CO Emission Prediction Based on Kernel Feature Space Semi-Supervised Concept Drift Detection in Municipal Solid Waste Incineration Process

Abstract

Carbon monoxide (CO) is a toxic pollutant emitted by municipal solid waste incineration (MSWI), which has a strong correlation with dioxins. In terms of the sustainable development of an ecological environment, CO emission concentration is strictly controlled by the environmental departments of various countries in the world. The construction of its prediction model is conducive to pollution reduction control. The MSWI process is affected by multi-factors such as MSW component fluctuation, equipment wear and maintenance, and seasonal change, and has complex nonlinear and time-varying characteristics, which makes it difficult for the CO prediction model based on offline historical data to adapt to the above changes. In addition, the continuous emission monitoring system (CEMS) used for conventional pollutant detection has unavoidable misalignment and failure problems. In this article, a novel prediction model of CO emission from the MSWI process based on semi-supervised concept drift (CD) detection in kernel feature space is proposed. Firstly, the CO emission deep prediction model and the kernel feature space detection model are constructed based on offline batched historical data, and the historical data set for the real-time construction of the pseudo-labeling model is obtained. Secondly, the drift detection for the CO emission prediction model is carried out based on real-time data by using unsupervised kernel principal component analysis (KPCA) in terms of feature space. If CD occurs, the pseudo-label model is constructed, the pseudo-truth value is obtained, and the drift sample is confirmed and selected based on the Page–Hinkley (PH) test. If no CD occurs, the CO emission concentration is predicted based on the historical prediction model. Then, the updated data set of the CO emission prediction model and kernel feature space detection is obtained by combining historical samples and drift samples. Finally, the offline history model is updated with a new data set when the preset conditions are met. Based on the real data set of an MSWI power plant in Beijing, the validity of the proposed method is verified.

1. Introduction

Municipal solid waste (MSW) refers to all kinds of solid waste generated in people’s daily production, life and consumption [1]. Long-term accumulation and management lack can easily cause urban living space pollution, occupation, virus transmission, and other hazards [2,3]. The latest World Bank statistics show that in the absence of action on MSW production, global MSW annual production will rise by 70% to 3.4 billion tonnes in 2050 [4]. The rapid growth of MSW production has led to the gradual increase in environmental risks of “Garbage siege” [5], which seriously restricts its sustainable and healthy development [6,7]. The report of China proposed that “building an ecological civilization is a millennium project for the sustainable development of the Chinese nation” and made a strategic deployment to accelerate the reform of the ecological civilization system and build a beautiful China [8], and clearly pointed out that the reasonable disposal of MSW is an important aspect of China’s ecological civilization construction [9].

MSW incineration (MSWI) technology has become the preferred technology for MSW disposal [10,11], which has advantages such as a high-quality reduction rate, high-capacity reduction rate, and energy recycling [12], as well as its potential value in economic and environmental protection [13,14]. At the same time, under the development background of the “carbon peaking and carbon neutrality” strategy, waste classification and “zero landfill” of the original MSW, MSWI is an indispensable part of sustainable development and green environmental protection of large cities [15]. Carbon monoxide (CO), as one of the toxic gases produced in the process of MSWI, is difficult to sense by the senses. When combined with hemoglobin, it will hinder blood oxygen delivery and even cause myocardial infarction in the human body [16,17], so it must be strictly controlled [18].

Due to the unknown uncertainty fluctuation of MSW components, the high-frequency intervention and experience difference of experts in the field in operation, the maintenance and aging of incineration equipment, and other factors, the operating condition of the MSWI process has complex dynamic time-varying characteristics [19]. This is difficult to overcome, and predictable operating condition fluctuation is reflected in the change in the distribution of data flow over time, which is called concept drift (CD) [20,21] in the academic field. Therefore, building a high-precision CO prediction model requires not only determining a suitable modeling algorithm, but also solving how to detect CD and update the historical model with samples that can characterize the drift. Although conventional pollutants, including CO, NOx, SO₂, etc., can be continuously detected by the continuous emission monitoring system (CEMS) located at the tail of the chimney, the system has unavoidable misalignment and failure phenomena [22]. The existing CO prediction models for the MSWI process include the method based on reduced depth feature and LSTM optimization [23], the method based on fixed window CD detection [24] and the heterogeneous ensemble prediction model method driven by virtual and real data [25]. However, these methods do not consider the prediction model updating when the CEMS system failure cannot provide the model truth value and CD occurs. Therefore, when updating the CO emission prediction model, it is also necessary to pay attention to how to effectively update the model without the truth value. Therefore, the constructed CO prediction model can replace these models in such a special situation. Moreover, to effectively control the MSWI process, domain experts typically predict changes in CO concentration to adopt more effective ‘air and material distribution’ control. Meanwhile, the CO prediction model is also necessary for realizing the intelligent optimization control of the MSWI process [26]. Therefore, the method proposed in this article not only has the functionality of a CEMS system in terms of CO emission concentration but also provides support for intelligent control and operational optimization of MSWI processes.

CD detection can be divided into two types: supervised and unsupervised [27]. The representative CD detection algorithm is the drift detection method (DDM), which defines the warning and drift levels according to the new sample measurement performance [28]. Frias et al. [29] based their study on the exponential weighted moving average method, and Mahdi et al. [30] based their study on the Page–Hinkley (PH) test method to confirm changes in the model accuracy to determine whether CD has occurred, but it is difficult to directly use the supervised CD detection method in the actual industrial process [31]. The representative algorithm of unsupervised CD detection [32] analyzes the distribution changes in feature space based on principal component analysis (PCA) [33]. The disadvantage of the PCA method is that it does not rely on the truth value in the CD detection stage, but samples labeled with truth value are still needed in the model update stage. It is difficult to make the model adapt to CD in a short period of time [34]. In addition, the influence of CD in complex industrial processes can be reflected in the comprehensive change in terms of output error space and input feature space. In order to make use of the large number of unlabeled samples of the process, the semi-supervised learning method has been used to model the process data. For example, Ji et al. [35] proposed a semi-supervised recursively weighted kernel regression algorithm for soft-sensing modeling of a semi-batch process, and Ge et al. [36] used a semi-supervised Bayesian method for soft-sensing modeling of a large number of unlabeled samples of the process. Zhu et al. [37] proposed a robust semi-supervised mixed rate principal component regression model and applied it to process soft measurement. This method can not only make use of a large number of unlabeled samples but also effectively solve the process outlier problem. Therefore, for the CO emission prediction problems of the MSWI process, it is very necessary to study how to carry out online prediction, real-time drift detection and model updating.

To solve the above problems, this article proposes a CO emission prediction model in the MSWI process based on semi-supervised CD detection in kernel feature space. It can be used to predict CO concentration and achieve emission reduction control of the MSWI process. The main innovations are as follows: (1) a semi-supervised CD detection framework is proposed, including historical CO prediction model and CD detection model construction, real-time CO prediction and CD detection, updated data set acquisition and historical model update modules; (2) an algorithm that uses unsupervised kernel principal component analysis (KPCA) identification to recognize CD in the feature space is proposed. When drift occurs, a pseudo-labeling model is constructed in real time to obtain pseudo-truth values, and drift samples are confirmed based on a PH test. (3) The updated data set of the model is obtained by combining historical samples and drift samples, and the strategy of using a new data set to update the historical model when the preset conditions are met is proposed.

The rest of the organizational structure of this article is arranged as follows. Section 2 describes the material and method, including the MSWI process for CO emission concentration, the overall algorithm and the implementation of the algorithm. In Section 3, the experimental research is carried out based on industrial application. Section 4 summarizes the research.

2. Materials and Methods

2.1. Materials

The process flow of a typical grate furnace MSWI is shown in Figure 1. It mainly includes six process stages: solid waste fermentation, solid waste combustion, waste heat exchange, flue gas cleaning, flue gas emission, and steam power generation [38].

For solid waste combustion, due to insufficient oxygen supply (part of the MSW cannot be completely oxidized), uneven combustion (part of the MSW cannot be fully exposed to oxygen) or insufficient temperature (insufficient incineration temperature, combustion reaction rate is reduced), the MSW will undergo incomplete combustion, resulting in the generation of CO. Therefore, the change in furnace temperature, flue gas oxygen content and other factors cause the CD in terms of CO emission concentration.

The traditional MSWI process can detect the CO emission concentration in real time through the CEMS [1], which is usually measured by full extraction or dilution extraction. However, when full extraction is carried out, it is easy to clog the extraction port in a positive pressure environment or when the extraction volume is too large. During dilution extraction, the measurement response is too long and the purity of dry compressed air is required to be high. In addition, CEMS has failure problems and requires qualified technicians to carry out regular maintenance [39]. Therefore, it is necessary to label the pseudo-truth values to analyze the CD phenomenon in the process when the truth values cannot be obtained.

Some detailed information about an MSW plant in Beijing is as follows. The capacity of MSW is 628.8 t/d. The length of the grate is 11 m and the width is 12.9 m. The airflow of the primary is 67,500 m³/h. The temperature of the air is 200 °C. Primary air enters the bed from four independent parts of the grate below. The flow rate of each part accounts for 24.31%, 43.35%, 19.27% and 13.07% of primary air, respectively.

The data was collected through an edge verification platform of an MSWI power plant equipped with a safety isolation acquisition device. The device facilitates one-way data transmission, mitigating interference issues during the data acquisition process, as shown in Figure 2.

Monitoring and responding to concept drift can help ensure that models remain accurate and reliable in dynamic environments such as the MSWI process. Concept drift is often manifested as a change in data distribution. It is assumed that all historical samples obey $F_{1,t}^{\mathrm{Htd}}$, and new time samples obey $F_k^{\mathrm{Otd}}$; if $F_{1,t}^{\mathrm{Htd}}=F_k^{\mathrm{Otd}}$, no drift occurs, and vice versa, $F_{1,t}^{\mathrm{Htd}}\ne F_k^{\mathrm{Otd}}$, drift occurs. In this article, the CO emission prediction model of the MSWI process is constructed. In total, 36 h of data were taken, and the average value of 5 min was taken as one sample for the final data, totaling 432 samples. The first two-thirds of the data set is divided into the training set and the last third into the testing set.

Figure 3 shows the histogram of CO emission concentration data in the actual MSWI process, which is represented by a series of longitudinal stripes or line segments with varying heights. Generally, the horizontal axis indicates the data type, and the vertical axis indicates the distribution.

According to the truth distribution of the training and testing sets, there are big differences in the distribution. It indirectly reflects the phenomenon of CD in the data set of the actual MSWI process.

2.2. Methods

The CO emission prediction strategy based on kernel feature space semi-supervised concept drift detection proposed in this article is shown in Figure 4.

The functions of each module of the proposed strategy are described as follows:

(1)

Historical CO prediction model and drift detection model construction module: historical samples are used to build prediction models and kernel feature space detection models and to obtain historical data sets to support pseudo-labeling model construction.

(2)

Real-time CO prediction model and drift detection module: When the CD detected value exceeds the KPCA control limit, it is considered that the sample has a drifting possibility. At this time, the sample is stored in the cache window to be labeled. When the number of samples in the window reaches the preset window capacity, a pseudo-labeling model is constructed in real time, and the drift samples are marked with pseudo-truth values. For the pseudo-truth values after labeling, the PH test method was used to determine whether the samples were drifting. If no CD occurs, the CO emission concentration is predicted based on the historical prediction model.

(3)

Update data set acquisition module: the samples confirmed to drift in the current cache window are recorded as the drift sample data set, and the historical data set is determined according to experience for the model update.

(4)

Historical model update module: The acquired updated data set is retrained for the CO prediction model and kernel feature space detection model, and the historical data set of the pseudo-labeling model is updated. At the same time, the cache window to be labeled is reset.

2.2.1. Historical CO Prediction Model and Drift Detection Model Construction Module

Prediction Model Construction Submodule

The purpose of model learning is to determine the CD prediction model with the minimum prediction error $f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)$. The structure of the used LSTM is as follows Figure 5.

The output of the LSTM-based CO emission prediction model constructed with offline samples can be expressed as follows:

$${{\widehat{y}}^{\prime }}_{\mathrm{Htd}}=f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(X_{\mathrm{std}}^{\mathrm{Htd}}\right)$$

(1)

Kernel Feature Space Detection Model Construction $D$ n Submodule

For high-dimensional process variables, the KPCA model is used to calculate the CD detection control limit for determining whether the MSWI process changes. At first, the historical training data is denoted as $X_{\mathrm{std}}^{\mathrm{Htd}}$. Then, we map the vector $x^{\mathrm{Htd}}$ in $X_{\mathrm{std}}^{\mathrm{Htd}}$ to a higher-dimensional space $F$ (denoted as dimensional) as follows:

$$\varphi \left(x^{\mathrm{Htd}}\right): R^M\to R^D, D\ge M$$

(2)

where $\varphi \left(\cdot \right)$ is a nonlinear mapping function, and $D$ is the mapped data dimension.

The high-dimensional mapping space is represented as follows:

$$\varphi \left(X_{\mathrm{std}}^{\mathrm{Htd}}\right)=\left[\varphi \left(x_1^{\mathrm{Htd}}\right), \varphi \left(x_2^{\mathrm{Htd}}\right),\cdots ,\right.\left. \varphi \left(x_N^{\mathrm{Htd}}\right)\right]$$

(3)

Then, the kernel function is introduced to calculate the kernel matrix (K). This article uses the sigmoid kernel function as follows:

$$K=f\left({\beta }_0{X}_{\mathrm{std}}^{\mathrm{Htd}}{\left(X_{\mathrm{std}}^{\mathrm{Htd}}\right)}^{\mathrm{T}}+{\beta }_1\right)$$

(4)

where $f\left(\cdot \right)$ uses the tanh function, ${\beta }_0$ and ${\beta }_1$ user-defined. Generally, ${\beta }_1$ is set as 0 by default.

By defining $1_N$ as 1 matrix, the kernel matrix is centralized as

$$U={\displaystyle \frac{1_N}{N}}$$

(5)

Then, the kernel matrix (K_c) after centralization is calculated as follows:

$$K_c=K^{N \times N}-U^{N \times N}{K}^{N \times N}-K^{N \times N}{U}^{N \times N}+U^{N \times N}{K}^{N \times N}{U}^{N \times N}$$

(6)

In this article, the eigenvalue decomposition method is used to solve the eigenvalue and eigenvector of the matrix, and the calculation is as follows:

$${\displaystyle \frac{K_c}{N}}=Q\Sigma Q^{-1}$$

(7)

where $Q$ is the eigenvector matrix and $\Sigma =diag\left({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N\right)$ is the diagonal matrix. In $K_c=K^{N \times N}-U^{N \times N}{K}^{N \times N}-K^{N \times N}{U}^{N \times N}+U^{N \times N}{K}^{N \times N}{U}^{N \times N}$, each element on the diagonal is an eigenvalue.

The eigenvalues are normalized and calculated as follows:

$$Q_s={\displaystyle \frac{Q}{{\left(\sqrt{N\lambda }\right)}^{\mathrm{T}}}}$$

(8)

where $Q_s$ is the matrix normalized by $Q$, and $\lambda$ is the eigenvector.

The cumulative feature contribution rate ($\eta$) and KPCA contribution threshold ${\delta }^{\mathrm{KPCA}}$ were used for the selection of the number of principal components, which are as follows:

$$\eta ={\displaystyle \frac{{\displaystyle {\sum }_{p=1}^p{\lambda }_p}}{{\displaystyle {\sum }_{m=1}^M{\lambda }_m}}}$$

(9)

where $p$ is the number of selected principal components, $M$ is the eigenvector number of $X^{\mathrm{Htd}}$, and $p$ is less than $M$.

Accordingly, the CD detection control limit for monitoring whether the operating condition changes is determined by P.

The nonlinear principal components are calculated by the obtained kernel principal components, which are shown as follows:

$$P=K_c{Q}_{s\_p}$$

(10)

where $Q_{s\_p}$ is the matrix of size $N\times p$ in $Q_s$, and $P$ is a nonlinear principal component matrix with size $N\times p$.

Statistical indicators $T^2$ and $SPE$ according to Hotelling are adopted as the control limits for the identification of operating condition drift. They are expressed as

$${(T_{\alpha }^2)}^{\mathrm{Htd}}={\displaystyle \frac{p(N-1)}{(N-p)}}{F}_{\alpha }\left(p,N-p\right)$$

(11)

$$SP{E}_{\alpha }^{\mathrm{Htd}}={\theta }_1{\left({\displaystyle \frac{c_{\alpha }\sqrt{2{\theta }_2{h}_0^2}}{{\theta }_1}}+1+{\displaystyle \frac{{\theta }_2{h}_0(h_0-1)}{{\theta }_1^2}}\right)}^{1/h_0}$$

(12)

where $F_{\alpha }(p,N-p)$ is an F distribution with $p$ and $(N-p)$ degrees of freedom and $c_{\alpha }$ represents the normal deviation from the upper $(1-\alpha )$ percentile. ${\theta }_i$ and $h_0$ are calculated as follows:

$$h_0=1-2{\theta }_1{\theta }_3/3{\theta }_1^2$$

(13)

$${\theta }_i={\displaystyle {\sum }_{m=p+1}^M{\left({\lambda }_m\right)}^i},i=1,2$$

(14)

Therefore, the historical CD detection model is constructed $f_{\mathrm{KPCA}}^{\mathrm{Htd}}\left(\cdot \right)$.

Pseudo-Labeling Model History Data Sets Obtain Submodules

The first-order difference components of each sample in error output and feature space in the historical sample set ($X_{1,t}^{\mathrm{Htd}}$) are calculated. By taking sample $x_t^{\mathrm{Htd}}$ as an example, they are calculated as follows:

$$\Delta x_t^{\mathrm{Htd}}=x_t^{\mathrm{Htd}}-x_{t-1}^{\mathrm{Htd}}$$

(15)

$$\Delta y_t^{\mathrm{Htd}}=y_t^{\mathrm{Htd}}-y_{t-1}^{\mathrm{Htd}}$$

(16)

where $x_t^{\mathrm{Htd}}$ and $x_{t-1}^{\mathrm{Htd}}$ represent the feature space of the sample respectively; $y_t^{\mathrm{Htd}}$ and $y_{t-1}^{\mathrm{Htd}}$ represent the true value of the sample, respectively.

According to Equations (15) and (16), the data set of first-order difference components of the error output space and the feature space of the historical sample is denoted as $\Delta y^{\mathrm{Htd}}$ and $\Delta X^{\mathrm{Htd}}$, respectively.

2.2.2. Real-Time CO Prediction Model and Drift Detection Module

Kernel Feature Space CD Real-Time Detection Submodule

In this article, we denoted the testing data as $X_k^{\mathrm{Otd}}$. Firstly, the vector $x_k^{\mathrm{Otd}}$ in $X_k^{\mathrm{Otd}}$ is mapped to a higher-dimensional space $F$, and the kernel function is introduced to calculate the kernel matrix $K_t$ (size $N_k\times N$). The used Sigmoid kernel function in terms of testing data is denoted as

$$K_t=f\left({\beta }_0{X}_k^{\mathrm{Otd}}{\left(X^{\mathrm{Htd}}\right)}^{\mathrm{T}}+{\beta }_1\right)$$

(17)

where $f\left(\cdot \right)$ uses the tanh function.

By defining $N_k\times N$ 1-matrix with size $N_k\times N$ as $1_{N_k}$, the kernel matrix is centralized as

$$U_t={\displaystyle \frac{1_{N_k}}{N}}$$

(18)

Then, the kernel matrix ($K_{t\_c}$) after centralization is calculated as follows:

$$K_{t\_c}=K_t^{N_k\times N}-U_t^{N_k\times N}{K}^{N\times N}-K_t^{N_k\times N}{U}^{N\times N}+U_t^{N_k\times N}{K}^{N\times N}{U}^{N\times N}$$

(19)

where the size of $K_{t\_c}$ is $N_k\times N$.

The kernel principal components in terms of testing data are calculated based on the obtained number of principal components, which are shown as follows:

$$P_t=K_{t\_c}{Q}_{s\_p}$$

(20)

where $Q_{s\_p}$ is the matrix of size $N\times p$, and $P_t$ is a matrix of size $N_k\times p$.

The CD detection value of the new sample is calculated as follows:

$$T_k^2=diag\left({\displaystyle \frac{P_t}{diag\left({\lambda }_p{P}_t\right)}}\right)$$

(21)

$$SP{E}_k={\displaystyle \sum {\left(K_{t\_c}{Q}_s\right)}^2}-{\displaystyle \sum P_t}$$

(22)

According to whether $T_k^2$ and $SP{E}_k$ of the samples in the window meet the following criterion, determine whether CD occurs.

$$\left\{\begin{array}{l}\mathrm{Drift},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if T_k^2>{(T_{\alpha }^2)}^{\mathrm{Htd}} \mathrm{or} SP{E}_k>SP{E}_{\alpha }^{\mathrm{Htd}}\\ \mathrm{Normal},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{other}\end{array}\right.$$

(23)

Prediction Submodule Based on the Historical Prediction Model without CD Drift

If no drift occurs, the prediction of CO emission concentration is made based on the historical prediction model $f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)$. The prediction results are denoted as follows:

$${\widehat{y}}_k^{\mathrm{new}}=f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(X_k^{\mathrm{Otd}}\right)$$

(24)

CD Pseudo-Labeling Model Real-Time Construction and Drift Sample Pseudo-Labeling Submodule

After KPCA filtering and the cache window to be labeled is filled, the sample set in the window is denoted as $X_{\mathrm{window} }=\left\{x_1^{\mathrm{window} },\cdots ,x_w^{\mathrm{window} }\right\}$, where w is the preset cache window sample capacity. The detailed pseudo-labeling process is as follows.

Firstly, a new set of first-order difference components is constructed as

$$\Delta y^{{\mathrm{Htd}}^{\mathbf{\prime }}}=\left\{\begin{array}{l}\Delta y^{\mathrm{Htd}}\hfill \\ \Delta y_1^{\mathrm{window} }\hfill \end{array}\right\}$$

(25)

$$\Delta X^{{\mathrm{Htd}}^{\mathbf{\prime }}}=\left\{\begin{array}{l}\Delta X^{\mathrm{Htd}}\hfill \\ \Delta X_1^{\mathrm{window} }\hfill \end{array}\right\}$$

(26)

where $y_1^{\mathrm{window} }$ and $X_1^{\mathrm{window} }$ are truth value and feature space, respectively; $\Delta y_1^{\mathrm{window} }$ and $\Delta X_1^{\mathrm{window} }$ are the first-order difference components obtained by $x_1^{\mathrm{window} }$ and the last time samples in the current training set.

By starting from $x_2^{\mathrm{window}}$, we calculate the first-order difference component of its feature space $X_1^{\mathrm{window}}$ as follows:

$$\Delta X_2^{\mathrm{window} }=X_2^{\mathrm{window} }-X_1^{\mathrm{window} }$$

(27)

Based on the idea of nearest neighbor, $\epsilon$ feature space difference components with the smallest distance between $\Delta X^{{\mathrm{Htd}}^{\prime }}$ and $\Delta X_2^{\mathrm{window} }$ are selected from $\Delta X^{{\mathrm{Htd}}^{\prime }}$ through Euclidean distance. By combining with their corresponding output space difference components, they are written as

$${\mathsf{\Omega }}_{\mathrm{nearest}}=\left\{\left(\Delta x_{\mathrm{nearest}\text{\_}1}^{{\mathrm{Htd}}^{\prime }},\Delta y_{\mathrm{nearest}\text{\_}1}^{{\mathrm{Htd}}^{\prime }}\right),\cdots ,\left(\Delta x_{\mathrm{nearest}\text{\_}\epsilon }^{{\mathrm{Htd}}^{\prime }},\Delta y_{\mathrm{nearest}\text{\_}\epsilon }^{{\mathrm{Htd}}^{\prime }}\right)\right\}$$

(28)

Based on ${\mathsf{\Omega }}_{\mathrm{nearest}}$, the pseudo-labeling model based on LSTM is established. The error output space difference component $\Delta {\widehat{y}}_2^{\mathrm{window}}$ of $x_2^{\mathrm{window}}$ is obtained as the following steps,

$${\mathsf{\Omega }}_{\mathrm{nearest}}\Rightarrow {\mathrm{LSTM}}_{\mathrm{nearest}}$$

(29)

$${\mathrm{LSTM}}_{\mathrm{nearest}}\left(\Delta X_2^{\mathrm{window} }\right)\Rightarrow \Delta {\widehat{y}}_2^{\mathrm{window} }$$

(30)

Then, the pseudo-truth value ${\dot{y}}_2^{\mathrm{window}}$ of $x_2^{\mathrm{window}}$ is calculated and denoted as

$${\dot{y}}_2^{\mathrm{window} }=\Delta {\widehat{y}}_2^{\mathrm{window} }+y_1^{\mathrm{window} }$$

(31)

Repeat the above process until the samples in the window are labeled with pseudo-truth values.

CD Sample Confirmation Submodule Based on PH Test

A reasonable analysis of the difference between the pseudo-truth value of the sample and the measured value is the key to confirming the sample CD. In the PH test method, given a series of observations $[l_1,l_2,\cdots l_m]$, the likelihood ratio statistic of the alternative hypothesis (there is a CD point $\gamma$ in the observations, that is, $1<\gamma <m$) against the null hypothesis (there is no drift in the observations, that is, $\gamma >m$) is calculated as [39]

$$L_{m,\gamma }=\frac{{\displaystyle \prod _{i=1}^{\gamma }}{f}_D\left(l_i\right){\displaystyle \prod _{i=\gamma +1}^m}{f}_D\left(l_i-\delta \right)}{{\displaystyle \prod _{i=1}^m}{f}_D\left(l_i\right)}$$

(32)

where ${\displaystyle \prod _{i=m+1}^m}{f}_D\left(l_i\right)=1$, ${\displaystyle \sum _{i=m+1}^m}{l}_i=0$; $f_D\left(\cdot \right)$ represents the distribution density function of the standard normal distribution $N(0,1)$; and $\delta$ means that the CD sample follows a normal distribution with a mathematical expectation of $\delta$.

The above Equation is expressed logarithmically as

$$Z_{m,\gamma }=\ln L_{m,\gamma }=\delta {\displaystyle \sum _{i=\gamma +1}^m\left(l_i-\frac{\delta }{2}\right)}$$

(33)

Accordingly, the log-likelihood ratio statistic of the alternative hypothesis (with CD) to the null hypothesis (without CD) is

$$Z_m=\underset{1\le \gamma <m}{\max }{Z}_{m,\gamma }=\max \left\{\delta {\displaystyle \sum _{i=\gamma +1}^m\left(l_i-\frac{\delta }{2}\right)}\right\}$$

(34)

By setting a threshold and comparing it with $Z_m$, we can determine whether CD exists in the current series of pseudo-truth values.

When all samples in the cache window to be labeled are marked with pseudo-truth values, this article adopts the PH test method to detect CD in the error output space of these samples. By taking the observed value Obs(t) at time t as an example, the detection process is as follows:

First, we calculate the cumulative variable ${\phi }_t$ for Obs(t):

$${\overline{Obs}}_{t-1}={\displaystyle \frac{1}{t-1}}{\displaystyle \sum _{m=1}^{t-1}Obs(m)}$$

(35)

$${\phi }_t={\displaystyle \sum _{m=1}^t(Obs(m)-{\overline{Obs}}_{m-1})}$$

(36)

where ${\overline{Obs}}_{t-1}$ represents the mean of all historical observations at the previous t − 1 moment; ${\phi }_t$ represents the difference between the current observation Obs(t) and the mean historical observations.

Then, the change index PH_t is calculated to determine whether the current observed value Obs(t) is abnormal:

$${\varphi }_t=\underset{m=1\cdots t}{\min }{\phi }_m$$

(37)

$$P{H}_t={\phi }_t-{\varphi }_t$$

(38)

where ${\varphi }_t$ represents the minimum cumulative variable value recorded from all current moments; $P{H}_t$ represents the difference between the cumulative variable ${\phi }_t$ and the minimum cumulative variable value at the current time $t$. When condition $P{H}_t>\lambda$ is satisfied, the observed Obs(t) is considered abnormal, where λ is the empirical threshold.

The sample set in the window is denoted as $X_{\mathrm{window}}^d=\left\{{\mathrm{x}}_1^d,\cdots ,{\mathrm{x}}_w^d\right\}$ when the cache window to be labeled is filled for the d time and all samples are labeled. We calculate the sample average measurement error ($Aveer{o}_d$) in the current window as follows:

$$Aveer{o}_d={\displaystyle \frac{1}{w}}\cdot {\displaystyle \sum _{m=1}^w\left|{\widehat{y}}_m^d-{\dot{y}}_m^d\right|}$$

(39)

where ${\widehat{y}}_m^d$ and ${\dot{y}}_m^d$ represent the true and pseudo-true values of the mth sample in the window, respectively.

On this basis, this article selects the observed value Obs(t) as the cumulative average measurement error of the samples in the window when the window is filled d times, namely

$$Obs(t){|}_{t=d}={\displaystyle \frac{Obs(t-1)\cdot (d-1)+Aveer{o}_d}{d}}(d\ge 1)$$

(40)

In this case, the cumulative variable (${\phi }_t$) represents the difference between the current cumulative average prediction error and the mean historical cumulative average prediction error. We denote ${\varphi }_t$ as the minimum ${\phi }_t$ value of the current record. When the cache window is filled for the first time, that is, when d = 1 and ${\varphi }_t={\phi }_t$. In this case, there is no basis for judging CD in the sample error output space, so this article represents ${\varphi }_t$ as

$${\varphi }_t{|}_{t=d}=\left\{\begin{array}{cc}\underset{m=1\cdots t}{\min }& {\phi }_m,d\ge 1\hfill \\ & {\varphi }_0,d=0\hfill \end{array}\right.$$

(41)

where ${\varphi }_0$ is the benchmark cumulative average measurement error, which is obtained according to the average measurement error of the verification sample. At the same time, this article sets $\lambda =0$, that is, when ${\phi }_t>{\varphi }_t$, which represents that the cumulative average prediction error in the next window is significantly higher than that in the historical sample, it is considered that the sample in the window can represent the CD, and it is used to construct a new training set.

2.2.3. Update Data Set Acquisition Module

When the samples in the cache window are confirmed to drift, this article builds a new training set based on the historical samples and the samples in the current window to update the prediction model. By taking the sample $X_{\mathrm{window}}^d$ in the window when the cache window is filled for the d time as an example, a new training set ($X_d^{\mathrm{newtd}}$) is constructed as follows:

$$X_d^{\mathrm{newtd}}=\left\{\begin{array}{l}{X}_{1,t}^{\mathrm{Htd}}\\ X_{1,w}^n\end{array}\right\}$$

(42)

$$y_d^{\mathrm{newtd}}=\left\{\begin{array}{l}{y}_{1,t}^{\mathrm{Htd}}\\ {\dot{y}}_{1,w}^n\end{array}\right\}$$

(43)

where $X_d^{\mathrm{newtd}}$ and $y_d^{\mathrm{newtd}}$ are the input feature space and truth value of the updated data set, respectively; $X_{1,w}^d$ and ${\dot{y}}_{1,w}^d$ are the input feature space and the pseudo-truth of the samples in the current window, respectively.

2.2.4. Historical Model Update Module

The updated data set is used to construct the CO prediction model and the kernel feature space CD detection model. The output of the new prediction model for CO emission based on LSTM by using $X_{\mathrm{window}}^d$ can be expressed as follows:

$${{\widehat{y}}^{\prime }}_{\mathrm{New}}^{\mathrm{newtd}}=f_{\mathrm{LSTM}}^{\mathrm{new}}\left(X_d^{\mathrm{newtd}}\right)$$

(44)

The KPCA-based CD model can be expressed as follows:

$$\{T_{\mathrm{new}}^2,SP{E}_{\mathrm{new}},\dots \}=f_{\mathrm{KPCA}}^{\mathrm{new}}\left(X_d^{\mathrm{newtd}}\right)$$

(45)

Replace the original model with a new mode and update the pseudo-token model history data set at the same time as follows:

$$\left\{\begin{array}{c}{f}_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)\leftarrow f_{\mathrm{LSTM}}^{\mathrm{new}}\left(\cdot \right)\\ f_{\mathrm{KPCA}}^{\mathrm{Htd}}\left(\cdot \right)\leftarrow f_{\mathrm{KPCA}}^{\mathrm{new}}\left(\cdot \right)\\ \Delta X^{\mathrm{Htd}}\leftarrow X_d^{\mathrm{newtd}}\\ \Delta y^{\mathrm{Htd}}\leftarrow y_d^{\mathrm{newtd}}\end{array}\right.$$

(46)

After the above update, the proposed prediction model can better adapt to the dynamic changes in the MSWI process.

2.3. Pseudo-Code

The pseudo-code of the pseudo-code algorithm used in this article is as follows

Table 1. Pseudo-code of LSTM-KPCA-PH algorithm.

Input: Data set XHtd, XOtd Output: ${{\widehat{y}}^{\prime }}_{\mathrm{Otd}}$

Set the parameters of the LSTM model and train the offline $f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)$ model;

Set KPCA model parameters, select sigmoid kernel function, F distribution, contribution rate;

Calculate the kernel matrix;

Centralize the kernel matrix;

Make the eigenvalue decomposition and the eigenvector normalization;

Calculate the nonlinear principal component;

Calculate control limit of statistical index ${(T_{\alpha }^2)}^{\mathrm{Htd}}$ and $SP{E}_{\alpha }^{\mathrm{Htd}}$;

Construct the offline KPCA model $f_{\mathrm{KPCA}}^{\mathrm{Htd}}\left(\cdot \right)$;

For n = 1 to Ntest

Based on $f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)$, the predicted value ${{\widehat{y}}^{\prime }}_n$ for each sample is obtained;

KPCA method is used to calculate the online statistical index $T_k^2$ and $SP{E}_k$ of the sample;

Determine whether the sample has drifted;

If drift occurs,

Store the sample in the cache window to be marked;

Check whether the cache window has reached its capacity;

If capacity is reached

Storage exception sample number;

Calculate the first-order difference component of the new sample;

Query the d difference components of the training set that are closest to the difference component of the new sample;

Mark the pseudo-truth value of the sample in the window;

Calculate the sum of sample errors in the current window;

Calculate the cumulative average forecast error so far in the current window;

The difference between the current cumulative average forecast error and the last cumulative average forecast error;

Judgment drift;

If drift occurs, the drift monitoring index and LSTM model are updated;

If no drift occurs, the next sample continues to be predicted using $f_{\mathrm{LSTM}}^{\mathrm{Htd}}\left(\cdot \right)$;    end   end

end

Predicted value of all samples after all loops is denoted as ${{\widehat{y}}^{\prime }}_{\mathrm{Otd}}$.

:

3. Results and Discussion

3.1. Performance Metrics

The evaluation index used in the article is RMSE. The former is calculated as follows:

$$RMSE=\sqrt{{\displaystyle \sum _{n=1}^N\frac{{\left(y_n-{\widehat{y}}_n\right)}^2}{N}}}$$

(47)

3.2. Experimental Results

3.2.1. Historical Model Construction

The CO prediction model adopts the LSTM model, which has 128 hidden layer units, 210 training times, 0.022 learning rate, 0.2 learning drop factor and 0.4 dropout rate.

Figure 6 shows a graph of the absolute value error of the training set. The horizontal axis shows the number of samples, and the vertical axis shows the absolute error predicted by the sample. The LSTM model has a certain effect on the construction of training samples in process data sets, and the error is small. Therefore, LSTM can be used to effectively build a CO emission prediction model.

The principal component contribution threshold of KPCA was set as 0.91, and five principal components were selected as the principal space for running CD detection. The calculated T² and SPE control limits are 9.4707 and 0.4246, respectively.

The curve of $\Delta y^{\mathrm{Htd}}$ in the pseudo-labeling model historical data set is shown in Figure 7.

The horizontal axis represents the number of samples, and the vertical axis represents the first-order difference component predicted by the sample. Figure 7 shows that most of the truth difference components are small, which shows that the method has a certain reliability.

3.2.2. Real-Time Prediction and CD Detection

The T² and SPE results based on KPCA CD detection are shown in Figure 8. The horizontal axis represents the number of samples, and the vertical axis represents the concept drift detection index T² and SPE calculated by this sample.

In this article, the window size is set as 3, and the parameters of the constructed pseudo-labeling LSTM model are the same as the constructed CO prediction model. The number of historical samples selected by Euclidean distance is 15. The comparison between the pseudo-truth labeling results and the actual truth values based on TD learning is shown in Figure 9. The horizontal axis represents the number of samples, the vertical axis represents the calculated value of this sample, the red line represents the pseudo-truth value, and the blue line represents the actual truth value.

Figure 9 shows that the variation trend in the pseudo-truth value is similar to the sample truth value. When the sample truth value is difficult to obtain completely, the pseudo-truth value can be used to approximate the CD in the sample error output space.

After the pseudo-truth value labeling of the samples with kernel feature space drift is completed, the PH test is used to detect the pseudo-label drift, as shown in Figure 10.

The horizontal axis represents the number of cache windows filled, and the vertical axis represents the cumulative average measurement error of samples within the window. Figure 10 shows the change in the cumulative average measurement error of samples in the window after each time the cache window to be labeled is filled, and the samples are labeled with pseudo-truth values. Among them, the cache window to be labeled is filled 10 times, and the cumulative average measurement error of samples in the window increases significantly in terms of CD. It becomes stable with the continuous update of the model, indicating that the proposed algorithm can effectively detect CD changes in the sample error output space.

3.2.3. Data Set and Model Update

According to the above test results, after the model is updated with a new training set composed of CD samples and historical samples, the predicted performance changes are shown in Figure 11. The horizontal axis indicates the number of samples, and the vertical axis indicates the absolute measurement error. The red line shows the original measurement error, and the blue line shows the measurement error after updating the model.

The above results show that the prediction error of the proposed CD detection algorithm in the prediction model is significantly reduced compared with the original historical model. Thus, the proposed algorithm can significantly improve the measurement performance of the model in the face of CD drift samples when the truth values of most drift samples are not labeled and can effectively improve the generalization performance of the CO emission prediction model in the MSWI process in the operating condition drift environment.

3.3. Method Comparison

In this article, each method was run 30 times, and the optimal results were selected, as shown in

Table 2. Comparison of prediction performance of different methods.

Detection Algorithm	Model Update Times	Update the Required Number of Truth Values	Model Prediction RMSE	Method
Unsupervised type	30	30	26.9249	Truth update is required
Supervised type	0	0	26.9690	Truth detection and updating are required
Textual algorithm	10	10	26.9444	Using pseudo-truth updates

. The predicted curves of different modeling methods are shown in Figure 12.

The horizontal axis indicates the number of samples, and the vertical axis indicates the absolute measurement error. The red line is the original measurement error, and the remaining color lines are the measurement error of the predicted value corresponding to the different models. Table 2 and Figure 11 show that the proposed method has few updating times and the optimal model performance. It can satisfy the CO prediction of the MSWI process when the CEMS system used for conventional pollutant detection has unavoidable misalignment and fault phenomenon.

3.4. Hyperparameter Analysis

This section analyzes the sensitivity of hyperparameters of the proposed method in terms of the RMSE index. The setting ranges of hyperparameters are shown in

Table 3. Range of hyperparameter settings.

Hyperparameter	Radius
Learning rate	0.001:0.001:0.03
dropout rate	0.1:0.1:0.9
Training times	50:5:350
Capacity of error output space detects the drift sample	3:1:20
F distribution	0.80:0.01:0.99
Contribution rate	0.80:0.01:0.99

, and the relationship between them and the RMSE index is shown in Figure 13. The horizontal axis represents the value of the hyperparameter, and the vertical axis represents the RMSE.

Figure 13 shows that

(1)

Learning rate: increasing this value can make the prediction model better fit the training data, and its change will cause fluctuations in the model performance, and 0.022 is preferable;

(2)

Dropout rate: both the training and test accuracy fluctuate within a small range with the change in this value, which is set to 0.4 in this article;

(3)

Training times: the predictive performance of the model is slightly worse with the increase in this value, and the model has the best performance when its value is 210;

(4)

The capacity of error output space detects the drift sample: the prediction performance of the model is slightly worse with the increase in this value, which is set to 3 in this article;

(5)

F distribution: With the setting of the F distribution rate, the model performance fluctuates greatly. In order to ensure the modeling performance, 0.9 is preferred;

(6)

Contribution rate: the smaller the number of selected features, the more appropriate contribution rate should be selected, which is 0.91 in this article.

In addition, considering the coupling between these hyperparameters, the multi-hyperparameter collaborative selection algorithm that can perform global optimization needs further research.

4. Conclusions

To solve the problem of CD in complex industrial processes and the difficulty of obtaining the truth value of predicted key process parameters in real-time, a prediction model of CO emission in the MSWI process based on semi-supervised CD detection in kernel feature space is proposed. The contribution of the proposed method is as follows: (1) the strategy of combining KPCA and PH test fully reflects the CD behavior of new samples in the input feature space and error output space; (2) the semi-supervised mechanism based on TD learning is used for pseudo-labeling of input feature space CD samples, which provides a new method for semi-supervised CD detection for industrial regression problems; (3) the feasibility of the proposed method in practical application is verified by using real MSWI process data set, and the performance of the proposed method is superior to that of the existing methods.

The limitations and potential problems include the following: (1) difficulty in selecting suitable kernel functions for KPCA and being able to update kernel functions based on data changes; (2) updating mixed pseudo-labeling modeling samples in the presence of truth values to improve model performance is not considered; (3) the processing of missing data and outliers in the actual MSWI process is not considered, and the generalization ability should be improved further.

Further research works include the following: (1) to improve the pseudo-labeling algorithm to improve the accuracy of labeling by using use auxiliary information or models; (2) in order to improve the accuracy of the prediction model in practical industrial applications, the mechanism of screening historical modeling samples and mixed pseudo-label modeling samples, and the processing of missing data and outliers would be considered, and a balancing strategy of these two types of samples should be addressed; (3) to determine the best time for model updating based on concept drift amplitude and time may be used in the further researches.