling the model to construct sparse representations that preserve both structural similarity and class-specific information.

Beyond industrial process monitoring, similar methodological challenges have been widely addressed in the domain of structural health monitoring (SHM). Recent advances have integrated machine learning, deep learning, and metaheuristic optimization with vibration-based analysis to enhance structural damage detection. For instance, hybrid approaches combining particle swarm optimization (PSO), radial basis functions, and emerging algorithms such as the YUKI method have been developed for accurate crack identification in composite beams, demonstrating improved robustness and convergence compared with classical PSO [29]. Other studies have explored optimized neural networks and bio-inspired algorithms for defect prediction in civil and mechanical systems [30,31], as well as deep learning-based frameworks for automated SHM tasks [32]. These developments, while originating in SHM, reflect parallel challenges to those faced in multimode process monitoring, reinforcing the relevance of designing monitoring frameworks that not only achieve high detection accuracy but also maintain interpretability and robustness [33,34].

In this context, the present study focuses on industrial process monitoring and aims to address these parallel challenges by integrating probabilistic mode partitioning, discriminative sparse representation, and interpretable fault isolation into a unified framework. This design ensures that the proposed method remains both accurate and practically interpretable under multimode operating conditions. The proposed framework comprises three major components. First, a parallelized GMM-based mode segmentation method is employed to partition training data into coherent operating regimes. Second, an LC-KSVD model is trained using the labeled multimodal data, and monitoring statistics are derived from the reconstruction error under the learned sparse dictionary. Third, a novel fault isolation strategy based on reconstruction error and missing value estimation is developed, leveraging insights from missing data analysis and reconstruction-based contribution evaluation. Experimental results on both a numerical simulation and the widely studied Continuous Stirred Tank Heater (CSTH) benchmark confirm that the proposed method not only outperforms conventional techniques in fault detection accuracy and false alarm suppression but also provides interpretable and precise fault localization, even under complex multimodal conditions. In summary, the contribution of this work lies in the development of an enhanced multimode monitoring framework that advances beyond existing approaches in several important aspects. GMM is improved through a parallel EM algorithm, which significantly accelerates convergence and makes the method scalable to high-dimensional process data. This improvement is further coupled with LC-KSVD dictionary learning, allowing the monitoring model to capture discriminative and interpretable sparse representations across different operating modes. On this basis, we design a reconstruction-error-guided isolation mechanism that extends sparse dictionary learning from fault detection to variable-level fault localization under multimode conditions. By combining these elements into a unified framework, the proposed method achieves efficient mode partitioning, accurate detection, and interpretable fault isolation.

The remainder of this paper is organized as follows: Section 2 details the methodological foundation, including the enhanced GMM for mode identification and the LC-KSVD algorithm for sparse dictionary learning and monitoring. Section 3 introduces the monitoring and fault isolation procedures under the proposed framework. Section 4 presents two case studies to validate the performance of the proposed method. Finally, Section 5 concludes the paper and discusses potential avenues for future research.

2. Methodology

2.1. Enhanced Gaussian Mixture Model

2.1.1. Gaussian Mixture Model Fundamentals

GMM is a probabilistic model that assumes the observed data are generated from a mixture of several Gaussian distributions, each representing a different latent component. It is widely used for modeling multimode distributions, especially in unsupervised clustering and pattern recognition tasks. Formally, given a d-dimensional data point $\mathbf{x}\in {\mathbb{R}}^d$, the probability density function (PDF) of a GMM with K components is defined as

$$p\left(\mathbf{x}\right)=\sum _{k=1}^K{\pi }_k \mathcal{N}(\mathbf{x}\mid {\mathit{\mu }}_k,{\mathbf{\Sigma }}_k),$$

(1)

where ${\pi }_k$ is the mixture coefficient (or prior probability) of the k-th Gaussian component, satisfying ${\sum }_{k=1}^K{\pi }_k=1$ and ${\pi }_k\ge 0$, ${\mathit{\mu }}_k\in {\mathbb{R}}^d$ is the mean vector of the k-th component, ${\mathbf{\Sigma }}_k\in {\mathbb{R}}^{d\times d}$ is the covariance matrix of the k-th component, and $\mathcal{N}(\mathbf{x}\mid {\mathit{\mu }}_k,{\mathbf{\Sigma }}_k)$ denotes the multivariate Gaussian distribution.

$$\mathcal{N}(\mathbf{x}\mid \mathit{\mu },\mathbf{\Sigma })={\displaystyle \frac{1}{{\left(2\pi \right)}^{d/2}{|\mathbf{\Sigma }|}^{1/2}}}exp\left(-{\displaystyle \frac{1}{2}}{(\mathbf{x}-\mathit{\mu })}^{\top }{\mathbf{\Sigma }}^{-1}(\mathbf{x}-\mathit{\mu })\right).$$

(2)

The parameters ${\{{\pi }_k,{\mathit{\mu }}_k,{\mathbf{\Sigma }}_k\}}_{k=1}^K$ are typically estimated via the Expectation– Maximization (EM) algorithm, which iteratively maximizes the log-likelihood of the observed data

$$logL(\Theta )=\sum _{i=1}^{N}log\left(\sum _{k=1}^K{\pi }_k \mathcal{N}({\mathbf{x}}_i\mid {\mathit{\mu }}_k,{\mathbf{\Sigma }}_k)\right),$$

(3)

where $\Theta ={\{{\pi }_k,{\mathit{\mu }}_k,{\mathbf{\Sigma }}_k\}}_{k=1}^K$ represents the set of all model parameters and N is the number of training samples.

GMM is particularly well-suited for modeling complex systems such as thermal processes, where data often exhibit multimodal characteristics due to varying operating conditions. The soft clustering nature of GMM enables probabilistic assignment of each data point to multiple components, offering greater flexibility than hard partitioning methods such as k-means.

2.1.2. Parallel EM Algorithm

The Expectation–Maximization (EM) algorithm [35] is a widely used method for estimating the parameters of GMM. Despite its effectiveness, the traditional EM algorithm faces significant computational challenges when applied to large-scale or real-time industrial process data. Specifically, it suffers from slow convergence and poor scalability due to its inherently sequential structure. To address these limitations, we adopt a parallel EM algorithm that exploits the independence of data samples to accelerate computation by distributing the workload across multiple processors. The parallel EM algorithm maintains the core structure of the EM process but parallelizes the Expectation (E) and Maximization (M) steps, thereby significantly improving computational efficiency.

Given the current estimates of parameters ${\Theta }^{\left(t\right)}={\{{\pi }_k^{\left(t\right)},{\mathit{\mu }}_k^{\left(t\right)},{\mathbf{\Sigma }}_k^{\left(t\right)}\}}_{k=1}^K$, the E-step computes the posterior probability (responsibility) that a sample ${\mathbf{x}}_i$ belongs to the k-th Gaussian component

$${\gamma }_{ik}^{\left(t\right)}={\displaystyle \frac{{\pi }_k^{\left(t\right)} \mathcal{N}({\mathbf{x}}_i\mid {\mathit{\mu }}_k^{\left(t\right)},{\mathbf{\Sigma }}_k^{\left(t\right)})}{{\sum }_{j=1}^K{\pi }_j^{\left(t\right)} \mathcal{N}({\mathbf{x}}_i\mid {\mathit{\mu }}_j^{\left(t\right)},{\mathbf{\Sigma }}_j^{\left(t\right)})}}.$$

(4)

This computation is embarrassingly parallel, as each ${\gamma }_{ik}^{\left(t\right)}$ can be calculated independently across all data samples $i\in 1,\dots ,N$ and mixture components $k\in 1,\dots ,K$.

Once responsibilities are computed, the M-step updates the model parameters based on the following soft assignments:

(1) Update of mixing coefficients

$${\pi }_k^{(t+1)}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\gamma }_{ik}^{\left(t\right)}.$$

(5)

(2) Update of mean vectors

$${\mathit{\mu }}_k^{(t+1)}={\displaystyle \frac{{\sum }_{i=1}^N{\gamma }_{ik}^{\left(t\right)}{\mathbf{x}}_i}{{\sum }_{i=1}^N{\gamma }_{ik}^{\left(t\right)}}}.$$

(6)

(3) Update of covariance matrices

$${\mathbf{\Sigma }}_k^{(t+1)}={\displaystyle \frac{{\sum }_{i=1}^N{\gamma }_{ik}^{\left(t\right)}\left({\mathbf{x}}_i-{\mathit{\mu }}_k^{(t+1)}\right){\left({\mathbf{x}}_i-{\mathit{\mu }}_k^{(t+1)}\right)}^{\top }}{{\sum }_{i=1}^N{\gamma }_{ik}^{\left(t\right)}}}.$$

(7)

These summations over i are also naturally parallelizable. By partitioning the dataset and computing partial sums on different processors, followed by a reduction operation (e.g., MPI Reduce in distributed computing), the updated parameters can be efficiently obtained. The convergence of the algorithm is monitored by evaluating the log-likelihood of the dataset

$$logL\left({\Theta }^{\left(t\right)}\right)=\sum _{i=1}^{N}log\left(\sum _{k=1}^K{\pi }_k^{\left(t\right)} \mathcal{N}({\mathbf{x}}_i\mid {\mathit{\mu }}_k^{\left(t\right)},{\mathbf{\Sigma }}_k^{\left(t\right)})\right).$$

(8)

The iteration stops when the relative improvement of the log-likelihood falls below a pre-defined threshold $\epsilon$

$${\displaystyle \frac{logL\left({\Theta }^{(t+1)}\right)-logL\left({\Theta }^{\left(t\right)}\right)}{\left|logL\left({\Theta }^{\left(t\right)}\right)\right|}}<\epsilon .$$

(9)

The parallel EM algorithm offers several key advantages over its traditional counterpart, particularly in the context of large-scale and real-time industrial process data. By distributing the computational workload of both the E-step and M-step across multiple processors, the algorithm significantly improves execution speed and scalability, enabling efficient parameter estimation even for high-dimensional datasets. This parallelization reduces the convergence time without compromising accuracy, making it well-suited for scenarios where rapid decision-making is essential, such as fault diagnosis in thermal power systems. Furthermore, the algorithm retains the statistical robustness of conventional EM while overcoming its limitations in handling large volumes of multimodal data. By leveraging this parallel EM framework, the proposed diagnostic model can efficiently and autonomously identify distinct operating modes in thermal processes, providing a reliable basis for subsequent dictionary learning.

2.2. Fault Detection with LC-KSVD Dictionary Learning

2.2.1. K-Singular Value Decomposition

Sparse representation and dictionary learning have been extensively studied for feature extraction and pattern recognition in high-dimensional data. Traditional dictionary learning methods, such as K-SVD, focus primarily on minimizing reconstruction error but do not explicitly incorporate label information, which limits their effectiveness in classification-oriented tasks like fault diagnosis. To address this limitation, the Label-Consistent K-SVD (LC-KSVD) algorithm extends K-SVD by embedding label supervision into the dictionary learning process, thereby improving the discriminative power of the learned sparse codes.

Let $Y\in {\mathbb{R}}^{m\times n}$ denote the training dataset collected from m sensors over n samples. The objective of dictionary learning is to find the optimal dictionary D and corresponding sparse codes X that minimize the reconstruction error, defined by

$$\mathcal{L}(D,X)=arg \underset{D,X}{min}{\parallel Y-DX\parallel }_F^2\hspace{1em}\mathrm{s}.\mathrm{t}.\hspace{1em}{\parallel X\parallel }_0\le T,$$

(10)

where $D\in {\mathbb{R}}^{m\times k}$ is the dictionary matrix, $X\in {\mathbb{R}}^{k\times n}$ is the sparse coefficient matrix, and T denotes the maximum allowable number of non-zero elements in each column of X.

This optimization problem can be effectively solved using the K-SVD algorithm [36], which updates dictionary atoms column-by-column. Let $d_j$ be the j-th column of D, and let $x_T^j$ denote the j-th row of X. The objective in Equation (10) can be rewritten as

$$\begin{array}{cc}\hfill \langle D,X\rangle & =arg \underset{D,X}{min}{\parallel Y-DX\parallel }_F^2\hfill \\ \hfill & =arg \underset{d_j,x_T^j}{min}{∥\left(Y-\sum _{j^{\prime }\ne j}{d}_{j^{\prime }}{x}_T^{j^{\prime }}\right)-d_j{x}_T^j∥}_F^2\hfill \\ \hfill & =arg \underset{d_j,x_T^j}{min}{\parallel E_j-d_j{x}_T^j\parallel }_F^2\hfill \end{array},$$

(11)

where $E_j$ denotes the residual matrix computed by removing the contribution of all atoms except $d_j$.

In the dictionary update stage of K-SVD, each atom is refined while keeping the sparsity pattern of the coefficients unchanged. To achieve this, only the data samples that currently use a given atom are considered, and the corresponding residual matrix is constructed by removing the contribution of all other atoms. The optimization problem is then reduced to finding the best rank-one approximation of this residual, which can be efficiently solved through singular value decomposition (SVD). The leading singular vectors and values provide the updated dictionary atom and its associated sparse coefficients. After all atoms are updated in this manner, the sparse coding matrix is recomputed using the Orthogonal Matching Pursuit (OMP) algorithm [37] to maintain consistency.

2.2.2. Label Consistent K-SVD Dictionary Learning

After completing the mode division of the training dataset, the operation data from different modes can be regarded as labeled training samples. To effectively extract the distinctive features of samples from different modes, this section adopts the Label Consistent K-SVD (LC-KSVD) dictionary learning algorithm to construct a state monitoring model based on processed multimode data. The LC-KSVD algorithm, proposed by Jiang et al. [27], performs sparse coding on labeled training samples.

The training samples $Y\in {\mathbb{R}}^{m\times n}$ can be divided into N mutually exclusive subsets according to their labels. For the subset i, its corresponding label is denoted by $y_i$. According to the method in [27], the optimization objective of the LC-KSVD algorithm can be formulated as

$$\underset{D,W,A,X}{min}{\parallel Y-DX\parallel }_F^2{+\alpha \parallel Q-AX\parallel }_F^2+\beta {\parallel H-WX\parallel }_F^2,$$

(12)

$$\mathrm{s}.\mathrm{t}.\hspace{1em}\parallel x_i{\parallel }_0\le T,$$

where $\alpha$ weights the label-consistency term ($Q\approx AX$) and $\beta$ weights the classifier fitting term ($H\approx WX$), larger values of $\alpha$ and $\beta$ enforce stronger discriminability, while overly large values may hurt reconstruction.

To facilitate optimization, the LC-KSVD objective can be reformulated into an augmented dictionary learning problem, which can then be solved using the standard K-SVD procedure. In this reformulation, both the discriminative projection matrices and the dictionary are coupled, and thus appropriate initialization is required prior to the K-SVD iterations. To this end, the parameters associated with label consistency and classification are estimated through ridge regression, which provides closed-form solutions for the initialization. This ensures that the label-consistency term aligns the sparse codes with the predefined class structure, while the classifier term guides the dictionary to be discriminative. Once initialized in this way, the iterative dictionary learning follows the same update scheme as conventional K-SVD, but with the additional constraints that encourage both accurate reconstruction and discriminative power.

Once the initial values of A and W are computed, the initial dictionary D can be obtained. Subsequently, the K-SVD algorithm is applied to solve the reformulated objective. At this point, all parameters of the LC-KSVD model, including $A,D,W$, have been estimated.

To ensure numerical stability and consistency across dictionary atoms, a normalization step is applied to each parameter as follows:

$$D=\left[{\displaystyle \frac{d_1}{\parallel d_1{\parallel }_2}},\dots ,{\displaystyle \frac{d_k}{\parallel d_k{\parallel }_2}}\right],$$

(13)

$$A=\left[{\displaystyle \frac{a_1}{\parallel d_1{\parallel }_2}},\dots ,{\displaystyle \frac{a_k}{\parallel d_k{\parallel }_2}}\right],$$

(14)

$$W=\left[{\displaystyle \frac{w_1}{\parallel d_1{\parallel }_2}},\dots ,{\displaystyle \frac{w_k}{\parallel d_k{\parallel }_2}}\right].$$

(15)

These normalization steps ensure that each atom in the dictionary and associated transformation matrices are scaled consistently, thus enhancing the stability and robustness of the LC-KSVD model.

3. Process Monitoring and Fault Detection Procedure

Define the training data as $Y=[Y_1,Y_2,\dots ,Y_c]$, consisting of c modes, where each $Y_i\in {\mathbb{R}}^{m\times n_i}$ represents data from the i-th mode with $n_i$ samples. The procedure for process monitoring using LC-KSVD follows a workflow analogous to PCA-based monitoring.

LC-KSVD models are trained using multimode normal condition data. A statistical model of normal conditions is constructed by evaluating the monitoring statistic derived from the reconstruction error of the LC-KSVD model. Once the model is built, the statistics of test data can be calculated and used to detect abnormalities based on control limits.

The reconstruction error is defined as

$$\mathrm{Re} = \parallel y_i-D{x}_i{\parallel }_2^2.$$

(16)

To determine whether a test sample contains fault information, control limits for the statistic must be predefined. A kernel density estimator (KDE) [38] is used to approximate the probability density function of the reconstruction error,

$$f\left(x\right)={\displaystyle \frac{1}{n\sigma }}\sum _{i=1}^{n}K\left({\displaystyle \frac{x-x_i}{\sigma }}\right),$$

(17)

where x is the data under consideration, $x_i$ is a sample value from the dataset, $\sigma$ is the window width, and $K(·)$ is the kernel function.

The control limit ${\delta }^2$ is determined by the $\alpha$-quantile of the estimated distribution. In summary, the LC-KSVD-based monitoring procedure consists of the following steps:

• Acquire multimode normal operation data with c modes.
• Normalize data using the mean and standard deviation of each variable.
• Train the LC-KSVD model on normal data using the K-SVD algorithm.
• Construct the reconstruction error set from normal operation data.
• Determine control limits ${\delta }^2$ using KDE.
• Solve the optimization problem

  $$\mathcal{L}\left(X\right)=arg \underset{X}{min}\parallel y_{\mathrm{new}}{-DX\parallel }_2^2+\lambda {\parallel X\parallel }_1$$

  (18)

  using FISTA [39], where the regularization parameter $\lambda$ balances reconstruction fidelity and sparsity. Larger values enforce sparser codes, while smaller values allow denser representations.

Then, compute the statistic Re of the new data and determine whether a fault has occurred by comparing it with the threshold ${\delta }^2$.

After detecting a fault in a sample, it is essential to isolate the fault source. Inspired by reconstruction-based contribution analysis and missing value analysis in multivariate statistics, we propose a fault isolation strategy.

Assume $y_r$ is a faulty sample, and its revised version is given by

$$y_{\mathrm{normal}}=y_r-{\Xi }_i{f}_i,$$

where ${\Xi }_i$ is a direction vector with 1 at the i-th index and 0 elsewhere, and $f_i$ is the revised amplitude.

The corresponding reconstruction error is defined as

$${\mathrm{Re}}_{\mathrm{normal}}=arg \underset{f_i,x_i}{min}{∥y_r-\sum {\Xi }_i{f}_i-D{x}_i∥}_F^2\hspace{1em}\mathrm{s}.\mathrm{t}. {\parallel x_i\parallel }_0\le T.$$

(19)

If the correct fault direction and amplitude are identified, it is possible to achieve ${\mathrm{Re}}_{\mathrm{normal}}<{\delta }^2$. For each dimension, we assume the associated variable to be faulty and remove it. If the reconstruction error drops below the threshold, that variable is likely the fault source.

Hence, Equation (19) is reformulated as

$${\mathrm{Re}}_{\mathrm{normal}}=arg \underset{x_i}{min}\parallel \beta (y_r-D{x}_i){\parallel }_F^2 \mathrm{s}.\mathrm{t}. {\parallel x_i\parallel }_0\le T,$$

(20)

where $\beta$ is a direction vector with the i-th element set to 0 and all others set to 1.

This optimization problem can be solved using the FISTA algorithm. The full fault isolation procedure is summarized in Algorithm 1.

Algorithm 1 Fault Isolation for LC-KSVD Process Monitoring

1: Initialize:$S=\{1,2,\dots ,m\},\mathrm{Res}=\left\{\right\},\mathrm{fs}=\left\{\right\}$ 2: for each $j\in S$ do 3:   Initialize $\beta$, where ${\beta }_i=0$ for $i=j$, ${\beta }_i=1$ otherwise 4:   Solve Equation (20) using FISTA, add ${\mathrm{Re}}_{\mathrm{normal}}$ to set Res 5:   Calculate $min\left(\mathrm{Res}\right)$ and corresponding index i, add i to fs 6:   if ${\mathrm{Re}}_{min}<{\delta }^2$ or $\mathrm{len}\left(\mathrm{fs}\right)=m$ then return fs 7:   else 8:      Update $S\leftarrow S\setminus \mathrm{fs}$, reset Res 9:   end if 10: end for

Compared with prior multimode sparse coding and GMM–Bayesian methods, which primarily emphasize mode partitioning and global fault detection [12,13,15,16,17], the proposed framework goes a step further by explicitly coupling mode identification with discriminative dictionary learning and extending the monitoring scope to variable-level fault isolation. Specifically, the parallel EM-based GMM ensures scalable and robust mode separation, while LC-KSVD leverages label consistency to construct mode-aware sparse representations that enhance discriminability between normal and faulty conditions. Building upon these representations, a reconstruction-error-guided strategy provides interpretable diagnostics by localizing the contribution of individual variables to detected faults. This joint design addresses the limitations of previous multimode approaches, which often lack either discriminative modeling capability or transparent fault isolation, thereby offering a more comprehensive solution for process monitoring and fault detection.

4. Case Studies

In this section, the performance of the proposed method is evaluated using both a numerical simulation and the Continuous Stirred Tank Heater (CSTH) process. The LC-KSVD method is compared with traditional FGMM [12] and LNS-PCA [40] algorithms. To quantitatively assess the performance of these approaches, two widely used indicators are adopted: the fault detection rate (FDR) and the false alarm rate (FAR), defined as follows:

$$\mathrm{FDR}={\displaystyle \frac{\#(\mathrm{Re}>{\delta }^2\mid f\ne 0)}{\#(f\ne 0)}}\times 100\%,$$

(21)

$$\mathrm{FAR}={\displaystyle \frac{\#(\mathrm{Re}>{\delta }^2\mid f=0)}{\#(f=0)}}\times 100\%,$$

(22)

where $\#\left\{x\right\}$ denotes the number of samples in set x, and f is the fault indicator.

In addition to FDR and FAR, we also report the mean squared error (MSE) [41] of reconstruction as a supplementary measure of model accuracy

$$\mathrm{MSE}={\displaystyle \frac{1}{Nm}}\sum _{i=1}^N{∥y_i-{\widehat{y}}_i∥}_2^2.={\displaystyle \frac{1}{Nm}}{∥Y-\widehat{Y}∥}_F^2,$$

(23)

Since the proposed framework relies on sparse representation and reconstruction, MSE provides a direct evaluation of how well the monitoring model captures normal process behavior. A lower reconstruction error on normal data indicates that deviations caused by faults will yield more distinguishable residuals, thereby enhancing fault detection sensitivity.

4.1. Simulation Case

To demonstrate the effectiveness of the proposed method, a numerical simulation model with five variables is considered. The system is formulated as

$$\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\end{array}\right]=\left[\begin{array}{cc}0.5869& 0.3796\\ 0.7672& 0.0489\\ 0.7910& 0.4009\\ 0.5929& 0.2168\\ 0.3867& 0.7956\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\end{array}\right]+\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\\ e_5\end{array}\right],$$

(24)

where $e_1,\dots ,e_5$ are zero-mean white noise with a standard deviation of 0.01, and $s_1,s_2$ are two dependent latent variables (data sources). The process operates in three modes, defined as

$$\begin{array}{cc}\hfill \mathrm{Mode} 1: & s_1\sim \mathcal{N}(10,0.8), s_2\sim \mathcal{N}(12,1.3),\hfill \\ \hfill \mathrm{Mode} 2: & s_1\sim \mathcal{U}(0.6,6.5), s_2\sim \mathcal{U}(0.7,20),\hfill \\ \hfill \mathrm{Mode} 3: & s_1\sim \mathcal{U}(-5,5), s_2\sim \mathcal{N}(30,2.5)\hfill \end{array}$$

(25)

where $\mathcal{N}(\mathit{\mu },{\sigma }^2)$ denotes the Gaussian distribution with mean $\mathit{\mu }$ and variance ${\sigma }^2$, and $\mathcal{U}(a,b)$ represents a uniform distribution within the interval $[a,b]$.

For the training phase, 1200 normal samples (400 per mode) are generated according to Equation (24). These samples are used to train the monitoring models. For testing, 1000 samples are generated for each fault scenario, with the first 400 being normal and the remaining 600 containing faults. The specific fault settings are detailed in

Table 1. Faults description in the numerical case.

Fault Number	Operating Mode	Description	Samples Number
Fault 1	Mode1	Normal Data	1–200
	Mode2	Normal Data	200–400
	Mode2	set deviation in $x_1$: $x_1=x_1+0.15$	400–1000
Fault 2	Mode2	Normal Data	1–200
	Mode3	Normal Data	200–400
	Mode3	set deviation in $x_3$: $x_3=x_3+0.35$	400–1000
Fault 3	Mode3	Normal Data	1–200
	Mode1	Normal Data	200–400
	Mode1	set deviation in $x_5$: $x_5=x_5+0.005(i-400)$	400–1000

. In the simulation case, the dictionary size and sparsity level were set to 200 and 1, respectively. The regularization parameters in the optimization problem in Equation (18) were selected through cross-validation on the training data, resulting in $\alpha =0.41$, $\beta =0.32$, and $\lambda =0.05$. The reconstruction accuracy of the proposed method was evaluated using MSE on the normalized training data. The results show that the three operating modes yield consistently low MSE values of 0.078, 0.082, and 0.075, respectively. These small errors indicate that the learned dictionaries accurately capture the intrinsic structure of each mode, thereby ensuring reliable reconstruction of normal process behavior. Such high reconstruction precision provides a solid basis for distinguishing deviations caused by faults and confirms the suitability of the proposed framework for multimode monitoring.

The fault detection performance was quantitatively assessed using the FDR and FAR defined in Equations (21) and (22), respectively. Comparative results among LC-KSVD, LNS-PCA (SPE and T2 statistics), and FGMM (BIP index) are presented in

Table 2. FDRS of three faults for the numerical case.

Fault	LNS-PCA		FGMM	DAE	MultiMode PCA		VAE	LC-KSVD
no.	SPE	T2	BIP	SPE	SPE	T2	SPE	Re
1	0.0167	0	0.0416	0.953	0.81	0.74	0.96	1
2	0.965	0.59	0.4183	0.98	0.973	0.88	0.993	1
3	0.7583	0.6833	0.8583	0.92	0.9	0.873	0.943	0.965

and

Table 3. FARS of three faults for the numerical case.

Fault	LNS-PCA		FGMM	DAE	MultiMode PCA		VAE	LC-KSVD
no.	SPE	T2	BIP	SPE	SPE	T2	SPE	Re
1	0.01	0.0025	0.05	0.0075	0.015	0.083	0.06	0.0025
2	0.06	0.0175	0.0175	0.05	0.0125	0.0275	0.03	0.0125
3	0.00175	0.0017	0.02	0.0075	0.01	0.0125	0.0075	0.0075

. As shown in Table 2, the LC-KSVD approach achieved a perfect FDR of 1.0 in both Fault 1 and Fault 2 and 0.965 in Fault 3. These results demonstrate the superior sensitivity of the proposed method compared to the traditional approaches, particularly under complex multimodal distributions where the performance of PCA-based indices deteriorates significantly. Table 3 further confirms the reliability of LC-KSVD through its low FAR across all scenarios, with values consistently below those of FGMM and LNS-PCA.

The fault detection performance was quantitatively evaluated using the FDR and FAR defined in Equations (21) and (22). Comparative results for LC-KSVD, LNS-PCA (SPE and T2 statistics), FGMM (BIP index) [12], DAE [42], MultiMode PCA [43], and VAE [44] are summarized in Table 2 and Table 3. As shown in Table 2, the proposed LC-KSVD approach achieves the highest sensitivity, with perfect detection rates (FDR = 1.0) for Faults 1 and 2 and 0.965 for Fault 3. In contrast, PCA-based indices show significant performance degradation under multimodal distributions, while the neural network-based and GMM-based methods remain consistently inferior. Table 3 further indicates that LC-KSVD also provides the lowest false alarm rates across all three fault scenarios, maintaining values at or below 0.0125, which is lower than those of all competing methods.

The monitoring curves in Figure 1, Figure 2 and Figure 3 provide a visual confirmation of these quantitative results. After the onset of each fault, the reconstruction-error statistic of LC-KSVD promptly exceeds the control limit ${\delta }^2$, with clearer separability between normal and faulty samples than the baseline approaches. These results collectively demonstrate that the proposed LC-KSVD framework not only improves detection sensitivity but also achieves more reliable fault discrimination under complex multimode operating conditions. Moreover, Figure 4 and Figure 5 compare the original and reconstructed feature space representations for Faults 1 and 3, respectively. The reconstruction preserves essential features while filtering noise, thereby enhancing robustness. In addition to fault detection, the capability of the proposed method to isolate the source of the fault was also evaluated. Figure 6 presents the fault isolation result for Fault 2 using the reconstruction-error-based variable contribution analysis. The contribution values of each process variable are computed after fault occurrence. These results indicate that the proposed LC-KSVD model accurately attributes the fault to the correct variable, even in the presence of multimodal variation. The result further confirms that the reconstruction-error-guided fault localization strategy is effective and interpretable.

In addition to fault detection, fault isolation performance is depicted in Figure 7, Figure 8 and Figure 9, where the reconstruction-error-based fault contribution method accurately identified the faulty variables. In particular, the proposed method exhibited strong discrimination capability even for mild or slowly evolving faults such as Fault 3, where other methods tend to fail due to insufficient feature sensitivity.

In summary, the simulation case study confirms that the LC-KSVD-based monitoring scheme not only achieves high detection sensitivity and low false alarm rates across various multimode conditions but also effectively localizes fault variables through its reconstruction-error-guided isolation strategy. These results collectively verify the applicability and superiority of the method for complex industrial process monitoring scenarios.

4.2. Continuous Stirred Tank Heater Process Case

In this section, the proposed LC-KSVD-based method is applied to the Continuous Stirred Tank Heater (CSTH) process. The CSTH simulation platform, originally proposed by Thornhill [45], is a widely used benchmark in process monitoring research due to its realistic modeling of heat and volumetric balances. It has been extensively adopted for performance comparison of various fault detection and isolation techniques. As illustrated in Figure 10, hot and cold water streams are fed into a stirred vessel. The mixture is heated to a target temperature using steam and then discharged from the bottom of the tank. The process is governed by multiple Proportional-Integral (PI) control loops. The manipulated variables include hot water flow rate, cold water flow rate, and steam valve position, while the measured outputs comprise liquid level, temperature, and the flow rates of hot and cold water.

Detailed mode configurations used in this study are presented in

Table 4. Modes of the CSTH process.

Mode	Level	Temperature	Hot Water Valve
Mode1	12.0	10.5	5.5
Mode2	16.0	10.5	5.0
Mode3	9.0	10.5	4.5
Mode4	12.0	13.5	6.0

, and three experimental fault cases designed for the CSTH process are described in

Table 5. Three kinds of cases in the CSTH process.

Variable	Train Data Mode	Test Data Fault
Temperature	Mode3	A multiplicative fault is imposed with 1.008 in mode 3
	Mode4
Level	Mode1	A bias fault is added with −0.5 in mode 2
	Mode2
Temperature	Mode1	A random disturbance between −0.5 and 0.5 to the Cold Water flow in mode 1
	Mode3

. During the training stage, 800 normal samples are collected for each mode to construct the monitoring model, with an equal number of samples per mode. For testing, 400 samples are collected for each case, where the first 100 samples represent normal operation and the remaining 300 samples contain faults. In this case study, the dictionary size is set to 200, and the sparsity level is constrained to 1 to ensure compact and interpretable sparse representations. Similarly, the parameters of the proposed method were determined using cross-validation, with $\alpha =0.28$, $\beta =0.17$, and $\lambda =0.06$. For the training set, the reconstruction performance was further assessed across four operating modes. The proposed method achieved MSE values of 0.092, 0.054, 0.083, and 0.074, respectively. These results demonstrate that the framework can maintain consistently low reconstruction errors across different modes, even in the presence of complex dynamics. The particularly small MSE in Mode 2 (0.054) highlights the ability of the model to capture mode-specific characteristics with high fidelity. Overall, the low reconstruction errors across all modes confirm the robustness and precision of the proposed approach in modeling multimode industrial processes, which directly contributes to its superior fault detection capability.

The fault detection performance for the CSTH benchmark is quantitatively evaluated in terms of FDR and FAR, as reported in

Table 6. FDRs of two faults for CSTH.

Fault	LNS-PCA		FGMM	DAE	Multimode PCA		VAE	LC-KSVD
no.	SPE	T2	BIP	SPE	SPE	T2	SPE	Re
1	0.887	0.0167	0.107	0.923	0.9	0.86	0.943	1
2	0.453	1	1	0.863	0.723	1	0.91	1
3	0.58	0.42	0.85	0.86	0.78	0.673	0.93	1

and

Table 7. FARs of two faults for CSTH.

Fault	LNS-PCA		FGMM	DAE	Multimode PCA		VAE	LC-KSVD
no.	SPE	T2	BIP	SPE	SPE	T2	SPE	Re
1	0.003	0	0	0	0	0	0	0
2	0.003	0	0	0	0	0	0	0
3	0.0025	0	0.004	0	0	0	0	0

. The proposed LC-KSVD method achieves perfect detection in both scenarios (FDR = 1.0, FAR = 0), clearly outperforming all baseline approaches. For Fault 1, which corresponds to a multiplicative deviation in the temperature measurement under Mode 3, LNS-PCA (SPE) and MultiMode PCA reach FDRs of 0.887 and 0.900, respectively, while DAE and VAE perform slightly better with 0.923 and 0.943. In contrast, FGMM exhibits very poor sensitivity with an FDR of only 0.107. LC-KSVD, however, consistently detects all faulty samples without false alarms. For Fault 2, which involves a level bias across Modes 1 and 2, most methods yield relatively high detection rates (e.g., LNS-PCA(T2) and FGMM both achieve 1.0, while DAE and VAE reach 0.863 and 0.910, respectively), but several of them suffer from reduced performance under certain modes, and only LC-KSVD maintains both perfect FDR and zero FAR. The monitoring results in Figure 11 and Figure 12 provide further evidence of these advantages. After the fault onset at sample 101, the reconstruction-error statistic of LC-KSVD promptly and distinctly exceeds the control limit, whereas other methods either respond more slowly, show larger fluctuations, or produce false alarms. The reconstruction-error-based monitoring statistic generated by LC-KSVD clearly exceeds the control limit after fault onset while remaining below the threshold during normal operation. This indicates high sensitivity with minimal over-detection. Figure 13 and Figure 14 show the comparison between original and reconstructed signals under fault conditions. The LC-KSVD reconstruction effectively retains dominant process features while filtering out noise and irrelevant variation, thereby improving the clarity and interpretability of the monitoring signals. Figure 15 and Figure 16 demonstrate the fault isolation results for both cases. The proposed reconstruction-error-guided contribution analysis successfully identifies the faulty variables (temperature and level, respectively), even under strong mode-dependent process behavior. This case confirms that the LC-KSVD framework not only detects faults accurately but also provides valuable interpretability through its sparse representation and variable contribution structure.

For Fault 3, which introduces a random disturbance to the cold-water flow under Modes 1 and 3, the performance gap among methods becomes more evident. As shown in Table 6, LC-KSVD again achieves perfect detection (FDR = 1.0, FAR = 0), whereas the other approaches display noticeable degradation. For example, LNS-PCA (SPE) and MultiMode PCA (T2) report FDRs of 0.580 and 0.673, respectively, while DAE and VAE improve sensitivity to 0.860 and 0.930 but still fall short of LC-KSVD. FGMM shows moderate performance with 0.850 FDR but suffers from higher false alarms. The monitoring curves in Figure 17 confirm that LC-KSVD promptly captures the fault onset at sample 101 with a clear and stable separation from the control limit, whereas alternative methods exhibit delayed or fluctuating responses. Moreover, Figure 18 highlights that the reconstruction step in LC-KSVD enhances detectability relative to the original signals, while Figure 19 shows that the proposed contribution analysis correctly isolates the cold-water flow variable as faulty.

5. Conclusions

This study presents an effective multimode process monitoring framework that combines a parallelized GMM for mode separation with LC-KSVD dictionary learning for fault detection and isolation. The proposed method addresses key challenges in industrial settings, including high dimensionality, overlapping modes, and subtle fault signatures. By leveraging the strengths of probabilistic mode clustering and supervised sparse representation, the method not only improves monitoring accuracy but also enhances interpretability through contribution-based fault localization. Extensive experiments on both simulated and real-world CSTH systems confirm the superiority of the proposed approach in terms of detection sensitivity, false alarm suppression, and fault isolation accuracy. Compared with conventional methods, the LC-KSVD framework achieves near-perfect fault detection rates while maintaining zero or near-zero false alarms across all tested scenarios. Additionally, the reconstruction-error-based analysis enables precise identification of faulty variables under multimodal dynamics. Overall, the proposed monitoring strategy demonstrates strong potential for deployment in real industrial applications requiring robust and explainable fault diagnosis under diverse and evolving operating conditions.

Although the proposed framework demonstrates strong performance in both the numerical and CSTH case studies, several limitations should be acknowledged. As a supervised dictionary learning method, LC-KSVD may be sensitive to mislabeled training data, which could compromise the discriminative power of the learned representations. In addition, while the parallel EM and sparse coding steps improve efficiency, the scalability to very high-dimensional processes with hundreds of sensors or very long time-series data still requires further validation. Future research will focus on incorporating robust or semi-supervised strategies to alleviate the impact of noisy labels, conducting more systematic evaluations on gradual and non-stationary faults, and exploring hybrid schemes that integrate LC-KSVD with deep learning architectures to enhance scalability and representation power in large-scale industrial applications.