Acidic Gas Prediction Modeling Based on Shared Features and Inverted Transformer of Municipal Solid Waste Incineration Processes

Abstract

Effective management of municipal solid waste is crucial for achieving sustainable development and maintaining a healthy ecological environment. Municipal solid waste incineration (MSWI) processes are highly nonlinear and exhibit strong coupling characteristics, which makes long-term stable control challenging. Accurate prediction of the various toxic and harmful acidic gases that will be generated during this process is crucial for supporting optimization and control research. This study proposes a predictive model for acidic gases using Random Forest (RF) and Inverted Transformer (ITransformer). First, the RF algorithm is used to identify feature variables that strongly correlate with the target variables, thereby facilitating the shared feature selection process for multiple acidic gases. These selected features are then fed into a multi-output ITransformer model, which predicts the target variables and generates multiple evaluation metrics. Finally, the model’s hyperparameters are optimized based on these metrics and the threshold ranges of the acidic gases. The experimental results using real data from a specific incineration plant show that 13 features remain after the shared feature selection process. Compared to other models, the proposed approach uses the fewest shared features while reducing computational costs. Moreover, the R² values for NO_x, SO₂, and HCl are 0.9791, 0.9793, and 0.9838, respectively.

1. Introduction

The acceleration of urbanization has led to a growing output of municipal solid waste (MSW) [1,2,3], placing tremendous pressure on urban management and the environment. The prolonged accumulation of MSW impacts residents’ lives, fosters the growth of bacteria, and can even damage soil and water resources [4]. Meanwhile, the growing output of MSW also hinders the sustainable development of the urban environment. As a result, finding effective solutions to the environmental issues caused by MSW accumulation has become a research hotspot [5,6,7]. Currently, the treatment technologies for MSW include landfilling, composting, incineration, etc. [8,9,10]. Although landfill treatment offers the advantage of simple operation, it is highly sensitive to factors such as site location and transportation costs. Moreover, it occupies a significant amount of land and can lead to a range of environmental problems, including leachate leakage and soil pollution [11]. Composting primarily decomposes the organic matter in MSW through microbial biochemical action, transforming it into usable organic fertilizer. It offers significant advantages in terms of safety and resource utilization, but it still poses challenges such as large land requirements and a high technical threshold [12]. MSW incineration (MSWI) [13] can significantly reduce the volume and weight of waste while simultaneously recovering energy [14]. However, MSWI processes generate various pollutants, and their emission concentrations must be controlled within the limits set by local environmental protection standards.

Unstable operation of MSWI can lead to the production of various pollutants at excessive concentrations, particularly acidic gases [9]. These gases, which include NO_x, SO₂, HCl, and HF, can irritate the respiratory mucosa, leading to symptoms such as coughing and difficulty breathing [15]. For example, prolonged exposure to low concentrations of acidic gases can increase the incidence of chronic obstructive pulmonary disease (COPD) and lung cancer, and raise the risk of asthma in children by 12% [16]. SO₂ and HCl can also cause conjunctivitis and keratitis, and in severe cases, may lead to corneal ulcers or permanent vision damage [17]. When SO₂ and NO_x combine with atmospheric water, they form sulfuric acid and nitric acid, which lower the pH of rainwater to below 5.6, leading to soil acidification and the death of vegetation [18]. Problems such as ozone layer depletion and photochemical smog also arise [19]. To address these issues and ensure that the emission concentration of acidic gases meets the standards, a key solution is the real-time detection and prediction of these processes. Acidic gases such as NO_x, SO₂, HCl, and HF can be detected by continuous emission monitoring systems (CEMSs) [20], enabling real-time detection. To address the need for intelligent optimization control of MSWI processes, an acidic gas prediction model should be constructed to provide effective support.

The emission concentrations of acidic gases from MSWI processes are influenced by multiple interconnected factors. To reduce model training time and resource consumption, it is essential to eliminate redundant variables through feature selection. Feature selection for multiple acidic gases involves removing ‘irrelevant features’ and identifying ‘shared features’. To eliminate irrelevant features, the correlation between individual factors in MSWI processes and the emission concentrations of NO_x, SO₂, and HCl should be considered. Common methods for calculating this include using the Pearson correlation coefficient, mutual information, and the maximal information coefficient [21,22,23,24,25]. To obtain ‘shared features’, factors related to multiple acidic gases should be considered for selection.

In actual MSWI processes, the generation of various acidic gases is usually not an isolated event and is the result of interactions between multiple process stages. For instance, the production of acidic gases like HCl, SO₂, and NO_x is influenced by various factors, including combustion temperature, oxygen content, and MSW composition. These factors interact in a complex, coupled manner. Specifically, the generation of SO₂ and NO_x is influenced by the same combustion conditions, and their concentration changes can be accurately predicted using a model with multiple inputs and outputs. This lays the foundation for coordinated control. Therefore, by establishing a prediction model that simultaneously forecasts the concentrations of different acidic gases, the intrinsic connections and interaction mechanisms between these pollutants can be effectively captured.

Numerous studies have developed prediction models for the emission concentrations of acidic gases. For example, Zhang et al. [26] used the Copula function to analyze the correlation between solid waste incineration operation control parameters and acid gas emission concentrations, and developed a BP neural network model. However, this BP model performs poorly on large datasets, lacks interpretability, and requires frequent parameter adjustments for long-term sequence prediction tasks, limiting its efficiency. Yao et al. [27] used MIC for feature selection and combined PCA with LSTM to propose a NO_x emission concentration prediction model for the SCR denitration system outlet. While their experiments demonstrated the model’s superiority, it lacks interpretability. Jin et al. [28] proposed a Long Short-Term Memory (LSTM) prediction model based on mutual information (MI) and the particle swarm optimization (PSO) algorithm, which successfully predicted SO₂ emission concentrations. However, the LSTM model lacks interpretability, and its parallel computing performance is limited, preventing it from effectively exploring the complex coupling relationships between industrial process variables. Fu et al. [29] developed a novel NO_x emission prediction framework that integrates fast attribute reduction (FAR) with a hybrid kernel random weight neural network (HKRWNN). While this approach can successfully predict NO_x emission concentrations, the HKRWNN algorithm has several limitations: a slow processing speed for large-scale time series data, high computational complexity, and limited scalability when handling large datasets. Meng et al. [30] proposed an emission prediction method based on an attention modular neural network (AMNN) to predict NO_x emission concentrations. However, the prediction results from the AMNN lack interpretability. Senem et al. [31] proposed a controlled autoregressive sliding average (ARMAX) model to predict HCl emission concentrations. However, ARMAX struggles with nonlinear data and large-scale datasets. Guo et al. [32] proposed an improved Transformer model that effectively predicted NO_x emission concentrations. Patricio et al. [33] proposed an LSTM-Conv1D model that effectively predicted SO₂ concentrations. However, the model lacks interpretability and may suffer from issues like overfitting when handling large-scale data. Despite this, research on the development of shared feature prediction models for the emission concentrations of multiple acidic gases (NO_x, HCl, and SO₂) from MSWI processes has not been reported.

To address the issues mentioned above, this study proposes a prediction model for the acidic gases emitted from MSWI processes. First, the RF algorithm is used to select feature variables that have a strong correlation with the target variables, thereby obtaining features shared by multiple acidic gases. Next, these selected features are used as inputs to a multi-output ITransformer model to predict the target variables and generate multi-output evaluation metrics. Finally, based on the above indicators and the threshold range for acidic gases, the model’s hyperparameters are optimized. The experiments show that the RF-ITransformer model has high prediction accuracy.

The innovative contributions of this article are as follows: (1) a novel prediction modeling framework for multiple acidic gases (NO_x, SO₂, and HCl) based on shared features and an ITransformer model is proposed for MSWI processes; (2) an RF-based algorithm for extracting shared features across multiple acidic gases is introduced; (3) the ITransformer multi-output model, utilizing newly defined multi-output evaluation indices, is developed.

To our knowledge, this is the first model for multiple acidic gases that are emitted from MSWI processes.

2. Materials and Methods

2.1. Materials

MSWI processes consist of several stages: solid waste fermentation, combustion, waste heat exchange, steam power generation, flue gas cleaning, and emission, as shown in Figure 1.

A detailed description is given below.

(1)

Solid waste fermentation stage: The high moisture content in raw MSW can lead to combustion instability. Therefore, biological fermentation is required as a pretreatment to improve the calorific value. The MSW undergoes 5–7 days of aerobic fermentation in a digester, where microorganisms reduce the moisture content through metabolic processes. Organic decomposition releases biothermal energy, increasing the lower heating value by approximately 30%, which helps meet the thermal requirements of the process. After fermentation is complete, the material is transferred by a bridge-type grab crane to a sealed hopper, from which it is periodically pushed into a grate system using hydraulic feeders to initiate the combustion process.

(2)

Solid waste combustion stage: The energy conversion process during combustion occurs in three primary thermodynamic stages. The first stage is drying and dehydration, where secondary radiant heat from the incinerator’s arch zone interacts with primary air preheated to 120–180 °C, causing rapid evaporation of surface moisture. This phase accounts for approximately 25% to 30% of the total combustion heat consumption. Next is the main combustion stage, where the MSW enters a gas–solid two-phase combustion process once its temperature reaches the volatile matter release threshold. Between 300 °C and 850 °C, volatile matter releases combustible gases (e.g., CO and CH4) through diffusion combustion, while at higher temperatures (850 °C to 1100 °C), fixed carbon undergoes surface and pore combustion for deep oxidation. To stabilize the process, the grate movement speed is regulated, and a staged supply of air (with a 60% to 70% primary air ratio) is implemented. This ensures that the oxygen concentration remains ≥6% and the flue gas residence time is ≥2 s, effectively decomposing toxic organic compounds such as dioxins. Finally, the incineration cooling phase begins as the MSW gradually combusts and the pyrolysis rates decrease. At this stage, the remaining combustible materials are primarily composed of coke. Under high temperatures and primary air, coke undergoes oxidation with O₂, releasing CO₂ and water vapor. This leads to the gradual accumulation of inert substances (CO, H2O, and ash) until the MSW is completely converted into ash, marking the termination of the combustion process. The incineration ash can then be used to create a new type of environmentally friendly building material.

(3)

Waste heat exchange stage: After the MSW undergoes high-temperature combustion (850–1100 °C) in the incinerator, it generates flue gases with a high enthalpy (approximately 800–1000 °C) that carry 60% to 80% of the total calorific value of the MSW. The high-temperature flue gas sequentially passes through the radiant heat exchange zone (upper furnace chamber) and the convective heat exchange zone (boiler tail) of a waste heat boiler, releasing thermal energy during the recovery phase. As the core equipment for thermal energy recovery, the waste heat boiler consists of three components: an economizer, evaporator, and superheater. During operation, the economizer first utilizes residual flue gas heat to warm water from 20–80 °C to 150–200 °C, improving the system’s thermal efficiency. Next, the high-temperature flue gas transfers heat to the water in the evaporator through radiation and convection, vaporizing it into saturated steam. Finally, the steam is further heated in the superheater to reach 350–450 °C, improving its quality.

(4)

Steam power generation stage: The high-temperature steam produced by the waste heat boiler drives the steam turbine generator, converting mechanical energy into electrical energy. Once self-sufficiency is achieved within the plant, surplus electricity can be exported, enabling resource recycling and generating economic benefits.

(5)

Flue gas cleaning and emission stage: First, the denitrification system removes nitrogen oxides (NO_x) from the flue gases. Next, semi-dry acid removal technology neutralizes the acidic gases in the flue gases. Subsequently, activated carbon adsorbs dioxins and heavy metals, while bag filters capture particulate matter, reactive substances, and adsorbed materials. Finally, the purified flue gases are discharged into the atmosphere through induced draft fans in compliance with national emission standards.

MSWI oxidizes the combustible components of MSW with oxygen in the air inside an incinerator, resulting in the destruction of hazardous substances through oxidation and pyrolysis at high temperatures. Through incineration, the volume of MSW can be reduced by 80–90% and its mass is reduced by 20–80%. Pathogens and hazardous substances are effectively eliminated, resulting in chemically stable, harmless ash and high-temperature flue gas. The flue gas can then be used in a boiler to generate high-temperature steam for electricity production via a steam turbine. The flue gases emitted from MSWI processes include particulate matter, acidic gases such as SO₂, HCl, and NO_x, heavy metals, and pollutants like dioxins, often referred to as the ‘poison of the century’. From an environmental protection perspective, establishing accurate predictive models for acidic gases helps reduce pollutant emissions from MSWI processes.

From a mechanistic perspective, the formation of SO₂ and NO_x requires chemical reactions involving oxygen. Whether combustion is complete or not influences the concentration of pollutants produced. The sulfur (S), nitrogen (N), and chlorine (Cl) content in MSW directly determines the concentrations of SO₂, NO_x, and HCl. Fuel-derived NO_x is produced by the oxidation of nitrogen compounds present in the fuel during combustion; its formation reaction is shown below:

N₂ + O₂ → 2NO

with thermal NO forming at high temperatures during combustion.

2NO + O₂ → 2NO₂

High temperatures promote sulfur oxidation, which is the main source of SO₂ generation:

S + O₂ → SO₂

Hypoxia promotes the formation of Cl2 while incomplete combustion of chlorine-containing organic compounds at low temperatures leads to increased production of HCl:

2Cl₂ + 2H₂O → 4HCl + O₂

To purify SO₂ and HCl, hydrated lime is injected into the reactor to neutralize the acidic gases, as shown below:

SO₂ + Ca(OH)₂ → CaSO₃ + H₂O

2HCl + Ca(OH)₂ → CaCl₂ + 2H₂O

Furthermore, the following oxidation reaction occurs, resulting in the formation of stable products:

2CaSO₃ + O₂ → 2CaSO₄

The Selective Non-Catalytic Reduction (SNCR) denitration principle in MSWI processes is as follows: a reductant, such as urea, is injected into the flue gas. Without a catalyst, the reductant selectively reacts with NO_x in the flue gas, reducing it to N2 and H2O. These reactions are shown below:

(NH₂)₂CO + H₂O → 2NH₃ + CO₂

4NH₃ + 4NO + O₂ → 4N₂ + 6H₂O

2.2. Methods

The modeling strategy proposed in this article primarily consists of an acidic gas shared feature selection module, an acidic gas multi-output modeling module, and an acidic gas modeling parameter selection module, as illustrated in Figure 2.

In Figure 2, $\mathit{X}\in {ℝ}^{N\times M}$ denotes the original sample input dataset ($n\in \{1,2,\dots N\}$, $m\in \{1,2,\dots M\}$); ${\theta }_q^{\min }$ and ${\theta }_q^{\max }$ denote the minimum and maximum values of the feature importance thresholds related to the target variables that were determined empirically; ${\mathit{X}}_1^{\mathrm{sel}}$, ${\mathit{X}}_2^{\mathrm{sel}}$, and ${\mathit{X}}_3^{\mathrm{sel}}$ denote the feature datasets selected by RF for the target variables NO_x, SO₂, and HCl, respectively; ${\theta }_1^{\mathrm{FI}}$, ${\theta }_2^{\mathrm{FI}}$, and ${\theta }_3^{\mathrm{FI}}$ denote the feature-importance thresholds for NO_x, SO₂, and HCl; ${\mathit{X}}_{\mathrm{in}}^{\mathrm{sel}}$ represents the shared feature set for ${\mathit{X}}_1^{\mathrm{sel}}$, ${\mathit{X}}_2^{\mathrm{sel}}$, and ${\mathit{X}}_3^{\mathrm{sel}}$; ${\mathit{X}}_{\mathrm{in},r}^{\mathrm{sel}}$ is the $r$th batch of sample inputs when building the ITransformer prediction model; $L$ is the number of encoder layers; ${\widehat{\mathit{Y}}}_Q$ is the output matrix containing the predicted values of NO_x, SO₂, and HCl in ${\widehat{\mathit{y}}}_1$, ${\widehat{\mathit{y}}}_2$, and ${\widehat{\mathit{y}}}_3$; $RRMS{E}_1$, $RRMS{E}_2$, and $RRMS{E}_3$ denote the relative root-mean-square errors (RRMSEs); $RRMSE$ is the RRMSE obtained using the arithmetic averaging method; and ${\theta }_{\mathrm{epochs}}$, ${\theta }_{\mathrm{learningrate}}$, ${\theta }_{\mathrm{model}}$, ${\theta }_{\mathrm{itransformer}}$, ${\theta }_{\mathrm{batchsize}}$, ${\theta }_{\mathrm{dropout}}$, and ${\theta }_{\mathrm{head}}$ are the iteration number, the learning rate, the model dimension, the encoder layer number, the batch size, the dropout rate, and the attention head number of the ITransformer model.

The functions of the different modules described below.

(1)

Acidic gas shared-feature selection module: First, the contribution of each feature to the model purity is tallied at every split node across all the decision trees in the RF, which is used to calculate an overall importance score for each feature. Next, low-importance features are filtered out to reduce input dimensionality, retaining the key features to enhance model interpretability. Finally, the shared-feature acquisition submodule extracts the common features subset for the three acidic gases.

(2)

Acidic gas multi-output modeling module: This module uses a many-to-many ITransformer prediction model. First, the data is embedded into a 3-D tensor that is processable by the system. Then, self-attention mechanisms are utilized to capture inter-variable relationships for feature extraction. Finally, the data is passed through a feed-forward neural network, which maps the input features to a high-dimensional space via linear transformation and applies activation functions for nonlinear transformation. This process allows the model to learn more complex and richer feature representations. The RRMSE is used as the evaluation metric to quantify prediction errors and guide hyperparameter selection for the acidic gas prediction model.

(3)

Acidic gas modeling parameter selection module: This module selects the appropriate feature-selection parameters based on the range of feature importance values to prevent model overfitting and improve generalization.

2.2.1. Shared-Feature Selection Module for Acidic Gases

Feature Selection Sub-Module Based on RF

RF is a machine learning algorithm that employs ensemble learning by integrating multiple decision trees. It enhances performance through random resampling (bootstrap) and random node splitting.

For feature selection using RF, the process begins by calculating the Feature Importance (FI) value, which quantifies the relationship between each feature in the process variables and the target variable. Out-of-bag (OOB) data is used to evaluate each feature’s importance. This method considers each feature’s contribution to the model’s predictive performance by assessing the changes in prediction errors between the original data and the data after adding perturbations.

By setting a feature importance threshold, low-importance features are identified and removed, thus completing the feature selection process [34,35,36,37].

Random backsampling is applied to the features of the original data ${\mathit{X}}_q$ to create the in-bag data. The data that is not sampled becomes the out-of-bag (OOB) data, while the in-bag data is used to train the decision tree $D{T}_t$. The optimal segmentation point is selected by splitting the current node’s sample into a left subtree ($x_{q,m}<{\theta }_q$) and a right subtree ($x_{q,m}>{\theta }_q$) based on the threshold ${\theta }_q$ of feature $x_{q,m}$ of the $q$th acidic gas, which are denoted as $S_{\mathrm{L},q}$ and $S_{\mathrm{R},q}$, respectively. The optimization process involves traversing all candidate features ${\mathit{X}}_q$ and thresholds ${\theta }_q$, selecting the combination $(m^{*},{\theta }_q^{*})$ that minimizes the weighted mean squared error ($MSE$), thereby completing the node split.

$$(m^{*},{\theta }_q^{*})=\arg \underset{m,{\theta }_q}{\min }[\frac{\left|S_{\mathrm{L},q}\right|}{Z}\cdot MSE(S_{\mathrm{L},q})+\frac{\left|S_{\mathrm{R},q}\right|}{Z}\cdot MSE(S_{\mathrm{R},q})]$$

(1)

where $\left|S_{\mathrm{L},q}\right|$ and $\left|S_{\mathrm{R},q}\right|$ represent the number of left subtrees and right subtrees of the characteristic partitioning node of the $q$th acidic gas, respectively; $Z$ represent the total number of samples of the current node; $MSE(S_{\mathrm{L},q})$ and $MSE(S_{\mathrm{R},q})$ represent the relative mean square errors of the left and right subtrees, respectively; and $m^{*}$ and ${\theta }_q^{*}$ represent the optimal splitting feature index and the optimal splitting threshold of the $q$th acidic gas, respectively.

$D{T}_t$ is used to predict the data outside the bag, and the original prediction error of the $t$th tree is recorded as $E_{\mathrm{original}}^{(t),q}$. The calculation is as follows:

$$E_{\mathrm{original}}^{(t),q}=\frac{1}{{\mathrm{D}}_{\mathrm{OOB}}^{(t),q}}{{\displaystyle \sum _{i\in {\mathrm{D}}_{\mathrm{OOB}}^{(t),q}}(y_{i,q}-{\widehat{y}}_{t,q}(x_{i,q}))}}^2$$

(2)

The feature with index $m$ in the OOB data of the $t$th tree is randomly replaced, and the prediction is re-made using the perturbed data. The calculation error $E_{\mathrm{perturbed}}^{(t,m_q),q}$ is determined as follows:

$$E_{\mathrm{perturbed}}^{(t,m),q}=\frac{1}{{\mathrm{D}}_{\mathrm{OOB}}^{(t),q}}{{\displaystyle \sum _{i\in {\mathrm{D}}_{\mathrm{OOB}}^{(t),q}}(y_{i,q}-{\widehat{y}}_{t,q}^{(m)}(x_{i,q}))}}^2$$

(3)

FI is calculated as follows:

$$F{I}_q^m=\sum _{t=1}^T(E_{\mathrm{perturbed}}^{(t,m)}-E_{\mathrm{original}}^{(t)})/T$$

(4)

where $T$ represents the total number of decision trees.

According to the set threshold ${\theta }_q^{\mathrm{FI}}$, the features that meet the conditions are selected based on the following criterion:

$${\zeta }_q^m==\left\{\begin{array}{c}1,F{I}_q^m\ge {\theta }_q^{\mathrm{FI}}\\ 0,F{I}_q^m<{\theta }_q^{\mathrm{FI}}\end{array}\right.$$

(5)

The feature corresponding to ${\zeta }_q^m=1$ is selected, and all the selected features are recorded as ${\mathit{X}}_q^{\mathrm{sel}}$. It should be noted that $Q=3$.

Redundant Feature Removal Sub-Module

To predict multiple acidic gases simultaneously, the shared feature set must satisfy the correlation requirements for all the acidic gases. The number of features in the three feature sets for acidic gases can be expressed by the following formula:

$$M_1^{\mathrm{sel}}=\mathrm{Len}({\mathit{X}}_1^{\mathrm{sel}})$$

(6)

$$M_2^{\mathrm{sel}}=\mathrm{Len}({\mathit{X}}_2^{\mathrm{sel}})$$

(7)

$$M_3^{\mathrm{sel}}=\mathrm{Len}({\mathit{X}}_3^{\mathrm{sel}})$$

(8)

where $\mathrm{Len}(\cdot )$ represents the length calculation function and $M_1^{\mathrm{sel}}$, $M_2^{\mathrm{sel}}$, and $M_3^{\mathrm{sel}}$ represent the number of features when NO_x, SO₂, and HCl are the target variables, respectively. The sets of feature variable names in the three feature sets are represented as follows:

$${\mathit{S}}_1^{\mathrm{sel}}=\{s_{1,1}^{\mathrm{sel}},\dots ,s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}},\dots ,s_{1,M_1^{\mathrm{sel}}}^{\mathrm{sel}}\}$$

(9)

$${\mathit{S}}_2^{\mathrm{sel}}=\{s_{2,1}^{\mathrm{sel}},\dots ,s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}},\dots ,s_{2,M_2^{\mathrm{sel}}}^{\mathrm{sel}}\}$$

(10)

$${\mathit{S}}_3^{\mathrm{sel}}=\{s_{3,1}^{\mathrm{sel}},\dots ,s_{3,m_3^{\mathrm{sel}}}^{\mathrm{sel}},\dots ,s_{3,M_3^{\mathrm{sel}}}^{\mathrm{sel}}\}$$

(11)

where ${\mathit{S}}_1^{\mathrm{sel}}$, ${\mathit{S}}_2^{\mathrm{sel}}$, and ${\mathit{S}}_3^{\mathrm{sel}}$ represent the sets of feature variable names with NO_x, SO₂, and HCl as the target variables, respectively, where $m_1^{\mathrm{sel}}\in \{1,2,\dots ,M_1^{\mathrm{sel}}\}$, $m_2^{\mathrm{sel}}\in \{1,2,\dots ,M_2^{\mathrm{sel}}\}$, and $m_3^{\mathrm{sel}}\in \{1,2,\dots ,M_3^{\mathrm{sel}}\}$.

When comparing the $m_1^{\mathrm{sel}}$th feature variable name $s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}$ in set ${\mathit{S}}_1^{\mathrm{sel}}$ with the $m_2^{\mathrm{sel}}$th feature variable name $s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}$ in set ${\mathit{S}}_2^{\mathrm{sel}}$, the following criteria are applied:

$${\zeta }_1=\left\{\begin{array}{c}1,s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}=s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\\ 0,s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}\ne s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\end{array}\right.$$

(12)

If ${\zeta }_1=1$, the $m_3^{\mathrm{sel}}$th feature variable name $s_{3,m_3^{\mathrm{sel}}}^{\mathrm{sel}}$ in set ${\mathit{S}}_3^{\mathrm{sel}}$ is compared. If they are still identical, $s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}$ is added to the set of shared feature variable names:

$${\mathit{S}}_{\mathrm{in}}^{\mathrm{sel}}={\mathrm{F}}_{\mathrm{append}}(s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}})$$

(13)

If ${\zeta }_1=0$, the variable name $s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}$ of the $m_1^{\mathrm{sel}}$th feature in ${\mathit{S}}_1^{\mathrm{sel}}$ is compared with the variable name $s_{2,m_2^{\mathrm{sel}}+1}^{\mathrm{sel}}$ of the $m_2^{\mathrm{sel}}+1$th feature in ${\mathit{S}}_2^{\mathrm{sel}}$, and the following criterion applies:

$${\zeta }_2=\left\{\begin{array}{c}1,s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}=s_{2,m_2^{\mathrm{sel}}+1}^{\mathrm{sel}}\\ 0,s_{1,m_1^{\mathrm{sel}}}^{\mathrm{sel}}\ne s_{2,m_2^{\mathrm{sel}}+1}^{\mathrm{sel}}\end{array}\right.$$

(14)

If the result of the comparison with $s_{2,m_2^{\mathrm{sel}}+M_2^{\mathrm{sel}}}^{\mathrm{sel}}$ is still ${\zeta }_2=0$, the $m_1^{\mathrm{sel}}+1$th characteristic variable name $s_{1,m_1^{\mathrm{sel}}+1}^{\mathrm{sel}}$ in ${\mathit{S}}_1^{\mathrm{sel}}$ is revisited for further comparison. The criteria are as follows:

$${\zeta }_3=\left\{\begin{array}{c}1,s_{1,m_1^{\mathrm{sel}}+1}^{\mathrm{sel}}=s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\\ 0,s_{1,m_1^{\mathrm{sel}}+1}^{\mathrm{sel}}\ne s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\end{array}\right.$$

(15)

If ${\zeta }_3=1$, the $m_3^{\mathrm{sel}}$th characteristic variable name $s_{3,m_3^{\mathrm{sel}}}^{\mathrm{sel}}$ in ${\mathit{S}}_3^{\mathrm{sel}}$ is further compared, and the criteria are as follows:

$${\zeta }_4=\left\{\begin{array}{c}1,s_{1,m_1^{\mathrm{sel}}+1}^{\mathrm{sel}}=s_{3,m_3^{\mathrm{sel}}}^{\mathrm{sel}}\\ 0,s_{1,m_1^{\mathrm{sel}}+1}^{\mathrm{sel}}\ne s_{3,m_3^{\mathrm{sel}}}^{\mathrm{sel}}\end{array}\right.$$

(16)

If the result of the comparison with $s_{3,m_3^{\mathrm{sel}}+M_3^{\mathrm{sel}}}^{\mathrm{sel}}$ is still ${\zeta }_4=0$, the comparison continues with the $m_1^{\mathrm{sel}}+2$th characteristic variable name in ${\mathit{S}}_1^{\mathrm{sel}}$. The comparison criteria are as follows:

$${\zeta }_5=\left\{\begin{array}{c}1,s_{1,m_1^{\mathrm{sel}}+2}^{\mathrm{sel}}=s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\\ 0,s_{1,m_1^{\mathrm{sel}}+2}^{\mathrm{sel}}\ne s_{2,m_2^{\mathrm{sel}}}^{\mathrm{sel}}\end{array}\right.$$

(17)

The above process is repeated until the same feature variable names are found in ${\mathit{S}}_1^{\mathrm{sel}}$, ${\mathit{S}}_2^{\mathrm{sel}}$, and ${\mathit{S}}_3^{\mathrm{sel}}$. Once this occurs, ${\mathit{S}}_{\mathrm{in}}^{\mathrm{sel}}$ is obtained. Furthermore, the shared feature ${\mathit{X}}_{\mathrm{in}}^{\mathrm{sel}}$ corresponding to ${\mathit{S}}_{\mathrm{in}}^{\mathrm{sel}}$ is obtained, which can be expressed as

$${\mathit{X}}_{\mathrm{in}}^{\mathrm{sel}}=\left\{\mathit{X}\right|{\mathit{S}}_1^{\mathrm{sel}}\cap {\mathit{S}}_2^{\mathrm{sel}}\cap {\mathit{S}}_3^{\mathrm{sel}}\}$$

(18)

where ${\mathrm{F}}_{\mathrm{append}}()$ represents the operation of adding characteristic variable names, where ${\mathit{X}}_{\mathrm{in}}^{\mathrm{sel}}\in {ℝ}^{N\times {\mathit{M}}_{\mathrm{in}}^{\mathrm{sel}}}$.

2.2.2. Acidic Gas Multi-Output Modeling Module

Multi-Output ITransformer Modeling Sub-Module

The multi-output ITransformer modeling sub-module structure [38] primarily consists of an embedding layer, multiple encoder layers, and other components. The input layer encodes sequential data, enabling the model to understand both the underlying meaning and features of the input. The encoder layers extract features from the input sequence through multi-layer stacking. Each encoder consists of a multi-head self-attention mechanism and a feed-forward network, which work together to capture long-term dependencies and variations within the sequence.

Let ${\mathit{X}}_{\mathrm{in},\mathrm{r}}^{\mathrm{sel}}\in {ℝ}^{N\times M}$ represent the sequence data in the $r$th batch, where $N$ denotes the sample dimension and $M$ represents the feature dimension.

(1)

Embedding Layer: The embedding layer maps scalar feature values into a high-dimensional semantic space, allowing the model to capture richer feature information. The computation formulas are as follows:

$${\mathit{X}}_{\mathrm{in},r}^{\mathrm{sel}}=\{\mathit{R},\mathit{M}\}$$

(19)

$${\mathit{X}}_{\mathrm{em},r}=\mathrm{Embedding}({\mathit{X}}_{\mathrm{in},r}^{\mathrm{sel}})$$

(20)

$${\mathit{X}}_{\mathrm{em},r}=\{\mathit{R},\mathit{M},\mathit{D}\}$$

(21)

where $R$ denotes the total number of batches; $D$ denotes the feature value dimension; and $\mathrm{Embedding}(\cdot )$ denotes the embedding operation.

Since the ITransformer model is insensitive to the order of the input data, positional encoding must be incorporated into the embedded representation to enable the model to capture the positional information of features within the sequence:

$$\mathit{P}=\{{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_M\}$$

(22)

$${\mathit{P}}_{i,2j}=\sin (\frac{i}{{10000}^{2j/T}})$$

(23)

$${\mathit{P}}_{i,2j+1}=\cos (\frac{i}{{10000}^{2j/T}})$$

(24)

where $i$ denotes the position of the feature in the sequence; $j$ denotes the dimension index; ${\mathit{P}}_{i,2j}$ denotes the positional encoding for even indices; and ${\mathit{P}}_{i,2j+1}$ denotes odd indices.

Positional encoding is added to the embedded representation to obtain an enriched representation that includes positional information:

$${\mathit{X}}_{\mathrm{pe},r}={\mathit{X}}_{\mathrm{em},r}+\mathit{P}$$

(25)

$${\mathit{X}}_{\mathrm{pe},r}=\{\mathit{R},\mathit{M},\mathit{D}\}$$

(26)

After the sequence data passes through the embedding layer and positional encoding, it proceeds through $L$ identical encoder layers.

$${\mathit{X}}_{\mathrm{pe},r}=\{{\mathit{X}}_{\mathrm{pe},r}^1,\dots ,{\mathit{X}}_{\mathrm{pe},r}^L\}$$

(27)

Each encoder layer consists of two sub-layers: self-attention and a feed-forward network.

(2)

Multi-Head Self-Attention Mechanism: The primary role of the multi-head self-attention mechanism is to enable the model to identify linear correlations and capture complex nonlinear dependencies. During the computation of each weighted output using the attention mechanism, the input data is first linearly transformed to generate the query $\mathit{Q}$, key $\mathit{K}$, and value $\mathit{V}$ matrices. Taking the $l$th encoder layer as an example, the process is as follows:

$$\mathit{Q}={\mathit{X}}_{\mathrm{pe},r}^l\cdot {\mathit{W}}^Q$$

(28)

$$\mathit{K}={\mathit{X}}_{\mathrm{pe},r}^l\cdot {\mathit{W}}^K$$

(29)

$$\mathit{V}={\mathit{X}}_{\mathrm{pe},r}^l\cdot {\mathit{W}}^V$$

(30)

where ${\mathit{W}}^Q$, ${\mathit{W}}^K$, and ${\mathit{W}}^V$ are the weight matrices associated with $\mathit{Q}$, $\mathit{K}$, and $\mathit{V}$, respectively.

The computation process of single-head self-attention is as follows:

$$\mathrm{Attention}(\mathit{Q},\mathit{K},\mathit{V})=\mathrm{softmax}(\frac{\mathit{Q}{\mathit{K}}^T}{\sqrt{d_K}})\mathit{V}$$

(31)

The output of the $i$th self-attention head in the $l$th encoder layer for the $r$th batch is denoted as

$${\mathit{X}}_{i,r}^{\mathrm{head},l}=\mathrm{Attention}(\mathit{Q}\cdot {\mathit{W}}_i^{\mathrm{Q}},\mathit{K}\cdot {\mathit{W}}_i^{\mathrm{K}},\mathit{V}\cdot {\mathit{W}}_i^{\mathrm{V}})$$

(32)

where $\mathrm{softmax}(\cdot )$ denotes the data normalization operation and $d_{\mathrm{K}}$ denotes the dimension of the key vector.

The multi-head self-attention mechanism is computed as follows:

$$\mathrm{MultiHead}(\mathit{Q},\mathit{K},\mathit{V})=\mathrm{Concat}({\mathit{X}}_{1,r}^{\mathrm{head},l},\dots ,{\mathit{X}}_{i,r}^{\mathrm{head},l})\cdot {\mathit{W}}^{\mathrm{O}}$$

(33)

The output of the multi-head self-attention mechanism in the $l$th encoder layer for the $r$th batch is denoted as

$${\mathit{X}}_{\mathrm{multihead},r}^l=\mathrm{MultiHead}(\mathit{Q},\mathit{K},\mathit{V})$$

(34)

where ${\mathit{W}}^{\mathrm{O}}$ is the output projection matrix used to integrate the outputs of all heads and $\mathrm{Concat}(\cdot )$ denotes the matrix concatenation operation.

(3)

First Residual Connection: Residual connections ensure that feature information is preserved and does not degrade as it passes through multiple encoder layers, enabling the model to effectively learn deep feature interactions. The computation formula is as follows:

$${\mathit{X}}_{1\mathrm{sc},r}^l={\mathit{X}}_{\mathrm{pe},r}^l+\mathrm{Dropout}({\mathit{X}}_{\mathrm{multihead},r}^l)$$

(35)

where $\mathrm{Dropout}(\cdot )$ denotes the regularization operation.

(4)

First Layer Normalization: Layer normalization ensures smooth convergence during training and improves stability. The computation formulas are

$${\mathsf{\mu }}_1=\frac{1}{N}\sum _{n=1}^N{\mathit{X}}_{1\mathrm{sc},r}^{n,l}$$

(36)

$${\mathsf{\sigma }}_1^2=\frac{1}{N}\sum _{n=1}^N{({\mathit{X}}_{1,r}^{n,l}-{\mathsf{\mu }}_1)}^2$$

(37)

$$\mathrm{LayerNorm}1({\mathit{X}}_{1\mathrm{sc},r}^l)={\mathsf{\gamma }}_1\cdot \frac{{\mathit{X}}_{1\mathrm{sc},r}^l-{\mathsf{\mu }}_1}{\sqrt{{\mathsf{\sigma }}_1^2+{\mathsf{\epsilon }}_1}}+{\mathsf{\beta }}_1$$

(38)

The output of the first layer normalization layer in the $l$th encoder layer for the $r$th batch is denoted as

$${\mathit{X}}_{\mathrm{layernorm}1,r}^l=\mathrm{LayerNorm}1({\mathit{X}}_{1\mathrm{sc},r}^l)$$

(39)

where ${\mathsf{\mu }}_1$ denotes the mean of ${\mathit{X}}_{1\mathrm{sc},r}^{n,l}$; ${\mathsf{\sigma }}_1^2$ denotes the variance of the first layer; ${\mathsf{\gamma }}_1$ denotes the scaling parameter of the first layer; ${\mathsf{\beta }}_1$ denotes the shift parameter of the first layer; and ${\mathsf{\epsilon }}_1$ is used in the first normalization layer to prevent division by zero.

(5)

Feed-Forward Network: The primary role of the feed-forward network is to apply nonlinear transformations and enhance features from the output of the self-attention mechanism. The computation formula is as follows:

$$\mathrm{FFN}({\mathit{X}}_{\mathrm{layernorm}1,r}^l)=\mathrm{ReLu}({\mathit{X}}_{\mathrm{layernorm}1,r}^l\cdot {\mathit{W}}_1+{\mathit{b}}_1){\mathit{W}}_2+{\mathit{b}}_2$$

(40)

The output of the feed-forward network in the $l$th embedding layer for the $r$th batch is denoted as

$${\mathit{X}}_{\mathrm{ffn},r}^l=\mathrm{FFN}({\mathit{X}}_{\mathrm{layernorm}1,r}^l)$$

(41)

where $\mathrm{ReLu}(\cdot )$ denotes the nonlinear activation function; ${\mathit{W}}_1$ and ${\mathit{W}}_2$ denote the weight matrices of the first and second linear layers in the feed-forward network, respectively; and ${\mathit{b}}_1$ and ${\mathit{b}}_2$ denote the bias terms.

(6)

Second Residual Connection: The second residual connection ensures stable information flow, facilitates gradient propagation, and works synergistically with layer normalization to support effective training of deep networks. Together with the first residual connection, it provides a dual safeguard for stable training and optimal performance in the ITransformer architecture. The computation formula is as follows:

$${\mathit{X}}_{2\mathrm{sc},r}^l={\mathit{X}}_{\mathrm{layernorm}1,r}^l+\mathrm{Dropout}({\mathit{X}}_{\mathrm{ffn},r}^l)$$

(42)

(7)

Second Layer Normalization: The computation formulas are

$${\mathsf{\mu }}_2=\frac{1}{N}\sum _{n=1}^N{\mathit{X}}_{2\mathrm{sc},r}^{n,l}$$

(43)

$${\mathsf{\sigma }}_2^2=\frac{1}{N}\sum _{n=1}^N{({\mathit{X}}_{2\mathrm{sc},r}^{n,l}-{\mathsf{\mu }}_2)}^2$$

(44)

$$\mathrm{LayerNorm}2({\mathit{X}}_{2\mathrm{sc},r}^l)={\mathsf{\gamma }}_2\cdot \frac{{\mathit{X}}_{2\mathrm{sc},r}^l-{\mathsf{\mu }}_2}{\sqrt{{\mathsf{\sigma }}_2^2+{\mathsf{\epsilon }}_2}}+{\mathsf{\beta }}_2$$

(45)

The output of the second layer normalization layer in the $l$th encoder layer for the $r$th batch is denoted as

$${\mathit{X}}_{\mathrm{layernorm}2,r}^l=\mathrm{LayerNorm}2({\mathit{X}}_{2\mathrm{sc},r}^l)$$

(46)

where ${\mathsf{\mu }}_2$ denotes the mean of ${\mathit{X}}_{2\mathrm{sc},r}^{n,l}$; ${\mathsf{\sigma }}_2^2$ denotes the variance of the second normalization layer; ${\mathsf{\gamma }}_2$ denotes the scaling parameter of the second normalization layer; ${\mathsf{\beta }}_2$ denotes the shift parameter of the second normalization layer; and ${\mathsf{\epsilon }}_2$ is used in the second normalization layer to prevent division-by-zero errors.

The outputs of $L$ identical encoder layers are concatenated; the computation formula is as follows:

$${\mathit{X}}_{\mathrm{multilayer},r}=\mathrm{Concat}({\mathit{X}}_{\mathrm{layernorm}2,r}^1,\dots ,{\mathit{X}}_{\mathrm{layernorm}2,r}^L)$$

(47)

The concatenated 3-D tensor is flattened into a 2-D matrix to facilitate input into the projection layer:

$${\mathit{X}}_{\mathrm{flat},r}=\mathrm{Flatten}({\mathit{X}}_{\mathrm{multilayer},r})$$

(48)

where $\mathrm{Flatten}\left(\right)$ denotes the flattening operation.

(8)

Projection Layer: A fully connected layer maps the flattened 2-D feature matrix to the final prediction target space, computing $Q$ output values ${\widehat{\mathit{Y}}}_{r,Q}$ for the $r$th batch as follows:

$${\widehat{\mathit{Y}}}_{r,Q}={\mathit{X}}_{\mathrm{flat},r}\cdot {\mathit{W}}_{\mathrm{y},Q}+{\mathit{b}}_{\mathrm{y},Q}$$

(49)

where ${\mathit{W}}_{\mathrm{y},Q}$ is the weight matrix for the multiple outputs and ${\mathit{b}}_{\mathrm{y},Q}$ is the bias matrix for the multiple outputs.

The above process is repeated for the remaining samples, and the final output is ${\widehat{\mathit{Y}}}_Q$, as shown below:

$${\widehat{\mathit{Y}}}_Q=\left\{{\widehat{\mathit{Y}}}_{r,Q},{\widehat{\mathit{Y}}}_{r+1,Q},\dots ,{\widehat{\mathit{Y}}}_{r+R,Q}\right\}$$

(50)

Based on Section 2.2.1, the model prediction process based on shared features $f_{\mathrm{ITransformer}}\left(\cdot \right)$ can be expressed as

$${\widehat{\mathit{Y}}}_Q=f_{\mathrm{ITransformer}}\left({\mathit{X}}_{\mathrm{in},r}^{\mathrm{sel}}\right)$$

(51)

Multi-Output Evaluation Metric Acquisition Submodule

This submodule quantitatively evaluates the predictions of the multi-output ITransformer model and generates objective performance metrics. Due to the significant differences in the concentration scales of the three acidic gases, traditional metrics inadequately reflect the model performance. Therefore, relative root-mean-square error (RRMSE) was used to assess the model, where $RRMS{E}_q$ denotes the relative root-mean-square error of the $q$th acidic gas and $y_{n,q}$ and ${\widehat{y}}_{n,q}$ denote the true and predicted values of the $q$th acidic gas, respectively.

$$RRMS{E}_q=\left(\sqrt{\frac{1}{N}{{\displaystyle \sum _{n=1}^N\left(\frac{y_{n,q}-{\widehat{y}}_{n,q}}{y_{n,q}}\right)}}^2}\right)$$

(52)

$$RRMSE=\frac{1}{Q}\sum _{q=1}^Q{RRMSE}_q$$

(53)

2.2.3. Acidic Gas Modeling Parameter Selection Module

During acidic gas shared-feature selection, appropriate parameter choices are made to filter out redundant features, reducing the input dimensionality and conserving computational resources. Similarly, in multi-output acidic gas modeling, selecting the right parameters is essential for improving model performance and generalization. Our parameter-selection strategy involves first choosing reasonable values as the initial settings and then experimentally adjusting them to find the combination that yields the best prediction. This process can be framed as solving the following optimization problem:

$$({\theta }_{\mathrm{epochs}}^{*},{\theta }_{\mathrm{dropout}}^{*},{\theta }_{\mathrm{learnrate}}^{*},{\theta }_{\mathrm{batchsize}}^{*},{\theta }_{\mathrm{model}}^{*},{\theta }_{\mathrm{head}}^{*},{\theta }_{\mathrm{itransformer}}^{*},{\theta }_1^{\mathrm{FI},*},{\theta }_2^{\mathrm{FI},*},{\theta }_3^{\mathrm{FI},*})=\arg \min [RRMSE]$$

(54)

$$st.\left\{\begin{array}{l}{\theta }_{\mathrm{epochs}}^{\min }<{\theta }_{\mathrm{epochs}}^{*}\le {\theta }_{\mathrm{epochs}}^{\max }\\ {\theta }_{\mathrm{dropout}}^{\min }<{\theta }_{\mathrm{dropout}}^{*}<{\theta }_{\mathrm{dropout}}^{\max }\\ {\theta }_{\mathrm{learnrate}}^{\min }\le {\theta }_{\mathrm{learnrate}}^{*}\le {\theta }_{\mathrm{learnrate}}^{\max }\\ {\theta }_{\mathrm{batchsize}}^{\min }\le {\theta }_{\mathrm{batchsize}}^{*}\le {\theta }_{\mathrm{batchsize}}^{\max }\\ {\theta }_{\mathrm{model}}^{\min }\le {\theta }_{\mathrm{model}}^{*}\le {\theta }_{\mathrm{model}}^{\max }\\ {\theta }_{\mathrm{itransformer}}^{\min }\le {\theta }_{\mathrm{itransformer}}^{*}\le {\theta }_{\mathrm{itransformer}}^{\max }\\ {\theta }_{\mathrm{head}}^{\min }\le {\theta }_{\mathrm{head}}^{*}\le {\theta }_{\mathrm{head}}^{\max }\\ {\theta }_1^{\min }\le {\theta }_1^{\mathrm{FI},*}\le {\theta }_1^{\max }\\ {\theta }_2^{\min }\le {\theta }_2^{\mathrm{FI},*}\le {\theta }_2^{\max }\\ {\theta }_3^{\min }\le {\theta }_3^{\mathrm{FI},*}\le {\theta }_3^{\max }\end{array}\right.$$

(55)

2.3. Pseudocode

The pseudocode of the algorithm used in this article is as follows (Algorithm 1):

Algorithm 1: RF-ITransformer
Step1:	Input: dataset $\left\{\mathit{X},\mathit{y}\right\}$, ITransformer model iterations ${\theta }_{\mathrm{epochs}}$, ITransformer model learning rate ${\theta }_{\mathrm{learningrate}}$, ITransformer model dropout rate ${\theta }_{\mathrm{dropout}}$;
Step2:	Calculate the feature importance threshold $F{I}_q^m$ based on the following Equations (1)–(4);
Step3:	According to Equation (5), the feature set ${\mathit{X}}_q^{\mathrm{sel}}$ satisfying the conditions is selected based on the set characteristic threshold ${\theta }_q^{\mathrm{FI}}$;
Step 4:	Calculate the shared features according to Equations (6)–(18), yielding ${\mathit{X}}_{\mathrm{in}}^{\mathrm{sel}}$;
Step 5:	For $r=1$ to $R$ Use the sequence data ${\mathit{X}}_{\mathrm{in},\mathrm{r}}^{\mathrm{sel}}$ from the $r$-th batch as the model input;
Step 6:	Based on Equations (19)–(27), the result after the embedding layer is calculated as ${\mathit{X}}_{\mathrm{pe},r}$;
Step 7:	Pass the ${\mathit{X}}_{\mathrm{pe},r}$ of the $r$-th batch through the self-attention mechanism of the $l$-th encoder layer, and compute the $i$-th head self-attention result ${\mathit{X}}_{i,r}^{\mathrm{head},l}$ according to Equations (28)–(32);
Step 8:	Based on Equations (33) and (34), the result of the multi-head self-attention mechanism is calculated as ${\mathit{X}}_{\mathrm{multihead},r}^l$;
Step 9:	According to Equation (35), the result of the first-layer residual connection is calculated as ${\mathit{X}}_{1\mathrm{sc},r}^l$;
Step 10:	Use Equations (36)–(39), compute the first-layer normalization result ${\mathit{X}}_{\mathrm{layernorm}1,r}^l$ in the $l$-th encoder layer for the $r$-th batch;
Step 11:	Use Equations (40) and (41), compute the feed-forward neural network output ${\mathit{X}}_{\mathrm{ffn},r}^l$ in the $l$-th embedding layer for the $r$-th batch;
Step 12:	Use Equation (42), compute the second-layer residual connection result ${\mathit{X}}_{2\mathrm{sc},r}^l$;
Step 13:	Use Equations (43)–(46), compute the second-layer normalization result ${\mathit{X}}_{\mathrm{layernorm}2,r}^l$ in the $l$-th encoder layer for the $r$-th batch;
Step 14:	Use Equations (47)–(49), compute the $Q$ predicted outputs ${\widehat{\mathit{Y}}}_{r,Q}$ for the $r$-th batch samples;
Step 15:	Use Equations (50) and (51), compute the final ITransformer model output as ${\widehat{\mathit{Y}}}_Q$;
Step 16:	Use Equations (52) and (53), quantify the multi-output ITransformer model with the computed $RRMSE$;
Step 17:	Adjust the appropriate hyperparameter combination according to Equations (54) and (55);
Step 18:	end;
Step 19:	The above steps yield the $f_{\mathrm{RF-ITransformer}}\left(\cdot \right)$ model;

3. Results

3.1. Performance Metrics

This study used $\mathrm{RRMSE}$, $\mathrm{RMAE}$, and ${\mathrm{R}}^2$ as the three evaluation metrics [39].

$$RMAE={\displaystyle \frac{1}{Q}}{\displaystyle \sum _{q=1}^Q\left(\frac{1}{N}{\displaystyle \sum _{n=1}^N\frac{\left|y_{n,q}-{\widehat{y}}_{n,q}\right|}{y_{n,q}}}\right)}$$

(56)

$$R^2=1-\frac{{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{q=1}^Q{\left(y_{n,q}-{\widehat{y}}_{n,q}\right)}^2}}{{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{q=1}^Q{\left(y_{n,q}-{\overline{y}}_{n,q}\right)}^2}}$$

(57)

where ${\overline{y}}_{n,q}$ denotes the mean of the true values.

3.2. Experimental Results

An industrial dataset for the municipal solid waste incineration (MSWI) processes was used in this study. The dataset was sourced from a municipal solid waste incineration plant in Beijing, with all the process variables collected at uniform time intervals, sampled every second. The hourly data was averaged, and the MSWI process dataset consists of two main components: 239 input variables and three target variables.

In this study, the dataset was divided into five equal parts. The first, third, and fifth parts were used as the training set, the second part was used as the validation set, and the fourth part was used as the test set.

In the following experiments, the concentration units of the three acidic gases are all mg/m³N.

3.2.1. Feature Selection

Using RF as the feature-selection method, the selected results were plotted and the feature importance was calculated separately for the target variables NO_x, SO₂, and HCl. When NO_x was the target, the RF model parameters were set as follows: min_samples_leaf = 5, n_estimators = 100, random_state = 0.42, and ${\theta }_1^{\mathrm{FI}}$ = 0.3. After feature selection, 82 features were selected (Appendix A 

Table A2. Feature variable information of the dataset with NO_x as the target variable after RF-based feature selection.

Num	Variable Name	Unit	Feature Importance
1.	Primary combustion chamber right flue gas temperature	1 °C	0.4192
2.	Primary combustion chamber left flue gas temperature 2	°C	0.4272
3.	Primary combustion chamber middle flue gas temperature 2	°C	0.4270
4.	Primary combustion chamber middle flue gas temperature 1	°C	0.3347
5.	Primary combustion chamber left flue gas temperature 3	°C	0.3490
6.	Primary combustion chamber middle flue gas temperature 3	°C	0.6407
7.	Top gas temperature at burnout grate 1	°C	0.3987
8.	Temperature of left primary combustion chamber	°C	0.3950
9.	Average temperature of primary combustion chamber	°C	0.4271
10.	Left inner temperature of drying grate	°C	0.4155
11.	Left outer temperature of drying grate	°C	0.5834
12.	Right outer temperature of drying grate	°C	0.3502
13.	Left inner temperature of combustion grate 1-1	°C	0.3950
14.	Right inner temperature of combustion grate 1-1	°C	0.3887
15.	Right outer temperature of combustion grate 1-1	°C	0.3177
16.	Left outer temperature of combustion grate 1-1	°C	0.4562
17.	Left inner temperature of combustion grate 1-2	°C	0.4714
18.	Left outer temperature of combustion grate 1-2	°C	0.4070
19.	Right outer temperature of combustion grate 1-2	°C	0.4378
20.	Left inner temperature of combustion grate 2-1	°C	0.4435
21.	Right outer temperature of combustion grate 2-1	°C	0.4663
22.	Right inner temperature of combustion grate 2-1	°C	0.3966
23.	Right outer temperature of combustion grate 2-2	°C	0.4980
24.	Left outer temperature of combustion grate 2-2	°C	0.5747
25.	Right inner temperature of combustion grate 2-2	°C	0.3073
26.	Left flue gas temperature of first pass	°C	0.5623
27.	Right flue gas temperature of first pass	°C	0.4112
28.	Left inlet flue gas temperature of protection tube	°C	0.3597
29.	Left inlet flue gas temperature of tertiary superheater	°C	0.3676
30.	Left inlet flue gas temperature of evaporator	°C	0.3255
31.	Right inlet flue gas temperature of evaporator	°C	0.3091
32.	Left inlet flue gas temperature of economizer	°C	0.4226
33.	Combustion grate inlet air temperature (burn 1-1, 2 + 2-1)	°C	0.3547
34.	Drying grate inlet air temperature	°C	0.3151
35.	Inlet biogas pressure	KPa	0.3146
36.	Drying section outlet air temperature	°C	0.5152
37.	Combustion zone outlet air temperature	°C	0.3785
38.	Primary superheater outlet steam temperature	°C	0.3731
39.	Secondary superheater outlet steam temperature	°C	0.4495
40.	Tertiary superheater outlet steam temperature	°C	0.5226
41.	Left drying grate 1 (air flow setpoint)	km3N/h	0.3615
42.	Left drying grate 2 (air flow setpoint)	km3N/h	0.3385
43.	Left combustion grate 1-2 (air flow setpoint)	km3N/h	0.3132
44.	Left combustion grate 2-1 (air flow setpoint)	km3N/h	0.3430
45.	Left combustion grate 2-2 (air flow setpoint)	km3N/h	0.3938
46.	Left drying grate 1 air flow	km3N/h	0.4580
47.	Right drying grate 1 air flow	km3N/h	0.4018
48.	Combustion zone grate right section 1-1 air flow	km3N/h	0.3646
49.	Combustion zone grate left section 1-1 air flow	km3N/h	0.3418
50.	Burnout zone right grate air flow	km3N/h	0.3413
51.	Boiler #2 secondary air total accumulation	km3N	0.4988
52.	Economizer total feedwater flow	t/h	0.6906
53.	Economizer No. 2 feedwater flow	t/h	0.4425
54.	Economizer No. 1 feedwater flow	t/h	0.4274
55.	Primary superheater cooling water flow	t/h	0.3788
56.	Primary superheater cooling water flow accumulation	t	0.3625
57.	Secondary superheater cooling water flow	t/h	0.4728
58.	Total primary and secondary superheater attemperation water	t/h	0.3935
59.	Boiler main steam total accumulation	t	0.3141
60.	Mixer feedwater flow A accumulation	kg	0.3978
61.	Mixer feedwater flow B accumulation	kg	0.4661
62.	FGD reactor inlet flue gas flow B	km3N/h	0.3427
63.	Urea solution supply flow	L/h	0.8573
64.	Urea solution supply flow accumulation	L	0.3281
65.	Total feed into lime feeder	kg	0.3904
66.	Total feed into activated carbon silo	kg	0.4174
67.	Amount of urea dilution solution injected into the furnace	L/h	0.4562
68.	Top temperature of grate in burnout zone 2	°C	0.4004
69.	Cumulative feed amount from activated carbon storage silo	kg	0.4174
70.	Furnace negative pressure	Pa	0.3109
71.	Boiler drum pressure	MPa	0.3017
72.	Steam pressure at outlet of tertiary superheater	MPa	0.4479
73.	Reactor outlet differential pressure A	Pa	0.3714
74.	Bag filter differential pressure B	Pa	0.3566
75.	Secondary air fan current	A	0.3326
76.	Grate hydraulic pump 2 current	A	0.3790
77.	Grate hydraulic pump 3 current	A	0.4338
78.	Primary air fan frequency	Hz	0.4058
79.	Flue gas oxygen concentration	%	0.4859
80.	Flue gas oxygen concentration	%	0.3906
81.	Flue gas oxygen concentration	%	0.7559
82.	Flue gas dust concentration	mg/m3N	0.4383

). When SO₂ was the target, the RF model parameters were as follows: min_samples_leaf = 5, n_estimators = 100, random_state = 0.42, and ${\theta }_2^{\mathrm{FI}}$ = 0.2. After feature selection, 57 features were selected (Appendix A 

Table A3. Feature variable information for the dataset with SO₂ as the target variable after RF-based feature selection.

Num	Variable Name	Unit	Feature Importance
1.	Primary combustion chamber left flue gas temperature 1	°C	0.2211
2.	Primary combustion chamber right flue gas temperature 1	°C	0.2559
3.	Primary combustion chamber middle flue gas temperature 2	°C	0.2398
4.	Primary combustion chamber left flue gas temperature 3	°C	0.3414
5.	Burnout grate top air temperature	°C	0.4033
6.	Burnout grate top air temperature	°C	0.2637
7.	Primary combustion chamber average temperature	°C	0.3556
8.	Drying grate left inner temperature	°C	0.3486
9.	Drying grate left outer temperature	°C	0.3200
10.	Drying grate right inner temperature	°C	0.2381
11.	Combustion grate 1-1 left outer temperature	°C	0.2521
12.	Combustion grate 1-1 right outer temperature	°C	0.2045
13.	Combustion grate 1-2 left inner temperature	°C	0.2173
14.	Combustion grate 1-2 left outer temperature	°C	0.2849
15.	Combustion grate 1-2 right outer temperature	°C	0.3437
16.	Combustion grate 2-1 left inner temperature	°C	0.2642
17.	Combustion grate 2-1 left outer temperature	°C	0.2178
18.	Combustion grate 2-2 left outer temperature	°C	0.3519
19.	Right slag outlet temperature	°C	0.2861
20.	Economizer outlet flue gas temperature	°C	0.2955
21.	Drying grate inlet air temperature	°C	0.5283
22.	Grate right cooling air outlet temperature	°C	0.2096
23.	Outlet biogas pressure	KPa	0.2874
24.	Drying section outlet air temperature	°C	0.2990
25.	Drying section biogas flow	m3/h	0.3797
26.	Inlet flue gas temperature 1	°C	0.2072
27.	Unit 2 primary air total flow	km3N	0.2791
28.	Left combustion grate 1-1 (air flow setpoint)	km3N/h	0.2122
29.	Right drying grate 1 (air flow setpoint)	km3N/h	0.2321
30.	Right drying grate 1 air flow	km3N/h	0.3079
31.	Left drying grate 2 air flow	km3N/h	0.2148
32.	Combustion grate left section 1-1 air flow	km3N/h	0.2690
33.	Unit 2 secondary air total flow	km3N	0.2661
34.	Secondary air flow	km3N/h	0.2507
35.	Secondary air fan branch 1 flow	km3N/h	0.3186
36.	Boiler main steam totalized flow	t	0.3169
37.	Boiler outlet main steam flow	t/h	0.3389
38.	Mixer feed water flow A accumulation	kg	0.3207
39.	Mixer feed water flow B accumulation	kg	0.3022
40.	Urea solvent supply flow accumulation	L	0.2990
41.	Furnace #2 urea solution accumulation	L	0.3106
42.	Lime feeder accumulation	kg	0.2586
43.	Activated carbon silo feed accumulation	kg	0.2899
44.	Primary air fan outlet air pressure	KPa	0.2786
45.	Secondary air fan outlet air pressure	KPa	0.3590
46.	Level 3 superheater outlet steam pressure	MPa	0.2844
47.	Reactor differential pressure B	Pa	0.2176
48.	Drying grate left outer speed	%	0.2214
49.	Secondary air fan inlet damper opening (actual value)	%	0.2242
50.	Secondary air fan current	A	0.2573
51.	Grate hydraulic pump 2 current	A	0.3516
52.	Secondary air fan frequency	HZ	0.2416
53.	Flue gas oxygen concentration	%	0.2579
54.	Flue gas oxygen concentration	%	0.2376
55.	Flue gas oxygen concentration	%	0.3130
56.	Flue gas dust concentration	mg/m3N	0.2639
57.	Reactor differential pressure B	Pa	0.2176

). When HCl was the target, the RF model parameters were as follows: min_samples_leaf = 5, n_estimators = 100, random_state = 0.42, and ${\theta }_3^{\mathrm{FI}}$ = 0.3. After feature selection, the number of selected features $M_3^{\mathrm{sel}}$ was 37 (Appendix A 

Table A4. Feature variable information for the dataset with HCl as the target variable after RF-based feature selection.

Num	Variable Name	Unit	Feature Importance
1.	Flue gas temperature 2, left side of primary combustion chamber	°C	0.3826
2.	Flue gas temperature 3, left side of primary combustion chamber	°C	0.4032
3.	Flue gas temperature 3, right side of primary combustion chamber	°C	0.3381
4.	Top air temperature at the grate end of the burnout zone	°C	0.3102
5.	Left side temperature of primary combustion chamber	°C	0.3023
6.	Average temperature of primary combustion chamber	°C	0.3457
7.	Drying grate left outer temperature	°C	0.3780
8.	Combustion grate 1-2 left inner temperature	°C	0.3627
9.	Combustion grate 1-2 right outer temperature	°C	0.3556
10.	Combustion grate 2-1 left inner temperature	°C	0.3858
11.	Combustion grate 2-2 left inner temperature	°C	0.5071
12.	Combustion grate 2-2 left outer temperature	°C	0.4332
13.	Combustion grate 2-2 right outer temperature	°C	0.3364
14.	Level 3 superheater left inlet flue gas temperature	°C	0.3254
15.	Economizer right inlet flue gas temperature	°C	0.4144
16.	Drying grate inlet air temperature	°C	0.3290
17.	Level 2 superheater outlet steam temperature	°C	0.4284
18.	Economizer outlet water temperature	°C	0.3366
19.	FGD bag filter B inlet temperature	°C	0.3209
20.	Left drying grate 2 (air setpoint)	km3N/h	0.3184
21.	Left drying grate 2 air flow	km3N/h	0.3173
22.	Combustion grate left section 1-1 air flow	km3N/h	0.3094
23.	Economizer No. 2 feedwater flow	t/h	0.3531
24.	Level 2 superheater cooling water flow	t/h	0.3123
25.	Total Level 1 and 2 superheater attemperation water	t/h	0.3082
26.	Boiler outlet main steam flow	t/h	0.4800
27.	Mixer feedwater flow A cumulative	kg	0.4024
28.	Urea solution supply flow	L/h	0.4342
29.	Secondary air fan outlet air pressure	KPa	0.3497
30.	Boiler drum pressure	MPa	0.4119
31.	Level 3 superheater outlet steam pressure	MPa	0.3492
32.	Left burnout grate air duct damper opening (actual)	%	0.3137
33.	Grate hydraulic pump 3 current	A	0.4138
34.	Primary air fan frequency	Hz	0.3590
35.	Flue gas oxygen concentration	%	0.3879
36.	Flue gas oxygen concentration	%	0.4601
37.	Flue gas temperature 2, left side of primary combustion chamber	%	0.5326

). The results are shown in Figure 3.

The shared features for the three target vectors $M_{\mathrm{in}}^{\mathrm{sel}}$ are shown in

Table 1. Shared feature statistical results.

Number	Feature Variable	Unit	Feature Importance Value
			NOx	SO2	HCl
1	Flue Gas Temperature 3 on Left Side of Primary Combustion Chamber	°C	0.3490	0.3414	0.4032
2	Gas Temperature at Top of Burnout Grate	°C	0.3987	0.4033	0.3102
3	Average Temperature of Primary Combustion Chamber	°C	0.4271	0.3556	0.3457
4	Temperature at Outer Left of Drying Grate	°C	0.5834	0.3200	0.3780
5	Temperature at Inner Left of Combustion Grate 1-2	°C	0.4714	0.2173	0.3627
6	Temperature at Outer Right of Combustion Grate 1-2	°C	0.4378	0.3437	0.3556
7	Temperature at Inner Left of Combustion Grate 2-1	°C	0.4435	0.2642	0.3858
8	Temperature at Outer Left of Combustion Grate 2-2	°C	0.5747	0.3519	0.4332
9	Inlet Air Temperature of Drying Grate	°C	0.3151	0.5283	0.3290
10	Left 1-1 Section Air Flow of Combustion Grate	Nm3/h	0.3418	0.2690	0.3094
11	Mixer Feedwater Flow A Accumulation	kg	0.3978	0.3207	0.4024
12	Flue Gas Oxygen Concentration	%	0.4859	0.2579	0.3879
13	Flue Gas Oxygen Concentration	%	0.3906	0.2376	0.4601

.

The feature importance values in Table 1 were plotted as a heat map, as shown in Figure 4.

The following are shown in Figure 3 and Figure 4 and Table 1: (1) The maximum feature importance for NO_x was 0.5834, the minimum was 0.3151, and the average was 0.4321. (2) For SO₂, the maximum feature importance was 0.5283, the minimum was 0.2173, and the average was 0.3239. (3) For HCl, the maximum feature importance was 0.4601, the minimum was 0.3094, and the average was 0.3741. (4) The proportion of shared features for NO_x, SO₂, and HCl was $\frac{13}{82}$, $\frac{13}{57}$, and $\frac{13}{37}$, indicating that some features were missing. The maximum importance values of these missing features were 0.7559, 0.4033, and 0.5326, respectively. This suggests that further feature improvement is required.

3.2.2. Prediction Model

The ITransformer network was configured with the following parameters: ${\theta }_{\mathrm{batchsize}}$ = 32, ${\theta }_{\mathrm{learnrate}}$ = 0.0004, ${\theta }_{\mathrm{epochs}}$ = 300, and ${\theta }_{\mathrm{dropout}}$ = 0.1, with the optimizer set to Adam. Additionally, ${\theta }_{\mathrm{model}}$ = 128, ${\theta }_{\mathrm{itransformer}}$ = 2, and ${\theta }_{\mathrm{head}}$ = 32 were applied. The corresponding loss curves are shown in Figure 5.

Figure 5 shows that when the iteration count reached 300, the loss function converged, and after that, the loss value remained almost unchanged with further iterations. The training and test loss curves closely overlapped, indicating that the model performed well on the test set and exhibited a strong generalization ability.

The comparison of the predicted and actual values for the test set is presented in Figure 6.

As shown in Figure 6 and

Table 2. Comparison of experimental results.

	NOx			SO2			HCl
	RMAE	RRMSE	R2	RMAE	RRMSE	R2	RMAE	RRMSE	R2
Training set	0.0244	0.0369	0.9853	0.1966	0.1504	0.9754	0.0351	0.0689	0.9902
Validation set	0.0266	0.0441	0.9788	0.1996	0.1447	0.9773	0.0364	0.0902	0.9824
Test set	0.0276	0.0495	0.9749	0.2239	0.1667	0.9699	0.0407	0.1035	0.9760

, the RF-ITransformer exhibited small error fluctuations across the three datasets, indicating strong model robustness and insensitivity to shifts in data distribution. This makes it well-suited for real-world applications.

The RF-Transformer model demonstrated strong predictive capabilities for emission concentrations. However, a key observation was that its accuracy tended to decrease during periods of peak emissions. This performance gap may be due to the highly nonlinear and fluctuating nature of industrial process data, where the model may not fully capture all critical dynamics under extreme operational conditions. Recognizing this limitation is essential, as improving peak prediction accuracy can significantly enhance both environmental performance and operational efficiency. This insight paves the way for future research focused on developing more reliable tools for sustainable industrial process management and reducing environmental impacts.

3.3. Method Comparison

Based on the feature selection results for the target variables in Section 3.2.1, the following models were established to predict the three target variables, NO_x, SO₂, and HCl: PCC-LSTM, MI-LSTM, RF-LSTM, PCC-ITransformer, MI-ITransformer, RF-ITransformer, PCC-BP, MI-BP, and RF-BP. The experimental results of these models were compared based on the RRMSE, RMAE, and R² evaluation metrics. The models were run 30 times, and the results for each method are shown in

Table 3. Comparison of experimental statistical results.

	NOx			SO2			HCl
	RMAE	RRMSE	R2	RMAE	RRMSE	R2	RMAE	RRMSE	R2
PCC-LSTM	0.0313 ± 7.89 × 10−6	0.0592 ± 4.658 × 10−5	0.9636 ± 7.848 × 10−5	0.2517 ± 4.705 × 10−4	0.3114 ± 2.454 × 10−3	0.8923 ± 1.745 × 10−3	0.0505 ± 3.711 × 10−5	0.1479 ± 3.043 × 10−4	0.9502 ± 1.556 × 10−4
MI-LSTM	0.0408 ± 4.06 × 10−6	0.0789 ± 1.657 × 10−5	0.9359 ± 4.744 × 10−5	0.2058 ± 3.282 × 10−4	0.2952 ± 8.639 × 10−4	0.9047 ± 3.674 × 10−4	0.0497 ± 3.477 × 10−5	0.1695 ± 2.284 × 10−4	0.9350 ± 1.338 × 10−4
RF-LSTM	0.0270 ± 9.23 × 10−6	0.0514 ± 5.22 × 10−6	0.9728 ± 5.87 × 10−6	0.1966 ± 2.641 × 10−4	0.1788 ± 4.493 × 10−4	0.9649 ± 7.287 × 10−5	0.0403 ± 1.579 × 10−5	0.0979 ± 1.469 × 10−4	0.9782 ± 3.576 × 10−5
PCC- ITransformer	0.0263 ± 2.9 × 10−6	0.0501 ± 1.418 × 10−5	0.9741 ± 1.635 × 10−5	0.3158 ± 6.802 × 10−4	0.3366 ± 8.078 × 10−4	0.8764 ± 4.875 × 10−4	0.0710 ± 5.75 × 10−5	0.1135 ± 2.918 × 10−4	0.9154 ± 2.352 × 10−4
MI- ITransformer	0.0419 ± 2.48 × 10−5	0.0830 ± 3.892 × 10−5	0.9289 ± 1.241 × 10−4	0.2295 ± 9.753 × 10−4	0.2835 ± 7.0387 × 10−4	0.9122 ± 2.93 × 10−4	0.0635 ± 4.158 × 10−5	0.1141 ± 3.562 × 10−4	0.9148 ± 2.798 × 10−4
RF- ITransformer	0.0247 ± 1.495 × 10−5	0.0449 ± 3.015 × 10−5	0.9791 ± 2.971 × 10−5	0.1369 ± 4.052 × 10−4	0.1357 ± 7.503 × 10−4	0.9793 ± 8.413 × 10−5	0.0291 ± 1.409 × 10−5	0.0839 ± 1.783 × 10−4	0.9838 ± 3.389 × 10−5
PCC-BP	0.0386 ± 4.218 × 10−5	0.0628 ± 9.115 × 10−5	0.9587 ± 1.767 × 10−4	0.2776 ± 5.264 × 10−4	0.3088 ± 8.879 × 10−4	0.8958 ± 4.405 × 10−4	0.0867 ± 8.051 × 10−5	0.1988 ± 4.320 × 10−4	0.9103 ± 3.646 × 10−4
MI-BP	0.0578 ± 6.362 × 10−5	0.0952 ± 1.1417 × 10−4	0.9059 ± 5.274 × 10−4	0.1835 ± 3.066 × 10−4	0.1872 ± 4.283 × 10−4	0.9616 ± 7.313 × 10−5	0.0786 ± 3.124 × 10−5	0.1939 ± 2.147 × 10−4	0.9151 ± 1.692 × 10−4
RF-BP	0.0362 ± 2.408 × 10−5	0.0585 ± 4.454 × 10−5	0.9645 ± 7.599 × 10−5	0.1896 ± 3.937 × 10−4	0.1854 ± 1.113 × 10−3	0.9616 ± 2.154 × 10−4	0.0556 ± 6.507 × 10−5	0.1134 ± 4.6712 × 10−4	0.9701 ± 1.481 × 10−4

.

As shown in Table 3, the RF-ITransformer model outperformed the other models in terms of the RRMSE, RMAE, and R² metrics, demonstrating strong generalization performance and stability, and yielding the best predictive results.

3.3.1. Experimental Comparison of Feature Selection

To evaluate the impact of various feature-selection methods on model prediction, the same dataset was used for both training and testing. After normalizing the data, mutual information (MI) and Pearson correlation coefficient (PCC) analyses were conducted on the features and target variables. The results showed that, using PCC, the three target variables shared 15 features; using MI, they shared 26 features; and using Random Forest (RF), they shared 13 features. The results are shown in Figure 7.

Figure 7 and Table 1 show that the three methods shared 15, 26, and 13 features, respectively. It was found that our method selected the fewest shared features, thereby reducing the computational cost.

3.3.2. Experimental Comparison of Prediction Models

To compare the performance of RF-ITransformer, the RF-LSTM and RF-BP models were used to predict each target variable, and comparison plots were generated. The BP parameters were set as follows: 300 epochs, learning rate = 0.005, hidden neurons = 2 × input features + 1, and dropout = 0.1. The LSTM parameters were set as follows: 300 epochs, learning rate = 0.005, 32 memory units, 2 hidden layers, and dropout = 0.1. The test-set curves are shown in Figure 8.

Figure 8 and Table 3 show that the features selected by RF adapted well to the models. Using the same feature selection method, the proposed model demonstrated a more stable performance and achieved the best predictive results among the three models. Despite using the same feature selection method, the three models exhibited different accuracies for the following reasons: First, there was a highly nonlinear coupling between the input and output features. The ITransformer model, by extracting multivariate features, fitted this relationship more accurately, while the BP and LSTM models, with fewer nonlinear layers, exhibited weaker fitting capabilities. Second, in terms of noise robustness, industrial data often contain noise. The self-attention mechanism of ITransformer helps mitigate the impact of noise, while LSTM’s update gate may fail to retain useful information precisely, and the BP model, which lacks a memory mechanism, is more sensitive to noise.

The advantages and disadvantages of the models used for predicting acidic gases are shown in

Table 4. Comparison of different models for predicting acidic gases.

Model	Advantages	Disadvantages
LSTM	1. Proficient in capturing temporal dependencies 2. Suitable for handling complex univariate sequence data	1. Poor interpretability 2. High complexity and many hyperparameters 3. Low computational efficiency and difficulty in parallelization 4. Not suitable for handling complex multivariate sequence data
ITransformer	1. Good interpretability 2. Fast training speed 3. Proficient in handling multivariate sequences and capturing correlations between variables 4. Suitable for handling complex multivariate sequence data	1. Large capacity, multiple parameters, and prone to overfitting on small datasets
BP	1. Simple structure and easy to implement 2. Fast training speed	1. Poor interpretability 2. Not suitable for handling complex multivariate sequence data

.

After thorough consideration, ITransformer was selected as the prediction model for this study.

3.4. Hyperparameter Analysis

The model’s hyperparameter settings are shown in

Table 5. Hyperparameter settings.

Related Parameter	Range	Step Size
Number of Iterations	[0, 2000]	100
Learning Rate	[0.0001, 0.0009]	0.0001
Batch Size	[32, 256]	32
Dropout Rate	[0, 1]	0.1
Model Dimension	8, 16, 32, 64, 128
Number of Encoder Layers	[1, 6]	1
Number of Self-Attention Heads	4, 8, 16, 32, 64, 128
RF Threshold (NOx)	[0.2, 0.6]	0.1
RF Threshold (SO2)	[0.2, 0.5]	0.1
RF Threshold (HCl)	[0.2, 0.5]	0.1

.

To investigate the impact of the ITransformer network parameters on prediction accuracy, the hyperparameters listed in Table 5 were selected for analysis using the R² evaluation metric. The results of the evaluation metrics are presented in Figure 9.

The following conclusions about the hyperparameters were obtained from Figure 9:

(1)

Number of iterations: Increasing the number of iterations lengthens the model’s runtime. However, the prediction accuracy did not improve significantly with additional training. Therefore, an optimal value should be chosen to minimize training time while maintaining high accuracy. A value of 300 iterations was found to be optimal.

(2)

Batch size: The batch size is a crucial hyperparameter that determines how many samples are used in each model update. Selecting an appropriate batch size is essential for training efficiency. A batch size of 64 yielded the best weight-update performance and highest prediction accuracy.

(3)

Learning rate: A suitable learning rate enables the objective function to converge to a local minimum within a reasonable time. Most networks initialize the learning rate at 0.01 or 0.001. In this study, learning rates from 0.001 to 0.01, in steps of 0.001, were tested. The highest accuracy was achieved at a learning rate of 0.004.

(4)

Model dimensions: The number of dimensions governs the model’s representational capacity, complexity, and computational efficiency. Increasing the number of dimensions only provided marginal performance gains while extending the runtime. Therefore, the dimension was set to 128.

(5)

Encoder layers: In ITransformer, the number of encoder layers directly impacts the model’s complexity and performance. By adjusting the layer count, the model can capture more complex patterns and fit intricate data distributions. Starting with a baseline of 1, the experiments added layers incrementally up to 5. The best predictive performance was achieved with 2 layers, so the model was accordingly set to 2.

(6)

Self-attention heads: This parameter determines the model’s ability to attend to different types of information simultaneously. In ITransformer, the head count directly influences the model’s capacity to capture complex relationships among multivariate features. The optimal performance was observed at 32 heads, so the value was set to 32.

(7)

Dropout rate: Dropout randomly deactivates neurons with a given probability, which enhances generalization. As dropout increased, there was a general decline in accuracy. Therefore, the dropout rate was set to 0.1.

(8)

RF threshold (NO_x): This key parameter for feature selection in Random Forests identifies strongly correlated variables, optimizing the prediction accuracy. The accuracy first increased and then decreased with rising thresholds, so the value was set to 0.3.

(9)

RF threshold (SO₂): The accuracy peaked when the threshold was set to 0.2.

(10)

RF threshold (HCl): The accuracy declined as the threshold increased, so the value was set to 0.3.

Therefore, different hyperparameters exhibit distinct characteristics. Finding an efficient way to search for the globally optimal parameters is a challenge that needs to be addressed.

4. Conclusions

This article presents an acidic gas prediction model for MSWI processes with three key contributions: (1) A shared-feature selection strategy based on RF was introduced to effectively assess the importance of features relative to target values. This approach reduces resource consumption and the lengthy training times typically associated with high-dimensional data in MSWI processes, thereby minimizing both financial and material costs in practical applications. (2) An ITransformer prediction model with multi-input and multi-output strategies was developed, validated, and demonstrated to have stable performance. The model utilizes a multi-head self-attention mechanism to capture feature correlations, enabling precise extraction of multivariate characteristics and enhancing model interpretability. By simultaneously predicting multiple acidic gas concentrations, the model captures the inherent relationships and interaction mechanisms between the pollutants, laying the groundwork for coordinated control strategies. (3) A multi-output evaluation metric was proposed to eliminate dimensionality discrepancies, quantify prediction errors, and provide a solid foundation for hyperparameter selection in acidic gas prediction models.

Despite its contributions, the study has several limitations, including the insufficient utilization of certain important features and the tedious nature of the hyperparameter tuning process. Future work will address these challenges by incorporating hyperparameter optimization algorithms. Additionally, underutilized features will be explored, and the proposed multi-input, multi-output prediction model will be extended into a more generalizable algorithm, which will be validated using public datasets. The evaluation framework will include multiple metrics, such as relative error, training duration, and model interpretability, to provide a comprehensive assessment of the model’s performance.