Flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning

Abstract

The invention discloses a flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and integrated learning, which comprises the following steps: for a flue gas SO2 concentration sequence collected in a flue gas desulfurization system, firstly decomposing the time sequence into a plurality of internal mode functions by using a variational mode decomposition method; then, a feedforward cascade neural network, a long short-term memory network and an autoregressive integral moving average model are used as base learners, and the dynamic characteristics of each modal function are learned; and finally, training element learners based on prediction results of all the base learners, and establishing an integrated model of each modal function. And recombining the predicted values of all the integrated models to realize the prediction of the SO2 concentration of the discharged flue gas. Accurate prediction of sulfur dioxide in discharged flue gas is beneficial to reduction of air pollution, prevention of formation of acid rain and protection of an ecological system and biological diversity. By adopting emission reduction measures and optimizing the production process in advance, enterprises can reduce the environmental protection treatment cost and improve the green production level.

Description

Technical Field

[0001] The invention belongs to the field of data processing and is a flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning. Background Art

[0002] The large amount of sulfur dioxide produced during coal combustion is one of the main sources of air pollution. Sulfur dioxide is highly corrosive and oxidizing, and can be converted into sulfate particles in the atmosphere to form acid rain, which seriously harms the environment and human health. Acid rain can cause soil and water acidification, destroy ecosystems, and endanger the survival of plants and animals. In addition, sulfur dioxide also poses a direct threat to human health, especially the respiratory and cardiovascular systems. Long-term exposure may cause asthma, chronic bronchitis and other diseases. Therefore, reducing sulfur dioxide emissions has become an important goal of global environmental protection, and the necessity of desulfurization processes in industry is self-evident. Governments of various countries have introduced strict emission standards and regulations, prompting relevant companies to adopt advanced desulfurization technologies to achieve the goal of sustainable development.

[0003] Although desulfurization technology has been widely used in practice, accurate modeling of its process still faces great challenges. First, the desulfurization process involves complex chemical reactions and physical transfer processes, including not only the absorption and oxidation of sulfur dioxide, but also the regeneration of absorbents and the treatment of by-products. These processes are highly nonlinear, time-varying and multi-scale, making it difficult for traditional mathematical modeling methods to accurately describe their dynamic behavior. In addition, the measurement data at industrial sites often contain noise and uncertainty, which further increases the difficulty of modeling. Therefore, establishing a model that can accurately reflect the dynamic characteristics of the desulfurization process is of great significance for optimizing the desulfurization process and improving the desulfurization efficiency.

[0004] At present, the commonly used desulfurization process modeling methods mainly include mechanism models and data-driven models. The mechanism model is built based on the basic principles of chemical reaction kinetics and transfer phenomena. This type of model has good explanatory power and can provide an in-depth understanding of the system behavior. However, since the desulfurization process involves complex chemical reactions and multiphase transfer phenomena, the model parameters are difficult to obtain accurately, and a large number of experimental verifications are required, which makes the practical application effect of the mechanism model limited, the calculation complexity is high, and the adaptability is poor. In contrast, the data-driven model uses historical data and is trained and predicted through machine learning algorithms. It has been widely used and studied in recent years. This type of model does not need to understand the internal mechanism of the system, but only relies on data for modeling. Commonly used data-driven models include interpolation, support vector machine regression (SVR), BP neural network, etc.

[0005] Interpolation method: The interpolation method uses the spatial distribution of known data points to achieve prediction by calculating the values ​​of unmeasured points. Although the interpolation method is simple and intuitive, due to the complexity of the desulfurization process and the high dimensionality of the data, the interpolation method shows obvious limitations when processing large-scale data, and the prediction accuracy and robustness are poor.

[0006] Support Vector Machine Regression (SVR): SVR is a regression model based on statistical learning theory, which makes predictions by finding the best hyperplane. SVR has advantages in processing small samples and high-dimensional data. However, for complex nonlinear systems such as the desulfurization process, the performance of SVR is limited by the selection of kernel functions and parameter settings, making it difficult to fully capture the dynamic characteristics of the system.

[0007] BP neural network: BP neural network is a commonly used artificial neural network model that realizes nonlinear mapping through a multi-layer perceptron structure. BP neural network has strong ability to process complex nonlinear relationships, but its training process is prone to fall into local optimality and requires a large amount of training data and computing resources. For a system like the desulfurization process with fast dynamic changes and many influencing factors, the prediction accuracy and stability of BP neural network still need to be improved.

[0008] Although data-driven models can provide better prediction results in some aspects, due to the high complexity and dynamic characteristics of the desulfurization process, traditional data-driven models are difficult to fully and accurately model and predict. This limitation in prediction performance may lead to improper setting of operating parameters of the desulfurization system, which in turn leads to serious consequences of excessive sulfur dioxide emissions. Therefore, a more accurate and robust prediction method is urgently needed to cope with the complexity and variability of the desulfurization process.

[0009] In view of the limitations of existing methods, the present invention proposes a flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and integrated learning. This method first uses variational mode decomposition technology to decompose the sulfur dioxide emission signal into multiple internal mode functions. Since the bandwidth of each internal mode function is within a certain range, it is easier to describe and analyze than the original sequence. Next, different types of prediction models are used to form an integrated model, and each sub-time series is learned and predicted. Finally, the prediction results of each sub-sequence are recombined to obtain the final sulfur dioxide concentration prediction value. This method not only improves the prediction accuracy, but also can better capture the dynamic change characteristics of sulfur dioxide emissions. It is of great significance to optimize the operating parameters of the desulfurization system and improve the desulfurization efficiency, thereby effectively reducing sulfur dioxide emissions and protecting the environment and public health. Summary of the invention

[0010] The present invention proposes a flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning. The method takes the flue gas desulfurization process in a coal-fired thermal power plant as the research object. For the collected flue gas sulfur dioxide concentration sequence, the variational mode decomposition method is used to decompose the time series into multiple internal mode functions. The original complex signal contains multiple frequency components, and directly modeling these complex signals may require consideration of the interaction of multiple frequencies. The decomposed sub-time series corresponds to a specific frequency range or scale, which can effectively simplify the processing of frequency or scale in the modeling process. Then, a feedforward cascade neural network, a long short-term memory network and an autoregressive integral moving average model are used to form a basic learner, and a multi-layer perceptron is selected as a meta-learner. The two types of learners are combined according to the hierarchical relationship to complete the construction of the ensemble model. The ensemble model is used to learn the dynamic characteristics of each internal mode function respectively, and the prediction sequence under the function is obtained. By reorganizing the prediction sequences of all ensemble models, the final prediction sequence of the exhaust gas SO2 concentration is obtained.

[0011] A flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning, characterized by comprising the following steps:

[0012] Step 1: The continuous emission monitoring system and information management system deployed in the flue gas desulfurization system respectively complete the real-time monitoring and data collection of the sulfur dioxide concentration in the flue gas.

[0013] Step 2: Use the variational mode decomposition method to decompose the time series corresponding to the flue gas sulfur dioxide concentration to obtain multiple internal mode functions with simpler frequency components.

[0014] Step 3: For each decomposed internal modal function, train multiple base learners respectively, and use the prediction results of all base learners as the training data of the meta-learner. Use the prediction results and original target values ​​generated above to train the meta-learner and establish an integrated model for each modal function. Then reorganize the prediction values ​​of all integrated models in the test set to obtain the final prediction result of the sulfur dioxide concentration in the exhaust gas.

[0015] Step 4: Use performance indicators (such as correlation coefficient, mean absolute percentage error, root mean square error) to test the generalization performance of the proposed prediction method.

[0016] The flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and integrated learning of the embodiment of the present invention. The original sulfur dioxide concentration sequence may contain multiple components with different frequencies, amplitudes and trends, which makes modeling complicated. Through mode decomposition, the complex signal is split into several single-mode subsequences, each of which usually represents a specific frequency or mode, thereby simplifying the structure and characteristics of each subsequence. Next, a feedforward cascade neural network, a long short-term memory network and an autoregressive integral moving average model are used to form an integrated model to learn and predict each internal modal function. Finally, the prediction results of each subsequence are reorganized to obtain the final predicted value of the sulfur dioxide concentration of the exhaust flue gas. This method not only improves the prediction accuracy, but also can better capture the dynamic change characteristics of sulfur dioxide emissions, which is of great significance for optimizing the operating parameters of the desulfurization system and improving the desulfurization efficiency, thereby effectively reducing sulfur dioxide emissions and protecting the atmospheric environment and public health.

[0017] In addition, the flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning according to this embodiment also has the following additional technical features:

[0018] In step 2, it is necessary to decompose the time series corresponding to the flue gas sulfur dioxide concentration using the variational mode decomposition method based on the actual desulfurization process data to obtain multiple internal mode functions. This method completes the extraction of modal signals by solving the following constrained variational problem.

[0019]

[0020] Among them, u k It represents the kth decomposition mode of the signal, and its corresponding center frequency is represented by ω k , the hyperparameter K represents the number of modes. All modes and corresponding center frequencies are represented by a set, and we get The optimization problem defined by equation (1) can be transformed into an unconstrained optimization problem by introducing a quadratic penalty term and a Lagrangian operator. The corresponding Lagrangian function is expressed as

[0021]

[0022] In the formula, α is a balance parameter used to ensure the true value of the data. The alternating direction multiplier method can be used to solve equation (2) to obtain the modal The frequency domain expression of is:

[0023]

[0024] in, and are u(t),u i (t),λ(t) and Since Equation (4) contains the Wiener filter structure, the time domain form of each mode can be obtained from the real part of the inverse Fourier transform of the filtered signal. The update formula for the center frequency corresponding to the mode is:

[0025]

[0026] Here, the final signal x(t) is decomposed into K internal mode functions.

[0027] The main steps of using variational mode decomposition method to obtain time series correlation are as follows:

[0028] (1) Parameter initialization, including the modal decomposition number K and the modal center frequency ω k , the penalty parameter α and the Lagrange multiplier λ of the constraint.

[0029] (2) The modal function and frequency are updated cyclically. For each modal function u k (t) and center frequency ω k Perform iterative updates until convergence.

[0030] (3) Lagrange multiplier update: Update the Lagrange multiplier to meet the signal reconstruction constraints.

[0031] (4) Convergence judgment: judge whether the changes of the modal function and the center frequency are less than the preset threshold. If so, stop the iteration.

[0032] (5) Output the results, generate and output each modal function u after decomposition k (t) and center frequency ω k .

[0033] In step 3, an integrated model is established based on different types of prediction models to complete the prediction of sulfur dioxide concentration in exhaust gas. Stacked ensemble learning is an integrated learning method that improves prediction performance by combining multiple basic models (first-level learners) and a meta-learner (second-level learner). The basic principle is to first train multiple different types of basic models and let these models predict the data, and then use these prediction results as new feature inputs and provide them to a meta-learner for training to generate the final prediction results. The advantage of stacked ensemble learning is that it can comprehensively utilize the advantages of multiple models, reduce the bias and variance of a single model, and thus improve the accuracy and robustness of the overall prediction. It is particularly effective in dealing with complex data sets and solving high-dimensional problems because it can capture the diversity and complementary information of different models.

[0034] For a univariate time series Y = [y1, y2, ..., y N ] T , assuming that the prediction sequence of the i-th prediction model is The mapping f(·) is used to merge n prediction sequences, then the final output of the stacked ensemble model can be expressed as

[0035]

[0036] The mapping f(·) is fitted by the meta-learner to the training samples Construct, the obtained meta-learner is the best combination of the outputs of the base learners.

[0037] According to the above calculation method, an integrated model is established for each internal mode function obtained by decomposition. The main steps of this integrated learning method are as follows:

[0038] (1) Data partitioning: divide the data set into training set and test set according to a certain ratio.

[0039] (2) Train the first-level learner. Select different types of models as the first-level learners and use the training set to train each first-level learner.

[0040] (3) Generate predictions for the first-level learner. For training data: each basic model predicts the training data and obtains prediction results. For test data: each basic model predicts the test data and obtains prediction results. These prediction results will be used as new features for the next step of model training.

[0041] (4) Construct a meta-training set to generate a new training set, called the meta-training set. The feature of this data set is the predicted output of the first-level learner, and the target variable is the true label of the original training set.

[0042] (5) Train the meta-learner. Select the model as the meta-learner and use the meta-training set to train the meta-learner.

[0043] (6) Generate the final prediction. The prediction output of the first-level learner on the test set is provided as input features to the trained meta-learner. The meta-learner generates the final prediction results based on these input features.

[0044] In step 4, the correlation coefficient (R 2 ), mean absolute percentage error (MAPE) and root mean square error (RMSE) are used to test the generalization performance of the proposed prediction method. The calculation expressions of the three performance indicators are:

[0045]

[0046] in y(i) and are the true value and predicted value at time i, respectively, and n is the total number of samples. The correlation coefficient ranges from 0 to 1, R 2The higher the value, the stronger the model's ability to fit the data. MAPE and RMSE are both statistical indicators that measure the difference between the model's predicted value and the actual value. The smaller the value of the two indicators, the higher the prediction accuracy of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a feedforward cascade neural network structure diagram.

[0048] Figure 2 This is a diagram of the long short-term memory network structure.

[0049] Figure 3 It is a diagram of the stacked ensemble learning structure.

[0050] Figure 4 It is a flow chart of the prediction method based on variational mode decomposition and ensemble learning. DETAILED DESCRIPTION

[0051] The embodiments of the present invention are described in detail below. This embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation method and a specific operation process are given, but the protection scope of the present invention is not limited to the following embodiments.

[0052] The technical solution adopted by the present invention is a flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning, which comprises the following steps:

[0053] Step 1: The continuous emission monitoring system and information management system deployed in the flue gas desulfurization system respectively complete the real-time monitoring and data collection of the sulfur dioxide concentration in the flue gas.

[0054] Step 2: Use the variational mode decomposition method to decompose the time series corresponding to the flue gas sulfur dioxide concentration to obtain multiple internal mode functions with simpler frequency components.

[0055] Step 3: According to the decomposed internal mode functions, multiple base learners are established respectively, and the prediction results of all base learners are used as training data for the meta-learner.

[0056] Step 4: Use the prediction results and original target values ​​generated above to train the meta-learner and build an integrated model for each modal function. Then reorganize the prediction values ​​of all integrated models to obtain the final prediction result of sulfur dioxide concentration in the exhaust gas.

[0057] According to the actual flue gas sulfur dioxide emission sequence of the desulfurization process, the modal components of the time series are analyzed. The variational modal decomposition method is used to extract different frequency components in the signal. This method completes the extraction of modal signals by solving the following constrained variational problem.

[0058]

[0059] Among them, u k It represents the kth decomposition mode of the signal, and its corresponding center frequency is represented by ω k , the hyperparameter K represents the number of modes. All modes and corresponding center frequencies are represented by a set, and we get The optimization problem defined by equation (1) can be transformed into an unconstrained optimization problem by introducing a quadratic penalty term and a Lagrangian operator. The corresponding Lagrangian function is expressed as

[0060]

[0061] In the formula, α is a balance parameter used to ensure the true value of the data. The alternating direction multiplier method can be used to solve equation (2) to obtain the modal The frequency domain expression of is:

[0062]

[0063] in, and are u(t),u i (t),λ(t) and Since Equation (4) contains the Wiener filter structure, the time domain form of each mode can be obtained from the real part of the inverse Fourier transform of the filtered signal. The update formula for the center frequency corresponding to the mode is:

[0064]

[0065] Among them, the final signal x(t) is decomposed into K internal mode functions, namely

[0066] The main steps of using variational mode decomposition method to obtain time series correlation are as follows:

[0067] Step 1: Parameter initialization, including the number of modal decompositions K and the center frequency ω of the mode k , the penalty parameter α and the Lagrange multiplier λ of the constraint.

[0068] Step 2: The modal function and frequency are updated cyclically. For each modal function u k (t) and center frequency ω k Perform iterative updates until convergence.

[0069] Step 3: Lagrange multiplier update: Update the Lagrange multiplier to meet the signal reconstruction constraints.

[0070] Step 4: Convergence judgment: judge whether the changes of the modal function and the center frequency are less than the preset threshold. If yes, stop the iteration.

[0071] Step 5: Output the results, generate and output each mode function u after decomposition k (t) and center frequency ω k .

[0072] According to the above method, the time series of sulfur dioxide concentration in the exhaust gas is analyzed and the corresponding internal mode function is extracted.

[0073] Different types of prediction models are used as basic learners and meta-learners to construct an integrated learning model with a hierarchical structure, learn the extracted internal modal functions, and complete the prediction of sulfur dioxide concentration in exhaust gas.

[0074] Stacked ensemble learning is an ensemble learning method that improves prediction performance by combining multiple base models (first-level learners) and a meta-learner (second-level learner). The basic principle is to first train multiple base models of different types and let these models predict the data. These prediction results are then used as new feature inputs and provided to a meta-learner for training to generate the final prediction results. The advantage of stacked ensemble learning is that it can comprehensively utilize the advantages of multiple models, reduce the bias and variance of a single model, and thus improve the accuracy and robustness of the overall prediction. It is particularly effective in dealing with complex data sets and solving high-dimensional problems because it can capture the diversity and complementary information of different models.

[0075] For a univariate time series Y = [y1, y2, ..., y N ] T , assuming that the prediction sequence of the i-th prediction model is The mapping f(·) is used to merge n prediction sequences, then the final output of the stacked ensemble model can be expressed as

[0076]

[0077] The mapping f(·) is fitted by the meta-learner to the training samples Construct, the obtained meta-learner is the best combination of the outputs of the base learners.

[0078] According to the above calculation method, an integrated model is established for each internal mode function obtained by decomposition. The main steps of this integrated learning method are as follows:

[0079] Step 1: Data division: divide the data set into training set and test set according to a certain ratio.

[0080] Step 2: Train the first-level learner. Select different types of models as the first-level learners and use the training set to train each first-level learner.

[0081] Step 3: Generate predictions for the first-level learner. For training data: each basic model predicts the training data and obtains prediction results. For test data: each basic model predicts the test data and obtains prediction results. These prediction results will be used as new features for the next step of model training.

[0082] Step 4: Construct a meta-training set to generate a new training set, called the meta-training set. The feature of this data set is the predicted output of the first-level learner, and the target variable is the true label of the original training set.

[0083] Step 5: Train the meta-learner, select the model as the meta-learner, and train the meta-learner using the meta-training set.

[0084] Step 6: Generate the final prediction. The prediction output of the first-level learner on the test set is provided as input features to the trained meta-learner. The meta-learner generates the final prediction results based on these input features.

[0085] In the designed integrated model, the basic learners are selected as long short-term memory network, feedforward cascade neural network and autoregressive integrated moving average model. The long short-term memory network can effectively maintain long-term information by introducing a gating mechanism, and can effectively capture and memorize important information in long time series. Assume x t is the input vector at time t, c t is the cell state, h t and h t-1 They represent the output of the LSTM network unit at time t and t-1 respectively. Equations (6)-(11) are the specific expressions of the LSTM network.

[0086] (1) Through the weight matrix W c and the bias vector b c Calculate candidate cell states

[0087]

[0088] (2) Input gate i t Use the output of the long short-term memory network unit at time t-1 and the external input x t The relevant weight matrix, bias vector and sigmoid function are calculated as W i ,b i andσ,i t The expression is

[0089] i t =σ(W i ·[h t-1 ,x t ]+b i ) (7)

[0090] (3) Forget gate f t To quantify the information retained in the cell state, use W f and b f represents the relevant weight matrix and bias vector, f t The expression is

[0091] f t =σ(W f ·[h t-1 ,x t ]+b f ) (8)

[0092] (4) The cell state c at the current moment t Use the cell state W at the previous moment o and b o Calculated

[0093]

[0094] (5) Output gate o t Determine the cell state c t The information that can be used as output, W o and b o Represents the relevant weight matrix and bias vector, expressed as

[0095] o t =σ(W o ·[h t-1 ,x t ]+b o ) (10)

[0096] (6) The output expression of the long short-term memory network unit is:

[0097] h t =o t *tanh(c t ) (11)

[0098] Where tanh(·) is the hyperbolic tangent function.

[0099] The feedforward cascade neural network has the characteristics of flexible structure, easy training, and excellent function approximation ability, so it is selected as the basic learner. Assuming that each hidden layer contains n nodes, the mathematical expression of the cascade network with h hidden layers is

[0100]

[0101] in and denote the input and output of the jth node in the hth hidden layer, respectively. and is the input and output of the jth node in the output layer. is the i-th input signal of the neural network at time t, given the time series So is the offset corresponding to the hth hidden layer, Represents the activation function of the jth node in the hth hidden layer.

[0102] The autoregressive integrated moving average model is a classic time series analysis model that has better short-term prediction capabilities than complex structure models. The model consists of three main parts: autoregression, difference, and moving average. The output of the model at any time in the future is equal to a linear combination of past values ​​and white noise. The difference part is used to remove the seasonality and trend of the time series and transform it into a stationary time series. The expression of the model is

[0103]

[0104] where Y t represents the value of the time series at time t, c is a constant term, φ i is the parameter of the autoregressive part, ε t is white noise (error term), θj is the parameter of the moving average part, Δ d Y t represents the time series after d times of difference. Assume y 1t ,y 2t and 3t They represent the predicted values ​​of the long short-term memory network, feedforward cascade neural network and autoregressive integrated moving average model at time t, respectively, and the multilayer perceptron NN is selected as the meta-learner, then the output expression of the corresponding integrated model is

[0105]

[0106] Example

[0107] Step 1: The continuous emission monitoring system and information management system deployed in the flue gas desulfurization system respectively complete the real-time monitoring and data collection of the sulfur dioxide concentration in the flue gas.

[0108] Step 2: Use the variational mode decomposition method to decompose the time series corresponding to the flue gas sulfur dioxide concentration to obtain multiple internal mode functions with simpler frequency components.

[0109] Step 3: For each decomposed internal modal function, train multiple base learners respectively, and use the prediction results of all base learners as the training data of the meta-learner. Use the prediction results and original target values ​​generated above to train the meta-learner and establish an integrated model for each modal function. Then reorganize the prediction values ​​of all integrated models in the test set to obtain the final prediction result of the sulfur dioxide concentration in the exhaust gas.

[0110] Step 4: Use performance indicators (such as correlation coefficient, mean absolute percentage error, root mean square error) to test the generalization performance of the proposed prediction method.

Claims

1. A flue gas sulfur dioxide concentration prediction method based on variational mode decomposition and ensemble learning, characterized in that: The method comprises the following steps: Step 1: The continuous emission monitoring system and information management system deployed in the flue gas desulfurization system respectively complete the real-time monitoring and data collection of the sulfur dioxide concentration in the flue gas; Step 2: Decompose the time series corresponding to the flue gas sulfur dioxide concentration using the variational mode decomposition method to obtain the internal mode functions of multiple frequency components; Step 3: According to the decomposed internal mode functions, multiple base learners are established respectively, and the prediction results of all base learners are used as training data of the meta-learner; Step 4: Use the prediction results and original target values ​​generated above to train the meta-learner and establish an integrated model for each modal function; then reorganize the prediction values ​​of all integrated models to obtain the final prediction result of sulfur dioxide concentration in the exhaust gas.

2. The method for predicting flue gas sulfur dioxide concentration based on variational mode decomposition and ensemble learning according to claim 1 is characterized in that: According to the actual flue gas sulfur dioxide emission sequence of the desulfurization process, the modal components of the time series are analyzed; the different frequency components in the signal are extracted using the variational modal decomposition method, which completes the extraction of modal signals by solving the following constrained variational problem; Among them, u k It represents the kth decomposition mode of the signal, and its corresponding center frequency is represented by ω k , the hyperparameter K represents the number of modes; all modes and corresponding center frequencies are represented by a set, and we get The optimization problem defined by equation (1) is transformed into an unconstrained optimization problem by introducing a quadratic penalty term and a Lagrangian operator. The corresponding Lagrangian function is expressed as In the formula, α is the balance parameter, which is used to ensure the true value of the data. The alternating direction multiplier method is used to solve equation (2) to obtain the modal The frequency domain expression of is: in, and are u(t),u i (t),λ(t) and Fourier transform form; Since equation (4) contains the Wiener filter structure, the time domain form of each mode is obtained by the real part of the inverse Fourier transform of the filtered signal; the update formula of the mode corresponding to the center frequency is Among them, the final signal x(t) is decomposed into K internal mode functions, namely The steps to obtain time series correlation using variational mode decomposition are as follows: Step 1: Parameter initialization, including the modal decomposition number K, the modal center frequency ω k , penalty parameter α and Lagrange multiplier λ of constraint conditions; Step 2: The modal function and frequency are updated cyclically. For each modal function u k (t) and center frequency ω k Perform iterative updates until convergence; Step 3: Lagrange multiplier update: update the Lagrange multiplier to meet the signal reconstruction constraint; Step 4: Convergence judgment: judge whether the changes of the modal function and the center frequency are less than the preset threshold; if so, stop the iteration; Step 5: Output the results, generate and output each mode function u after decomposition k (t) and center frequency ω k ; According to the above method, the time series of sulfur dioxide concentration in the exhaust gas is analyzed and the corresponding internal mode function is extracted.

3. The method for predicting flue gas sulfur dioxide concentration based on variational mode decomposition and ensemble learning according to claim 1 is characterized in that: Using different types of prediction models as basic learners and meta-learners, an integrated learning model with a hierarchical structure is constructed to learn the extracted internal modal functions and complete the prediction of sulfur dioxide concentration in exhaust gas; Using stacked ensemble learning for training and prediction: For a univariate time series Y = [y1, y2, …, y N ] T , assuming that the prediction sequence of the i-th prediction model is The mapping f(·) is used to merge n prediction sequences, i = 1, 2, …, n, then the final output of the stacked ensemble model is expressed as The mapping f(·) is fitted by the meta-learner to the training samples Build; An integrated model is established for each internal mode function obtained by decomposition; the learning method steps of the integrated model are as follows: Step 1: Data division: divide the data set into training set and test set according to a certain ratio; Step 2: Train the first-level learner, select different types of models as the first-level learners, and use the training set to train each first-level learner; Step 3: Generate predictions for the first-level learner. For training data, each basic model predicts the training data and obtains prediction results. For test data, each basic model predicts the test data and obtains prediction results. These prediction results will be used as new features for the next step of model training. Step 4: Construct a meta-training set to generate a new training set, called the meta-training set. The feature of this data set is the predicted output of the first-level learner, and the target variable is the true label of the original training set. Step 5: Train the meta-learner, select the model as the meta-learner, and train the meta-learner using the meta-training set; Step 6: Generate the final prediction. The prediction output of the first-level learner on the test set is provided as input features to the trained meta-learner. The meta-learner generates the final prediction results based on these input features.

4. The method for predicting flue gas sulfur dioxide concentration based on variational mode decomposition and ensemble learning according to claim 3 is characterized in that: In the designed integrated model, the basic learners are selected as long short-term memory network, feedforward cascade neural network and autoregressive integrated moving average model respectively; the long short-term memory network can effectively maintain long-term information by introducing a gating mechanism, and can effectively capture and memorize important information in long time series; let x t is the input vector at time t, c t is the cell state, h t and h t-1 They represent the output of the LSTM network unit at time t and t-1 respectively; Equations (6)-(11) are the specific expressions of the LSTM network; (1) Through the weight matrix W c and the bias vector b c Calculate candidate cell states (2) Input gate i t Use the output of the long short-term memory network unit at time t-1 and the external input x t The relevant weight matrix, bias vector and sigmoid function are calculated as W i ,b i andσ,i t The expression is i t =σ(W i ·[h t-1 ,x t ]+b i ) (7) (3) Forget gate f t To quantify the information retained in the cell state, use W f and b f represents the relevant weight matrix and bias vector, f t The expression is f t =σ(W f ·[h t-1 ,x t ]+b f ) (8) (4) The cell state c at the current moment t Use the cell state W at the previous moment o and b o Calculated (5) Output gate o t Determine the cell state c t The information used as output, W o and b o Represents the relevant weight matrix and bias vector, expressed as the t =σ(W o ·[h t-1 ,x t ]+b o ) (10) (6) The output expression of the long short-term memory network unit is: h t =o t *tanh(c t ) (11) Among them, tanh(·) is the hyperbolic tangent function; In a feedforward cascade neural network, assuming that each hidden layer contains n nodes, the mathematical expression of a cascade network with h hidden layers is in and denote the input and output of the jth node in the hth hidden layer, respectively. and is the input and output of the jth node in the output layer; is the i-th input signal of the neural network at time t, given the time series So is the offset corresponding to the hth hidden layer, represents the activation function of the jth node in the hth hidden layer; The autoregressive integrated moving average model consists of three parts: autoregression, difference, and moving average. The output at any time in the future is equal to the linear combination of the past value and white noise. The role of the difference part is to remove the seasonality and trend of the time series and transform it into a stationary time series. The expression of the autoregressive integrated moving average model is: where Y t represents the value of the time series at time t, c is a constant term, φ i is the parameter of the autoregressive part, ∈ t is white noise, θj is the parameter of the moving average part, Δ d Y t represents the time series after d differences; assuming y 1t ,y 2t and 3t They represent the predicted values ​​of the long short-term memory network, the feedforward cascade neural network and the autoregressive integrated moving average model at time t, respectively, and the multilayer perceptron NN is selected as the meta-learner. The output expression of the corresponding integrated model is

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