Mixed Student’s T-Distribution Regression Soft Measurement Model and Its Application Based on VI and MCMC

Abstract

The conventional diagnostic techniques for ethylene cracker furnace tube coking rely on manual expertise, offline analysis and on-site inspection. However, these methods have inherent limitations, including prolonged inspection times, low accuracy and poor real-time performance. This makes it challenging to meet the requirements of chemical production. The necessity for high efficiency, high reliability and high safety, coupled with the inherent complexity of the production process, results in data that is characterized by multimodal, nonlinear, non-Gaussian and strong noise. This renders the traditional data processing and analysis methods ineffective. In order to address these issues, this paper puts forth a novel soft measurement approach, namely the ‘Mixed Student’s t-distribution regression soft measurement model based on Variational Inference (VI) and Markov Chain Monte Carlo (MCMC)’. The initial variational distribution is selected during the initialization step of VI. Subsequently, VI is employed to iteratively refine the distribution in order to more closely approximate the true posterior distribution. Subsequently, the outcomes of VI are employed to initiate the MCMC, which facilitates the placement of the iterative starting point of the MCMC in a region that more closely approximates the true posterior distribution. This approach allows the convergence process of MCMC to be accelerated, thereby enabling a more rapid approach to the true posterior distribution. The model integrates the efficiency of VI with the accuracy of the MCMC, thereby enhancing the precision of the posterior distribution approximation while preserving computational efficiency. The experimental results demonstrate that the model exhibits enhanced accuracy and robustness in the diagnosis of ethylene cracker tube coking compared to the conventional Partial Least Squares Regression (PLSR), Gaussian Process Regression (GPR), Gaussian Mixture Regression (GMR), Bayesian Student’s T-Distribution Mixture Regression (STMR) and Semi-supervised Bayesian T-Distribution Mixture Regression (SsSMM). This method provides a scientific basis for optimizing and maintaining the ethylene cracker, enhancing its production efficiency and reliability, and effectively addressing the multimodal, non-Gaussian distribution and uncertainty of the coking data of the ethylene cracker furnace tube.

1. Introduction

The significance of ethylene in the petrochemical industry is increasing in tandem with the ongoing advancement of society and the economy [1,2,3]. In the context of industrial production, ethylene is derived from naphtha and other raw materials under conditions of anaerobic or low oxygen content. This is achieved through the process of cracking, whereby carbon–carbon and carbon–hydrogen bonds are broken at elevated temperatures. In order to enhance the safety and efficacy of ethylene cracker production, it is imperative to implement a comprehensive monitoring system for the ethylene production process. However, the elevated temperature, pressure, flammability and explosivity inherent to the production of ethylene columns render direct measurement of certain operational parameters, such as furnace temperature, challenging. With the development of artificial intelligence technology, people have started to try to model the ethylene cracker modeling and they are trying to monitor it through artificial intelligence (AI) methods. By simulating the coking process, establishing a detailed mechanism model and exploring the optimal control strategy as well as the simulation and automatic control technology of the coke clearing process, these results demonstrate the progress made in improving the operating efficiency of cracker furnaces, prolonging the service life of furnace tubes and achieving an accurate diagnosis, and at the same time point out the directions that still need further research and improvement in practical applications [4,5,6]. Therefore, it is imperative that fast, accurate and stable ethylene cracking stovepipe models are developed in order to advance production intelligence [7]. Nevertheless, the high-temperature cracking catalytic generation process of ethylene is inherently complex, rendering it more challenging to develop a comprehensive model of the ethylene cracking process.

The challenge and uniqueness of directly measuring the pertinent variables may impact the precision of sample matching. Soft measurement technology enables the real-time monitoring of crucial process parameters without the necessity for costly sensors or intricate measurement apparatus, thereby enhancing the accuracy and efficiency of data processing. This technology is extensively utilized in the chemical industry and other domains [8,9,10]. The estimation of the dominant variable through the use of auxiliary variables is a common practice in soft measurement techniques. However, when the auxiliary variables are multidimensional and exhibit inconsistent effects, the resulting prediction error is likely to be amplified [11,12,13]. In general, soft measurement methods are classified into two main categories: mechanistic model-based and data-driven [14]. In the case of the ethylene cracker, the mechanistic modeling approach is, by its very nature, strongly interpretative and complex. Furthermore, it requires precise parameters that are difficult to measure and model [15,16]. In contrast, data-driven approaches employ historical data and machine learning algorithms that are not contingent on a priori knowledge and are therefore well-suited to the analysis of complex and uncertain systems [17,18]. Consequently, mechanistic model are typically well-suited for elucidating the fundamental mechanisms, whereas data-driven methodologies are more appropriate for addressing the intricacies and uncertainties inherent to the system [19]. However, the data obtained from the ethylene cracker production process are characterized by multimodality, nonlinearity, non-Gaussianity and strong noise [20], which necessitates the utilization of advanced methods and techniques to facilitate its analysis. A comparison of the mechanistic and data-driven models is shown in

Table 1. Comparison of mechanistic and data-driven models.

Comparison Term	Mechanistic Model	Data-Driven Model
define	model based on laws of physics and known processes	model based on statistical laws of data and machine learning algorithms
modeling methodology	constructing model using a priori knowledge and laws of physics	build model by learning from large amounts of data
suitability	for systems with well-defined physical mechanisms	for complex systems that are difficult to physically model
explanatory	model is usually well interpreted	model may be weakly interpreted
precision	depends on how accurately the model captures the physical mechanisms	depends on the quality and quantity of data
generalization capability	strong generalization ability in the presence of similar physical mechanisms	large amounts of multi-sample data are required to ensure generalizability
development difficulty	requires deep domain knowledge and physics understanding	data preprocessing, feature engineering and model tuning skills required

.

In the field of statistical modeling and machine learning, the mixed Student’s t-distribution regression models have been the subject of considerable interest due to their resilience to outliers and their capacity to accommodate multimodal data effectively. The model is capable of capturing both heteroskedasticity and the heavy-tailed nature of the data. Therefore, it is also widely used in a number of fields, including finance, bioinformatics and signal processing.

The prevalence of outliers in industrial data renders traditional analyses based on Gaussian distribution vulnerable to interference. In contrast, the Student’s t-distribution, due to its thicker tails and lower sensitivity to outliers, is better suited to handle this type of data, thus improving the robustness and accuracy of the analysis [21]. Nevertheless, the estimation of its parameters necessitates the performance of intricate integration operations, rendering the application of traditional maximum likelihood estimation a challenging undertaking. To overcome this challenge, researchers have proposed a variety of Bayesian inference methods, among which VI and MCMC are two mainstream solutions. While VI and MCMC each possess distinctive advantages in addressing complex models, they also exhibit inherent limitations. As a Bayesian approximation method, VI is efficient but contingent upon the precision of the variational distribution, potentially inadequate for accurately approximating the posterior distribution. Conversely, MCMC, while theoretically capable of accurately recovering the posterior probabilities, is computationally inefficient, may require an extended convergence period, and is costly to process large data sets [22,23]. In light of the aforementioned considerations, this paper puts forth a novel approach that integrates VI and MCMC. This integration aims to enhance the efficacy and precision of parameter estimation in mixed Student’s t-distribution regression models [24].

This paper presents an overview of the Student’s T-Distribution Mixture Regression model and its associated parameter estimation challenges. It also outlines a proposed alternating iteration method and provides experimental verification of its advantages in terms of parameter estimation, prediction accuracy, and computational efficiency.

In summary, the main contributions of this paper are as follows:

• We developed an algorithm called variational Bayesian MCMC (VM), which consists of VI and MCMC;
• Our VM introduces the Student’s t-distribution, which provided robust classification results;
• In order to determine whether the proposed method is effective, it is necessary to use two sets of data that contain different categories as a basis for the experiment.

2. Related Works

In recent years, soft measurement techniques have assumed an increasingly pivotal role in industrial process control, particularly in the estimation of critical quality parameters that are challenging to measure directly. Conventional soft measurement techniques, such as those based on inference and control through mechanistic modeling, can satisfy the aforementioned demand to a certain extent. However, their efficacy is frequently constrained by the precision and practicality of the model employed. For example, Hongye Jiang et al. (2023) [25] proposed a new corrosion prediction mechanism model for predicting the corrosion rate of a gathering pipeline in a carbon dioxide environment. This model was validated by comparing it with experimental data, and it demonstrated good prediction accuracy. However, the accuracy of this model depends on the availability of high-quality experimental data. Conversely, Shaochen Wang et al. (2023) [26] integrated deep learning methodologies with a model based on the distillation process mechanism. This approach has the potential to markedly enhance the safety, stability, and reliability of the production system. However, it necessitates substantial computational resources, which may present a challenge in practical applications. The advancement of artificial intelligence has led to a growing interest in data-driven soft measurement methods based on data. To illustrate, the soft sensor framework proposed by Runyuan Guo et al. (2021) [27] employs a generative adversarial network (GAN) and gated recurrent unit (GRU) to enhance the precision and resilience of prediction. However, the intricacy of the model may compromise the dependability of its prediction outcomes. Similarly, the Long Short-Term Memory–Deep Factorization Machine (LSTM-DeepFM) model, a data-driven self-supervised approach proposed by Ren L et al. (2022) [19], combines the advantages of LSTM and DeepFM to capture data features with greater accuracy, thereby enhancing prediction and monitoring precision. However, the model structure is more complex and necessitates a substantial amount of data for effective training. The data-driven soft sensor approach proposed by He Z et al. (2022) [28] combines multiple machine learning algorithms, which enhances the efficiency and quality control of the papermaking process by monitoring key quality parameters in real time. However, the integration of multiple machine learning algorithms also elevates the complexity of the model and the challenge of maintenance. Notwithstanding the efficacy of these methodologies in addressing intricate data characteristics, such as nonlinearity, non-Gaussianity, and uncertainty, there remain a few challenges and constraints.

In the field of parameter estimation for mixed Student’s t-distribution regression model, researchers have primarily concentrated on the utilization of these methods to enhance model robustness and inference efficiency. For example, the Bayesian mixed model based on Student’s t-distribution proposed by Markus Svense’n et al. (2005) [29] displays good robustness in addressing outliers and is able to automatically determine the number of components in the mixed model. The mixed regression model (SMR) based on a Student’s t-distribution, as proposed by Jingbo Wang et al. (2018) [30], serves to enhance the robustness of the model to outliers; however, the computational complexity is considerable. The matching method based on a Bayesian mixed Student’s t-distribution model, as proposed by Lijuan Yang et al. (2018) [31], further improves the robustness to outliers through the optimization of the variational Bayesian framework. The semi-supervised robust modeling method (SsSMM), which is based on Student’s t-distribution mixture model and was proposed by Weiming Shao et al. (2019) [32], employs unlabeled data to address the issue of multimodal industrial processes. However, it should be noted that the method is more susceptible to parameter selection.

In recent years, there have been notable advancements in the field of Bayesian inference, particularly in the utilization of VI and MCMC, in addressing intricate statistical models. While VI and MCMC are commonly employed for parameter estimation in mixed Student’s t-distribution regression models, they each have their limitations.

The approximation accuracy of VI may be affected when dealing with a highly complex model. To illustrate, the variational Bayesian Gaussian Mixture Regression (VBGMR) method proposed by Jinlin Zhu et al. (2016) [17] is more efficient in model selection. However, the choice of prior distribution and model parameters significantly affects the results. The variational Bayesian Monte Carlo (VBMC) method proposed by Luigi Acerbi et al. (2018) [33] combines the VI advantages and reduces the number of required samples. Nevertheless, it may require fine parameter tuning and optimization. In a further development of the variational Bayesian approach, Jingbo Wang et al. (2019) [34] proposed a robust inference sensor development method based on variational Bayesian mixed Student’s t-distribution regression (VBSMR). This method is effective in identifying and reducing the impact of outliers on the prediction performance, but also requires consideration of its computational complexity and parameter selection challenges.

In contrast, while MCMC can provide accurate posterior estimates, the speed of convergence and the computational cost are often limiting factors. For example, Leandro Passos de Figueiredo et al. (2019) [35] proposed a method based on Gaussian mixture modeling and Markov Chain Monte Carlo to solve the linear ground seismic inversion problem. The method can effectively deal with non-Gaussian noise and complex data distributions through the Gaussian mixture model, which improves the accuracy of the inversion results. However, due to the need for a large number of samples, the computational cost of the method is high and the convergence speed is slow, which limits its efficiency in practical applications to some extent. Jonas Gregor Wiese et al. (2023) [36] proposed a new method for efficient MCMC sampling in Bayesian neural networks, which reduces computation by exploiting the symmetry of the parameter a posteriori density landscapes while maintaining the accuracy of the inference. However, this approach still requires a significant amount of computational resources. The learning method proposed by Marcel Hirt et al. (2023) [24] to accelerate the variational autoencoder (VAE) using MCMC, although improves the generation performance, has a higher computational complexity when dealing with large-scale datasets with high computational complexity. In the 2024 study on the application of MCMC to deep learning and big data, Rohitash Chandra et al. (2024) [37] highlight that for some complex models, the MCMC chains may take a considerable amount of time to converge, which can lead to inefficiency. Similarly, in their 2024 study, Zeji Yi et al. (2024) [38] proposed the MCMC-based method that improves the sample efficiency of high-dimensional Bayesian optimization. However, this method may still require a significant amount of computational resources when dealing with extremely high-dimensional data.

This paper presents a novel approach that combines VI with the MCMC. This integration aims to enhance the computational efficiency of VI while leveraging the advantages of accurate inference offered by the MCMC. The objective is to facilitate more precise and efficient parameter estimation for the mixed Student’s t-distribution regression model, which are often a complex statistical model.

3. Theoretical Background

This section will delve into the underlying theory that underpins our analysis, including an overview of the Student’s t-distribution, and principles of VI and MCMC, which together provide the necessary theoretical background for the advancement of our proposed methodology.

3.1. Student’s T-Distribution

The Student’s t-distribution can be viewed as a mixture of an infinite number of Gaussian distributions with the same mean and different variances [39,40], i.e.,

$$\mathit{St}\left(x|\mu ,\lambda ,\nu \right)={\displaystyle \frac{\Gamma \left(\frac{\nu }{2}+\frac{1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}}{\left({\displaystyle \frac{\lambda }{\pi \nu }}\right)}^{\left({\displaystyle \frac{1}{2}}\right)}{\left[1+{\displaystyle \frac{\lambda {\left(x-\mu \right)}^2}{\nu }}\right]}^{-{\displaystyle \frac{\nu }{2}}-{\displaystyle \frac{1}{2}}},$$

(1)

where the parameter $\mu$ represents the mean of the Student’s t-distribution, the parameter $\lambda$ denotes the precision, and the parameter $\nu$ signifies the degree of freedom. When $\nu =1$, the Student’s t-distribution becomes a Cauchy distribution. Conversely, when $\nu \to \infty$, the Student’s t-distribution $\mathit{St}\left(x|\mu ,\lambda ,\nu \right)$ becomes a Gaussian distribution $\mathcal{N}\left(x|\mu ,{\lambda }^{-1}\right)$ with mean $\mu$ and precision $\lambda$ as shown in Figure 1 [30,41].

Generalized to the multivariate Student’s t-distribution and using the same methodology as for the univariate variable [29,42], i.e.,

$$\mathit{St}\left(x|\mu ,\Lambda ,\nu \right)={\displaystyle \frac{\Gamma \left(\frac{\mathit{D}}{2}+\frac{\nu }{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)}}{\displaystyle \frac{{|\Lambda |}^{-\frac{1}{2}}}{{\left(\pi \nu \right)}^{\frac{\mathit{D}}{2}}}}{\left[1+{\displaystyle \frac{{▵}^2}{\nu }}\right]}^{-{\displaystyle \frac{\mathit{D}}{2}}-{\displaystyle \frac{\nu }{2}}},$$

(2)

where $\mathit{D}$ is the dimension of and ${▵}^2$ denotes the squared Mahalanobis distance, which is defined as follows:

$${▵}^2={\left(x-\mu \right)}^T\Lambda \left(x-\mu \right),$$

(3)

The Student’s t-distribution is a formal generalization of the Gaussian distribution. When $\nu \to \infty$, the Student’s t-distribution can be simplified to a Gaussian distribution with a mean of $\mu$ and a precision of $\Lambda$. Furthermore, the maximum likelihood parameter values of the Student’s t-distribution are more robust to outliers than other distributions.

3.2. Variational Inference

VI is a method for approximate inference of probabilistic model that approximates the true posterior distribution $p\left(z|x\right)$ [43] by optimizing an easy-to-handle variational distribution $q_{\theta }\left(z\right)=q_{\theta }\left(z|x\right)$, which is a family of distributions with parameters. VI transforms an inference problem into an optimization problem that minimizes the difference between the variational distribution $q_{\theta }\left(z\right)$ and the true posterior distribution $p\left(z|x\right)$ by adjusting the variational parameter $\theta$, which is usually measured by the Kullback–Leibler ($\mathit{KL}$) scattering measure [33,44].

$$\mathit{KL}\left(q_{\theta }\left(z\right)p\left(z|x\right)\right)=E_{q_{\theta }\left(z\right)}\left[log{q}_{\theta }\left(z\right)-logp\left(z|x\right)\right],$$

(4)

sign

$$\mathit{L}\left(q_{\theta }\left(z\right)\right)=E_{q_{\theta }\left(z\right)}\left[logp\left(x,z\right)\right]-E_{q_{\theta }\left(z\right)}\left[log{q}_{\theta }\left(z\right)\right],$$

(5)

The $\mathit{KL}$ scatter can be simplified to the following form:

$$\mathit{KL}\left(q_{\theta }\left(z\right)\left|\right|p\left(z|x\right)\right)=logp\left(z|x\right)-\mathit{L}\left(q_{\theta }\left(z\right)\right),$$

(6)

Since $\mathit{KL}\left(q_{\theta }\left(z\right)\left|\right|p\left(z|x\right)\right)\ge 0$, it follows that $\mathit{L}\left(q_{\theta }\left(z\right)\right)$ is a lower bound on the log-likelihood function $logp\left(z|x\right)$, i.e., minimizing the $\mathit{KL}$ scatter $\mathit{KL}\left(q_{\theta }\left(z\right)\left|\right|p\left(z|x\right)\right)$ is equivalent to maximizing the evidence lower bound $\mathit{L}\left(q_{\theta }\left(z\right)\right)$ [45], it is common in VI to approximate the true posterior distribution $p\left(z|x\right)$ by optimizing $\mathit{L}\left(q_{\theta }\left(z\right)\right)$ [46].

3.3. Markov Chain Monte Carlo

The fundamental principle of the MCMC is the construction of a Markov Chain for the purpose of approximating complex probability distribution [38,47]. A Markov Chain can be defined as a stochastic process in which the probability of a given state depends solely on the previous state and is independent of all preceding states. Monte Carlo, on the other hand, estimates numerical solutions by repeated random sampling. Among them, the Metropolis–Hastings algorithm is a representative algorithm for MCMC.

Assuming the state $x_{i-1}=x$ at time $\left(i-1\right)$, the probability distribution to be sampled is $p\left(x\right)$, the Metropolis–Hastings algorithm [48] uses a Markov Chain with a transfer kernel $p\left(x,x^{\prime }\right)$:

$$p\left(x,x^{\prime }\right)=q\left(x,x^{\prime }\right)\alpha \left(x,x^{\prime }\right),$$

(7)

where $q\left(x,x^{\prime }\right)$ and $\alpha \left(x,x^{\prime }\right)$ are referred to as the suggestion distribution and acceptance distribution, respectively. $q\left(x,x^{\prime }\right)$ is the transfer kernel of another Markov Chain and its probability value is not zero. A candidate state $x{⠀}^{\prime }$ is randomly selected according to $q\left(x,x^{\prime }\right)$, while the acceptance probability is

$$\alpha \left(x,x^{\prime }\right)=min\left\{1,{\displaystyle \frac{p\left(x^{\prime }\right)q\left(x{}^{\prime },x\right)}{p\left(x\right)q\left(x,x^{\prime }\right)}}\right\},$$

(8)

Then the transfer kernel is

$$p\left(x,x^{\prime }\right)=\left\{\begin{array}{c}q\left(x,x^{\prime }\right),p\left(x^{\prime }\right)q\left(x{}^{\prime },x\right)\ge p\left(x\right)q\left(x,x^{\prime }\right)\hfill \\ q\left(x{}^{\prime },x\right){\displaystyle \frac{p\left(x^{\prime }\right)}{p\left(x\right)}},p\left(x^{\prime }\right)q\left(x{}^{\prime },x\right)<p\left(x\right)q\left(x,x^{\prime }\right)\hfill \end{array}\right.,$$

(9)

A number u is randomly selected from the interval (0,1) according to a uniform distribution, and if $u\le \alpha \left(x,x^{\prime }\right)$, it means accept $x^{\prime }$ and decided to transfer to state $x^{\prime }$ at moment i, i.e., $x_i=x^{\prime }$; otherwise, it means rejecting $x^{\prime }$ and deciding to stay in state x at moment i, i.e., state $x_i=x$ [49].

4. Methods

To overcome the influence of outliers and improve the classification accuracy of the model, a symplectic MCMC Student’s t-distribution hybrid regression model combining the flexibility of the hybrid model and the thick-tailed property of the Student’s t-distribution is constructed. The model can enhance the robustness of leptokurtic points in the data and improve classification accuracy by adjusting the number of components in the mixture model and the parameters of the Student’s t-distribution [50,51]. The hybrid model permits the capture of multiple distribution patterns of the data, and the properties of the t-distribution facilitate the accommodation of heavy-tailed distribution and outliers in the data, thereby ensuring the reliability of classification results despite the presence of outliers.

4.1. Mixed Student’s T-Distribution Regression Based on VI and MCMC (VMSTMR)

Assuming that x obeys the mixture distribution of K components, the Student’s t-distribution mixture model takes the form of

$$p\left(x|\mu ,\Lambda ,\nu ,\pi \right)=\sum _{k=1}^K{\pi }_k\mathit{St}\left(x|{\mu }_k,{\Lambda }_k,{\nu }_k\right),$$

(10)

where the mixing coefficients $\pi =\left\{{\pi }_1,{\pi }_2,\dots ,{\pi }_K\right\}$, ${\pi }_k$ are the a priori probability values of the $kth$ component with $0\le {\pi }_k\le 1$ and ${\displaystyle \sum _{k=1}^K}{\pi }_k=1$ for all k.

Assume that the input variables X and Y obey a linear relationship, i.e.,

$$y_n=x_n^T{\omega }_k+{\epsilon }_k,$$

(11)

where ${\omega }_k$ is the regression coefficient between $y_n$ and $x_n$, ${\epsilon }_k$ is the measurement noise, and ${\epsilon }_k\sim \mathcal{N}\left(0,{\sigma }_k\right)$.

Therefore, it is possible to obtain

$$p\left(y_n|x_n,z_n,\omega ,\sigma \right)=\prod _{k=1}^K\mathcal{N}\left(y_n|x_n^T{\omega }_k,{\sigma }_k\right),$$

(12)

where $\omega =\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_k\right\}$ and $\sigma =\left\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_k\right\}$. When $z_{nk}=1$, it represents that $x_n$ belongs to the $kth$ component, otherwise $z_{nk}=0$.

Since VI is an approximation method, there are inevitably some errors between the variational distribution it obtains and the true posterior distribution. This error may have an impact on the subsequent inference and classification accuracy, but the advantage of VI is that it improves computational efficiency. In contrast, the MCMC can approximate the true posterior distribution more accurately, but it has the disadvantage of requiring a longer running time to ensure that the sample distribution gradually approaches the true posterior distribution.

In Section 3.2, it is mentioned that the initial variational distribution $q_{\theta }\left(z\right)$ is used to approximate the true posterior distribution $p\left(z|x\right)$. First, $q_{\theta }\left(z\right)$ is optimized using VI. Then, $q_{\theta }\left(z\right)$ is further improved by running an MCMC for a fixed number of iterations $t\left(t\in N\right)$. Specifically, suppose that after several iterations of VI, $q_{\theta }\left(z\right)$ is refined using an MCMC to obtain an approximation $q_{\theta }^{\left(t\right)}\left(z\right)$ that is closer to the true posterior distribution, i.e.,

$$q_{\theta }^{\left(t\right)}\left(z\right)=\int Q^{\left(t\right)}\left(z|z_0\right)q_{\theta }\left(z_0\right)d{z}_0,$$

(13)

where $Q^{\left(t\right)}\left(z|z_0\right)$ is the overall transition kernel [52]. Since $q_{\theta }^{\left(t\right)}\left(z\right)$ is an improvement of $q_{\theta }\left(z\right)$, $q_{\theta }^{\left(t\right)}\left(z\right)$ is closer to $p\left(z|x\right)$ than $q_{\theta }\left(z\right)$, i.e.,

$$\mathit{KL}\left(q_{\theta }\left(z\right)\left|\right|p\left(z|x\right)\right)\ge \mathit{KL}\left(q_{\theta }^{\left(t\right)}\left(z\right)\left|\right|p\left(z|x\right)\right)\ge 0,$$

(14)

The equality sign holds if and only if $q_{\theta }\left(z\right)=p\left(z|x\right)$ [53].

To eliminate $log{q}_{\theta }^{\left(t\right)}\left(z\right)$, add a term $\mathit{KL}\left(q_{\theta }^{\left(t\right)}\left|\right|q_{\theta }\left(z\right)\right)$ to Equation (14), i.e.,

$${\mathit{L}}_{\mathit{VM}}\left(\theta \right)=\mathit{KL}\left(q_{\theta }\left(z\right)\left|\right|p\left(z|x\right)\right)-\mathit{KL}\left(q_{\theta }^{\left(t\right)}\left(z\right)\left|\right|p\left(z|x\right)\right)+\mathit{KL}\left(q_{\theta }^{\left(t\right)}\left|\right|q_{\theta }\left(z\right)\right),$$

(15)

This scatter can be expressed as

$${\mathit{L}}_{\mathit{VM}}\left(\theta \right)=-E_{q_{\theta }\left(z\right)}\left[f_{\theta }\left(z\right)\right]+E_{q_{\theta }^{\left(t\right)}\left(z\right)}\left[f_{\theta }\left(z\right)\right],$$

(16)

As $t\to \infty$, $q_{\theta }^{\left(t\right)}\left(z\right)$ converges to $p\left(z|x\right)$, and ${\mathit{L}}_{\mathit{VM}}\left(\theta \right)$ converges to the symmetric $\mathit{KL}$ dispersion of the variational distribution with the posterior distribution.

4.2. Parameter Estimates

In the Bayesian framework, the original variational distribution is first parametrically evaluated using VI. The posterior probability is first calculated

$$\gamma \left(z_{nk}|x_n,y_n\right)=f\left(y_n|x_n,z_{nk}=1\right)f\left(x_n|{\theta }_k\right){\displaystyle \frac{{\pi }_k}{{\displaystyle \sum _{j=1}^K}{\pi }_{j}f\left(x_n|{\theta }_j\right)}},$$

(17)

The expectation for the intermediate variable ${\eta }_{nk}$

$$E\left[{\eta }_{nk}\right]={\displaystyle \frac{\gamma \left(z_{nk}|x_n,y_n\right)}{f\left(y_n|x_n,\theta \right)}},$$

(18)

The expectation for the intermediate variable $ln\left({\eta }_{nk}\right)$

$$E\left[ln\left({\eta }_{nk}\right)\right]=lnf\left(y_n|x_n,z_{nk}=1\right)+lnf\left(x_n|{\theta }_k\right)-ln\left(\sum _{j=1}^K{\pi }_{j}f\left(x_n|{\theta }_j\right)\right),$$

(19)

Update weights

$${\pi }_k={\displaystyle \frac{{\displaystyle \sum _i}\gamma \left(z_{ik}|x_i,y_i\right)}{N_k}},$$

(20)

Updating the mean

$${\mu }_k={\displaystyle \frac{{\displaystyle \sum _i}\gamma \left(z_{ik}|x_i,y_i\right)·{\eta }_{ik}·x_i}{{\sum }_i{R}_{ik}·{\eta }_{ik}}},$$

(21)

Update precision

$${\Lambda }_k={\displaystyle \frac{{\displaystyle \sum _i}\gamma \left(z_{ik}|x_i,y_i\right)·{\eta }_{ik}·\left(x_i-{\mu }_k\right)·{\left(x_i-{\mu }_k\right)}^T}{N_k}},$$

(22)

Updating the degrees of freedom

$$v_k={\displaystyle \frac{\gamma {\left(z_{nk}|x_n,y_n\right)}^T\left(E\left[ln\left({\eta }_{nk}\right)\right]-E\left[{\eta }_{nk}\right]\right)}{N_k}},$$

(23)

Updated regression coefficients

$${\omega }_k={\displaystyle \frac{x_{wan}^T·\gamma {\left(z_{nk}|x_n,y_n\right)}^T·y}{x_{wan}^T·\gamma {\left(z_{nk}|x_n,y_n\right)}^T·x_{wan}}},$$

(24)

Update noise

$${\sigma }_k^2={\displaystyle \frac{{\displaystyle \sum _i}\gamma \left(z_{ik}|x_i,y_i\right){\left(y_i-x_i·{\omega }_k\right)}^2}{N_k}},$$

(25)

This was followed by parameter optimization of the variational distribution using MCMC. The log-likelihood function was calculated for each data point $x_n$

$$logp\left(x_n|\theta \right)=\sum _{k=1}^K{\alpha }_{k}logp\left(x_n|{\mu }_k,{\Lambda }_k,{\nu }_k\right)+logp\left(y_n|x_n,\theta \right),$$

(26)

where ${\alpha }_k$ is the mixing coefficient of the $kth$ component, ${\mu }_k$ is the mean vector, ${\Lambda }_k$ is the precision vector, ${\nu }_k$ is the degrees of freedom, and $y_n$ is the observation.

$$logp\left(x_n|{\mu }_k,{\Lambda }_k,{\nu }_k\right)=-{\displaystyle \frac{1}{2}}log\left|{\Lambda }_k\right|-{\displaystyle \frac{1}{2}}{\left(x_n-{\mu }_k\right)}^T{\Lambda }_k^{-1}\left(x_n-{\mu }_k\right)-{\displaystyle \frac{1}{2}}log\left(2\pi \right)-{\displaystyle \frac{\mathit{D}}{2}}log\left({\nu }_k\right),$$

(27)

where $\mathit{D}$ is the data dimension.

$$logp\left(y_n|x_n,\theta \right)=log\left(\int \mathcal{N}\left(y_n|x_n,\theta \right)d\theta \right)=log\left(\mathcal{N}\left(y_n|x_n,\theta \right)\right),$$

(28)

where $\mathcal{N}\left(y_n|x_n,\theta \right)$ is the probability density function of the Student’s t-distribution.

To avoid numerical overflow

$$log\sum _i{e}^{log{x}_i}=\underset{i}{max}log{x}_i+log\sum _j{e}^{log{x}_j-log{x}_i},$$

(29)

The Metropolis–Hastings criterion decides whether to accept the new parameter values and calculate the acceptance probability

$$\alpha =min\left(1,{\displaystyle \frac{p\left({\theta }^{\prime }\right)}{p\left(\theta \right)}}\right),$$

(30)

where $p\left({\theta }^{\prime }\right)$ is the probability of the new parameter ${\theta }^{\prime }$ and $p\left(\theta \right)$ is the probability of the old parameter $\theta$. A number u is randomly drawn from the interval (0,1) according to a uniform distribution with parameter ${\theta }_i={\theta }^{\prime }$ if $u\le \alpha \left(\theta ,{\theta }^{\prime }\right)$ and parameter ${\theta }_i=\theta$ otherwise.

The HMC method proposes new parameter values

$${\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}t}}={\displaystyle \frac{\partial H}{\partial \theta }},$$

(31)

$$\theta \left(T\right)=\theta \left(0\right)+{\int }_0^T{\displaystyle \frac{\partial H}{\partial \theta }}\mathrm{d}t,$$

(32)

where $H\left(\theta \right)$ is the Hamiltonian function, $\theta \left(0\right)$ is the initial value, and $\theta \left(T\right)$ is the proposed new parameter value. To ensure optimization of the HMC parameters, the step size must be adaptively adjusted according to the acceptance rate. If the acceptance rate is too low, the step size is reduced to 0.9 times the original acceptance rate, and if the acceptance rate is too high, the step size is increased to 1.1 times the original acceptance rate to explore the parameter space more efficiently. By continuously adjusting the step size, the goal is to stabilize the acceptance rate between approximately 0.65 and 0.75, which is the recommended acceptance rate range. The number of jump steps can also be adjusted if the step size has been adjusted to the appropriate range but the acceptance rate is still unsatisfactory. In the experiments, the number of jump steps will be adaptively adjusted according to the performance of the algorithm, and the strategy of adaptively adjusting the distribution of proposals will be used to improve the mixing efficiency of the Markov Chain, thus improving the overall performance of the algorithm.

After each iterative process is completed, the difference in the variational Bayesian MCMC ${\mathit{L}}_{\mathit{VM}}\left(\theta \right)$ is calculated, and the difference is compared with the specified threshold $\epsilon$. When the difference is lower than $\epsilon$, the algorithm is considered to have converged, i.e.,

$$|{\mathit{L}}_{\mathit{VM}}{\left(\theta \right)}_t-{\mathit{L}}_{\mathit{VM}}{\left(\theta \right)}_{t-1}|<\epsilon ,$$

(33)

When the difference is higher than $\epsilon$, the algorithm proceeds to the next round of iterations if the number of iterations does not exceed the specified range.

4.3. Overall Flow of the Model

The classification process is divided into two phases: offline modeling and online prediction.

In the offline modeling phase, data such as cracker tube outlet temperature and tube inlet temperature are first collected from a petrochemical ethylene plant, the obtained data are cleaned, and the cleaned data are stored in a CSV file in the format of (samples, features) for subsequent processing. A VI is used to optimize a variational distribution to approximate the true posterior distribution. At the same time, Markov Chains are constructed using the MCMC to generate samples whose distribution will eventually converge to the true posterior. In the method of this paper, the variational distribution learned by VI is used as the initial distribution of the MCMC, the variational distribution is improved by iterating the specified number of rounds through the MCMC, and the parameters of VI are updated using the improved variational distribution. The offline modeling phase is completed when the VI and the MCMC are alternately iterated to the specified number of rounds or when the model converges.

In the online prediction phase, for the collected historical data, the values of its probability density function under the mixture components are calculated and the probability density of each mixture component is converted into a posteriori probabilities. By combining the features of the historical data with the a posteriori probabilities and multiplying them by a weight matrix, an output is obtained that represents the relative position of that data in each category. The category to which the data most likely belongs is determined by comparing the distance of the output to the category label. When the true value of the online analysis output is finally obtained, it is stored in the historical data. It can be used for continuous learning and improvement of the model. Offline modeling is carried out periodically or as needed to update the model. The hyperparameters are set to be the prior distribution of the model. The flowchart of VMSTMR is shown in Figure 2.

Algorithm 1 gives the pseudo-code for the VMSTMR-based soft measurement model.

Algorithm 1 Soft measurement model based on VMSTMR

1: Input the training set x, y, set the parameters $\pi$ , $\mu$ , $\Lambda$ , $\nu$ , $\omega$ , $\sigma$, and the thresholds $\epsilon$ 2: Initialize the variational distribution of the model $q_{\theta }\left(\mathit{z}\right)$ 3: Set the number of iterations m such that $i=0$: 4: while  $i<m$  do 5:     $i\Leftarrow i+1$ 6:     while $k=1,\dots ,K$; $n=1,...,N$ do 7:           Compute the posterior probability $\gamma \left(z_{nk}|x_n,y_n\right)$ 8:           Compute the expectation of the intermediate variable $E\left[{\eta }_{nk}\right]$ 9:           Update the parameters ${\pi }_k$, ${\mu }_k$, $v_k$, ${\omega }_k$, ${\sigma }_k$ 10:         Calculate the log-likelihood function for the data points $logp\left(x_n|\theta \right)$ 11:         Calculation of acceptance probability $\alpha$ 12:         if $u\le \alpha$ then 13:                Proposed new parameter values $\theta \left(T\right)$ 14:         end if 15:         Calculating the variational bayesian MCMC ${\mathit{L}}_{\mathit{VM}}\left(\theta \right)$ 16:         if ${\mathit{L}}_{\mathit{VM}}\left(\theta \right)$ converges then 17:               break 18:         end if 19:       end while 20: end while 21: Input the test set and compute the posterior probability $\gamma \left(z_{nk}|x_{test},y_n\right)$ 22: Compute the conditional probability density function $p\left(\widehat{y}|x_{test},z,\omega ,\sigma \right)$ of $\widehat{y}$ 23: Calculate the degree of coking ${\widehat{y}}_{new}$ corresponding to the test set $x_{test}$

4.4. Evaluation Index

In this paper, a total of four model evaluation metrics are used, which are root mean square error (RMSE), accuracy (Accuracy), mean absolute error (MAE), and mean absolute percentage error (MAPE) [54,55]. Among them, RMSE is usually used to measure the accuracy of the model, which reflects the overall magnitude of the difference between the output values and the true values, where the smaller the value of RMSE, the better the classification effect of the model; Accuracy is usually used to assess the performance of the classification model, which reflects the ratio of the number of samples correctly classified by the model to the total number of samples, where the bigger the value of Accuracy, the better the classification performance of the model; MAE is usually used to measure the error of the model, which reflects the average of the error between the output value and the real value, where the smaller the value of MAE, the smaller the classification error of the model; MAPE is usually used to measure the performance of the model, which reflects the extent of the deviation between the output value and the real value expressed as a percentage, where the smaller the value of MAPE, the better the classification accuracy of the model. To summarize, RMSE, MAE, and MAPE are all indicators used to measure the classification error of the model, so the smaller the value, the better, indicating that the output result of the model is closer to the real value; meanwhile, Accuracy is an indicator used to measure the performance of the classification model, and the bigger the value, the better, indicating that the model is more capable of classification.

The formulas for the four evaluation metrics are as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{m}}\sum _{i=1}^m{\left[{\widehat{y}}^{\left(i\right)}-y^{\left(i\right)}\right]}^2},$$

(34)

$$Accuracy={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}},$$

(35)

$$MAE={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left|{\widehat{y}}^{\left(i\right)}-y^{\left(i\right)}\right|,$$

(36)

$$MAPE={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left|{\displaystyle \frac{y^{\left(i\right)}-{\widehat{y}}^{\left(i\right)}}{y^{\left(i\right)}}}\right|\times 100,$$

(37)

In the above equation, $\widehat{y}$ denotes the output value, y represents the true value, TP is the number of correctly categorized positive samples, TN is the number of correctly categorized negative samples, FP is the number of incorrectly categorized positive negative samples, FN is the number of incorrectly categorized negative positive samples, and m denotes the number of sample observations.

5. Results

In this experiment, PLSR [56], GPR [57], GMR [58], STMR [22], and SsSMM [32] are selected as benchmarks for the comparative analysis to evaluate the performance of different models by comparing the RMSE, Accuracy, MAE and MAPE values to estimate the performance of the different models.

To avoid the effect of initial parameter randomness, 50 simulations of PLSR, GPR, GMR, STMR, SsSMM, and VMSTMR algorithms were performed and averaged for each evaluation metric.

5.1. Simulation Experiments

Assuming the existence of a two-dimensional input variable $x={\left(x_1,x_2\right)}^T$ and an output variable y, the input variables assume a mixed Gaussian distribution, and the specific parameter settings of the model are shown in

Table 2. Parameters of the three Gaussian distribution components.

	$\mathit{k}=1$	$\mathit{k}=2$	$\mathit{k}=3$
${\pi }_k$	0.3	0.3	0.4
${\Lambda }_k$	$\left[\begin{array}{cc}2.5& 1.0\\ 1.0& 1.5\end{array}\right]$	$\left[\begin{array}{cc}2.5& 1.0\\ 1.0& 2.0\end{array}\right]$	$\left[\begin{array}{cc}2.5& -1.0\\ -1.0& 2.0\end{array}\right]$
${\mu }_k$	${\left[\begin{array}{cc}-3.5& 1.0\end{array}\right]}^T$	${\left[\begin{array}{cc}2.5& 3.5\end{array}\right]}^T$	${\left[\begin{array}{cc}2.0& -2.0\end{array}\right]}^T$
${\omega }_k$	${\left[\begin{array}{ccc}1.0& 1.0& -2.0\end{array}\right]}^T$	${\left[\begin{array}{ccc}1.0& -1.0& 0.0\end{array}\right]}^T$	${\left[\begin{array}{ccc}-1.0& 1.0& 1.0\end{array}\right]}^T$
${\sigma }_k$	0.25	0.25	0.25

.

A total of 2000 samples were randomly generated into three data sets, i.e., training, validation and test sets, each containing three categories, Model1, Model2 and Model3, and the sample ratio was distributed according to 6:2:2 to ensure that the training set contained 1200 samples and the test and validation sets contained 400 samples each. The training set is used to train the parameters of the VMSTMR model to be able to make an accurate classification of unseen data; the validation set is used to adjust the parameters of the VMSTMR model to avoid overfitting; and the test set is used to evaluate the final performance of the model after the model training is completed. The distribution of the noiseless data is shown in Figure 3.

To verify the robustness of the model, outlier points of 1%, 3%, 5%, and 7% were introduced in the input data samples. These outlier points are generated by transforming the eigenvalues of randomly selected sample data to values far from the center of their normal distribution. For example, introducing 1%, 3%, 5%, and 7% outlier points in a training set containing 2000 samples means adding 20 (1%), 60 (3%), 100 (5%), and 140 (7%) outlier points, respectively, to that data set. The distribution of the added random noise data is shown in Figure 4.

To reduce random errors and improve the reliability of the results, a fixed random seed was used to ensure that the randomness remained consistent from one experiment to the next. The hyperparameters of the PLSR, GPR, GMR, STMR, SsSMM and VMSTMR models were then optimized using a grid search algorithm. After determining the optimal hyperparameters of the models, 50 repetitions of the simulation experiments were performed and a cross-validation technique was applied to increase the efficiency of data usage and reduce the chance of evaluation results. The evaluation metrics for each experiment were recorded in detail. In order to obtain a stable estimate of the model performance, the evaluation metrics of all experiments were averaged and the specific results of the obtained average performance metrics are shown in

Table 3. Mean values of each indicator for each model on randomized data.

Outlier	Method	Evaluation Indicators
		RMSE	Accuracy	MAE	MAPE
0%	PLSR	0.4583	0.7950	0.2067	11.6111
	GPR	0.4472	0.8000	0.2000	10.6389
	GMR	0.2972	0.9417	0.0683	4.2500
	STMR	0.2708	0.9367	0.0667	4.6389
	SsSMM	0.2739	0.9400	0.0650	4.3333
	VMSTMR	0.2646	0.9450	0.0600	3.6944
1%	PLSR	0.5154	0.7855	0.2294	12.7063
	GPR	0.5122	0.7871	0.2261	11.7437
	GMR	0.3471	0.9356	0.0809	3.2728
	STMR	0.3374	0.9257	0.0875	5.1293
	SsSMM	0.3250	0.9274	0.0825	5.0193
	VMSTMR	0.2695	0.9472	0.0594	3.4791
3%	PLSR	0.6258	0.7557	0.2848	15.6419
	GPR	0.6322	0.7460	0.2929	15.1025
	GMR	0.4691	0.8948	0.1392	6.7826
	STMR	0.4388	0.9126	0.1181	5.6499
	SsSMM	0.4238	0.9191	0.1084	4.6926
	VMSTMR	0.3530	0.9288	0.0858	4.1667
5%	PLSR	0.6796	0.690	0.3508	21.8254
	GPR	0.6761	0.6810	0.3556	21.2566
	GMR	0.5492	0.8905	0.1619	6.3492
	STMR	0.4595	0.9000	0.1349	6.0450
	SsSMM	0.4508	0.9111	0.1238	5.2381
	VMSTMR	0.3695	0.9206	0.0952	4.6032
7%	PLSR	0.7267	0.6651	0.3879	23.4813
	GPR	0.7426	0.6729	0.3863	24.0654
	GMR	0.7159	0.5950	0.4408	34.5145
	STMR	0.5131	0.8910	0.1511	7.0093
	SsSMM	0.4882	0.8879	0.1480	7.3079
	VMSTMR	0.4569	0.9065	0.1277	6.5421

.

By comparing the mean values of the indicators, it can be observed that the Accuracy of the PLSR model is the lowest regardless of whether outliers are added or not, or how many outliers are added, which indicates that it has the worst classification effect, and at the same time, the RMSE, MAE, and MAPE of the PLSR model are the largest, which indicates that it has the worst model performance. Comparatively, the Accuracy of the VMSTMR model is always the highest and provides the best classification effect. Its RMSE, MAE, and MAPE are the smallest, indicating the best model performance.

After calculation, it can be obtained that the RMSE of PLSR, GPR, GMR, STMR, and SsSMM models increased by 0.0571, 0.0650, 0.086, 0.0639, and 0.0511, respectively, and that of VMSTMR model only increased by 0.0049 when 1% outliers were added, which indicates that the classification accuracy of VMSTMR model in extreme cases is more stable and improved the dispersion of the error; the MAE of the VMSTMR model was reduced by 74.11%, 73.73%, 36.88%, 32.11%, and 28% compared to the PLSR, GPR, GMR, STMR, and SsSMM models, which significantly reduced the overall sample error; the MAPE was decreased by 72.62%, 70.37%, 12.76%, 32.17%, and 30.69% respectively, which indicates that the output values of the VMSTMR model deviate from the true values to a lesser extent during classification.

Similarly, the increase in RMSE as well as the relative degree of MAE and MAPE of each model after adding 3%, 5% and 7% outlier points were calculated, and it can be concluded that the VMSTMR model has the smallest increase in RMSE, which indicates that the VMSTMR model is more stable than the PLSR, GPR, GMR, STMR and SsSMM models in dealing with outlier points. At the same time, the relative increase in MAE and MAPE of the VMSTMR model is also the smallest, which means that the classification error of the VMSTMR model is relatively small when outliers are introduced, and the deviation between its classification results and the actual values is the smallest. Therefore, the VMSTMR model performs best in terms of robustness and its output results are closer to the real values.

The VMSMTR model combines VI and MCMC techniques to effectively overcome some of the limitations of the traditional model. Compared with PLSR, although PLSR can effectively deal with multicollinearity, it is not as flexible as the VMSMTR model in dealing with complex data distributions and outliers. The VMSMTR model is able to capture the diversity and complexity of the data more efficiently by mixing multiple Student’s t-distributions and is therefore more robust to outliers.

Compared to the GPR and GMR models, which use a Gaussian distribution as the base model, the VMSTMR model is less sensitive to outliers. This is because the VMSTMR model uses a mixture of multiple Student’s t-distributions, which allows it to adapt to non-Gaussian and multimodal data, thus improving the classification accuracy of the model.

Compared to the STMR and SsSMM models, the VMSTMR model achieves more stable parameter estimation by combining the VI and MCMC techniques, thus solving the approximation inaccuracy problem of the STMR and SsSMM models that can occur when a single VI technique is used to estimate model parameters. As a result, the VMSTMR model is able to capture the underlying distribution of the data more accurately and improve the generalization ability of the model.

The VMSTMR model excels at handling complex industrial data by combining VI and MCMC techniques, but this also leads to an increase in model run time. Specifically, the CPU execution time of the VMSTMR model is approximately one hundred and ten seconds. In contrast, PLSR, a linear regression method, is more computationally efficient with a CPU execution time of approximately seventy seconds. When dealing with large datasets, the CPU execution time of the GPR model increases significantly to approximately eighty seconds, and the CPU execution time of the GMR model is typically shorter than that of the VMSTMR model, which is approximately eighty-five seconds. Due to the omission of the MCMC step, the STMR model has a shorter CPU execution time than the VMSTMR model, approximately ninety seconds. The SsSMM model has a similar CPU execution time to the VMSTMR model, approximately one hundred seconds, when processing both labeled and unlabeled data.

5.2. Actual Industrial Data

Actual operating data from an ethylene cracker at a major petrochemical company are used in this section. When screening the characteristics, the experience of local experts and the mechanism model of catalytic cracking are combined. Finally, six key characteristics are selected, including the furnace tube outlet temperature, the furnace tube inlet temperature, the adiabatic pressure ratio, the cross-section pressure, and the venturi pressure, and the framework of the diagnostic reasoning system for the coking of the furnace tube of an ethylene cracking furnace tube is shown in Figure 5. In addition, the furnace tube coking condition is categorized into four normal levels mild coking, moderate coking and severe coking, according to the degree of furnace tube coking.

As the unprocessed output of the model is a floating-point value and the coking level of the furnace tube is an integer value, the output of the model is processed as follows in order to enable the model to more accurately predict the coking level of the furnace tube.

$${\widehat{y}}_{new}=\left\{\begin{array}{c}{j}_n,|\widehat{y}-j_n|<0.5\hfill \\ j_{n+1},\widehat{y}-j_n=0.5\hfill \end{array}\right.,$$

(38)

where $j_n$ is the degree of coking in the furnace tube, $\widehat{y}$ is the output of the unprocessed model, and ${\widehat{y}}_{new}$ is the processed output.

The dataset used in this section contains 5000 samples, of which the training set accounts for 3500 samples and the test set accounts for 1500 samples. RMSE, Accuracy, MAE, and MAPE are used as evaluation metrics to measure the prediction accuracy and robustness of each model.

A large number of iterations were performed to ensure that the MCMC chain could adequately mix and explore the parameter space. At the same time, an algorithm for adaptively adjusting the proposal distribution was used to improve the mixing efficiency of the chain. Under this configuration, the acceptance rate during MCMC iterations was stabilized at around 0.7, which is in the ideal acceptance rate interval and helps to maintain the efficiency and exploratory power of the algorithm. The variation in the acceptance rate during MCMC iterations is shown in Figure 6.

Table 4. Average values of each indicator for each model on industrial data.

Outlier	Method	Evaluation Indicators
		RMSE	Accuracy	MAE	MAPE
0%	PLSR	0.2041	0.9583	0.0417	2.8938
	GPR	0.1875	0.9648	0.0352	2.2846
	GMR	0.1909	0.9635	0.0365	2.2628
	STMR	0.1552	0.9759	0.0241	1.2946
	SsSMM	0.1712	0.9707	0.0293	1.4143
	VMSTMR	0.1531	0.9766	0.0234	1.2728
1%	PLSR	0.3546	0.9446	0.0729	4.1801
	GPR	0.3330	0.9626	0.0554	2.5695
	GMR	0.3537	0.9330	0.0825	5.1336
	STMR	0.3311	0.9491	0.0658	2.7149
	SsSMM	0.3232	0.9620	0.0554	2.6126
	VMSTMR	0.3047	0.9691	0.0464	1.8908
3%	PLSR	0.5451	0.8906	0.1580	9.5427
	GPR	0.5262	0.8963	0.1492	8.1432
	GMR	0.5416	0.8635	0.1795	11.4174
	STMR	0.5103	0.9172	0.1290	5.1700
	SsSMM	0.4908	0.8881	0.1473	8.5763
	VMSTMR	0.4391	0.9159	0.1132	4.8638
5%	PLSR	0.6625	0.8202	0.2505	15.2113
	GPR	0.6431	0.8357	0.2325	13.3468
	GMR	0.5656	0.8134	0.2232	14.0615
	STMR	0.5732	0.8407	0.2083	13.0982
	SsSMM	0.5590	0.8983	0.1612	7.0851
	VMSTMR	0.4638	0.8977	0.1333	5.5314
7%	PLSR	0.8600	0.6928	0.4185	28.0640
	GPR	0.8101	0.7348	0.3692	23.0335
	GMR	0.7489	0.6801	0.3869	23.4451
	STMR	0.7024	0.8193	0.2683	14.9848
	SsSMM	0.7473	0.8157	0.2895	16.5396
	VMSTMR	0.4569	0.8802	0.1721	6.4939

demonstrates the average values of the evaluated metrics, and according to the computational analysis, it can be concluded that the VMSTMR model exhibits superior performance compared to the PLSR, GPR, GMR, STMR, and SsSMM models in all cases, regardless of whether or not outliers are introduced or how the number of outliers varies.

Based on the experimental results in Table 4, it can be seen that the VMSTMR model performs best, followed by the SsSMM model. Further, by looking at Figure 7, it can be found that in Figure 7a, the overlap between the true and output values of the VMSTMR model is lower than that in Figure 7b. Therefore, it can be concluded that the VMSTMR model is better than the SsSMM model for good classification, i.e., the VMSTMR model performs the best in terms of classification effectiveness.

In order to better evaluate the performance of the VMSTMR model and to explore different variants of the MCMC kernel function to improve its sampling efficiency, uniform kernels, Gaussian kernels and Laplacian kernels were first compared, and then the effectiveness of the Random Walk Metropolis(RWM) as an alternative proposal kernel was tested. A 0% data set was used in the experiments for testing. The test results show that the RMSE, Accuracy, MAE and MAPE of uniform kernels are 0.2001, 0.9596, 0.0404 and 1.7524, respectively, the RMSE, Accuracy, MAE and MAPE of Gaussian kernels are 0.1804, 0.9675, 0.0326 and 1.621, respectively. The RMSE, Accuracy, MAE and MAPE of the Laplacian kernels are 0.2165, 0.9531, 0.0469 and 2.2193, respectively. Compared to the VMSTMR model, there is a time saving of approximately twenty to forty seconds for both the uniform, Gaussian and Laplacian kernels. However, this time advantage did not translate into improved experimental results, i.e., although sampling speed was improved, accuracy and reliability were sacrificed. To further validate the optimality of the adopted method, the RWM method was parameterized and tested as an alternative proposal kernel. The test results show that the performance of the parametrically tuned RWM on the four metrics of RMSE, Accuracy, MAE and MAPE are 0.215, 0.9538, 0.0462 and 4.4278, respectively. Despite the parametric tuning, RWM still falls short of VMRM on the three performance metrics of RMSE, MAE and MAPE. This suggests that although the uniform, Gaussian and Laplace kernels have advantages in sampling efficiency, these kernels and the RWM do not perform as well as the VMSTMR model in terms of accuracy and reliability. Therefore, the proposed kernel selected for the VMSTMR model is a more appropriate choice for the application of ethylene coke diagnosis.

6. Conclusions

This paper addresses the limitations of traditional ethylene cracker furnace tube coking diagnostic methods in terms of detection period, accuracy and real-time, as well as the multimodal, nonlinear, non-Gaussian and strong noise problems of the production process data. To this end, a novel soft measurement method is proposed: the mixed Student’s t-distribution regression soft measurement model based on VI and MCMC. In order to assess the reliability of VMSTMR, outlier points at 1%, 3%, 5% and 7% were incorporated into the experimental data set. The experimental results demonstrate that while the classification performance of all methods declines with an increase in the proportion of outlier points, VMSTMR exhibits a relatively lower decline. VMSTMR outperforms the PLSR, GPR, GMR, STMR and SsSMM methods in classifying the coking situation of ethylene cracker tubes, indicating that VMSTMR possesses a high level of robustness in the presence of outliers.

In actual industrial production, the application of the VMSTMR model can realize real-time monitoring of the coking condition of the ethylene cracker furnace tube. This is of great significance for optimizing the production process, reducing equipment downtime and extending the service life of the furnace tube, which in turn improves the production efficiency of ethylene and the economic benefits of the enterprise. Therefore, the introduction of the VMSTMR model can not only become a key technology for intelligent maintenance and fault prediction of ethylene crackers but is also expected to promote the technological progress and sustainable development of the entire petrochemical industry.

7. Future Research Directions

In machine learning and statistics, nonlinear data processing is the handling of nonlinear relationships between variables, which includes techniques such as data transformation, nonlinear model application, and feature engineering. For example, in support vector machines (SVMs), the choice of kernel function is critical in dealing with nonlinear problems. Commonly used kernel functions include linear kernel, polynomial kernel, radial basis function (RBF) kernel, and sigmoid kernel, etc., and their choice depends on the data characteristics and the specific needs of the problem. In addition, a hybrid model improves model performance by combining the advantages of different models, but this may also increase the complexity and training difficulty of the model.

Despite the significant achievements of the VMSTMR model in improving robustness, it still faces challenges in dealing with nonlinear data. To further improve the VMSTMR model’s ability to handle nonlinear data, future research will focus on addressing the model’s limitations in this area. Specifically, the research will explore the introduction of stacked self-encoders to deal with nonlinear problems. The stacked self-encoder is able to map raw data to a feature space that better reflects its intrinsic structure and complex relationships through a multilayer nonlinear transformation. This approach first learns an efficient representation of the data through layer-by-layer pre-training, and then optimizes the entire network through global fine-tuning to capture the complex nonlinear relationships in the data. This strategy not only improves the classification accuracy and generalization ability of the VMSTMR model in complex industrial data environments but also increases the robustness and applicability of the model.