Soft sensing method for dioxin emissions in MSWI process based on integrated TS fuzzy regression tree

Abstract

The present invention proposes a soft measurement method for dioxin emissions from the MSWI process based on an integrated T-S fuzzy regression tree. Dioxins (DXN), a highly toxic pollutant produced during the municipal solid waste incineration (MSWI) process based on a grate furnace, are key environmental indicators for achieving optimal control of the process operation. First, a TSFRT model for dioxin emissions based on a screening layer and a fuzzy reasoning layer is constructed; then, a variety of parameter update learning algorithms for the antecedent and consequent parts of fuzzy reasoning are proposed, and five TSFRT models for dioxin emissions, namely TSFRT-Ⅰ, TSFRT-Ⅱ, TSFRT-Ⅲ, TSFRT-Ⅳ and TSFRT-V, are obtained; finally, taking the TSFRT-Ⅲ model for dioxin emissions as an example, an integrated TSFRT (EnTSFRT) model with TSFRT-Ⅲ as the base learner is constructed to achieve high-precision modeling of dioxin emission concentrations. Experimental results on a real DXN dataset demonstrate the effectiveness and rationality of the proposed method.

Description

Technical Field

[0001] The invention belongs to the technical field of soft measurement. Background Art

[0002] Municipal solid waste (MSW) treatment aims to achieve harmlessness, volume reduction, and resource utilization. Furthermore, grate-based MSW systems (MSWs) are crucial for reducing future DXN emissions in my country. Therefore, high-precision soft sensing of DXN emissions is a top priority for MSW process control.

[0003] Data-driven soft sensing techniques can effectively address these issues. These techniques use machine learning or deep learning to establish a mapping relationship between easily measurable process variables and DXN emission concentrations. This typically requires determining a mapping function to predict DXN emissions. However, existing methods often lack interpretability, struggle to handle the uncertainties inherent in real processes, and require further improvement in their generalization performance.

[0004] Fuzzy decision trees (FDTs) use a branch-and-bind backtracking mechanism to construct classification decision models capable of handling inherently uncertainties. Subsequently, crisp decision trees (CDTs) such as CART, ID3, and C4.5 were proposed. Studies have shown that CDT models are highly robust and can become extremely convenient and interpretable white-box algorithms simply by adjusting hyperparameters. In addition, fuzzy set theory, as a major pattern recognition technique, has attracted widespread attention, leading to numerous studies on fuzzy classification trees (FCTs). For example, fuzzy partitioning is used to construct FCT models for data with clear semantic information, and FCT methods that combine ID3 trees with fuzzy approximate reasoning include a complete FCT model with growth, pruning, fine-tuning, and testing processes. Therefore, pattern recognition techniques that combine fuzzy theory with CDT, namely FCT algorithms, have become one of the research hotspots capable of handling problems with uncertain characteristics. Currently, FCT is widely used in sign language recognition, partial discharge pattern classification, sorting tasks, sample selection, data mining, visual classification, distributed computing, and other classification tasks, but it is difficult to directly apply to the regression task of DXN emission concentration prediction addressed by the present invention.

[0005] This paper addresses the soft-sensing problem of DXN emission concentrations. First, a TSFRT model for dioxin emissions based on a screening layer and a fuzzy inference layer is constructed. Next, multiple parameter update learning algorithms for the antecedent and consequent parts of the fuzzy inference are proposed, resulting in five TSFRT models for dioxin emissions: TSFRT-I, TSFRT-II, TSFRT-III, TSFRT-IV, and TSFRT-V. Finally, using the TSFRT-III model as an example, an integrated TSFRT (EnTSFRT) model is constructed based on TSFRT-III to achieve high-precision modeling of dioxin emission concentrations. Experimental results on a real DXN dataset demonstrate the effectiveness and rationality of the proposed method.

[0006] The MSWI process includes process stages such as solid waste storage and transportation, solid waste incineration, waste heat boiler, steam power generation, flue gas purification and flue gas emission. Take the grate-type MSWI process with a daily processing capacity of 800 tons as an example.

[0007] Combining the entire process of DXN decomposition, generation, adsorption and emission, the main functions of each stage are described as follows:

[0008] 1) Solid Waste Storage and Transportation: Sanitation vehicles transport MSW from various urban collection points to the MSWI power plant. After weighing and recording, the waste is dumped from the unloading platform into the unfermented area of ​​the solid waste storage tank. A solid waste grabber then mixes and stirs the waste before transferring it to the fermentation area. After 3-7 days of fermentation and dehydration, the MSW is incinerated to ensure its low calorific value. Research has shown that raw MSW contains trace amounts of DXN (approximately 0.8 ng TEQ / kg) and contains various chlorine-containing compounds required for DXN formation reactions.

[0009] 2) Solid Waste Incineration Stage: A solid waste grab bucket deposits fermented MSW into the feed hopper, which pushes the MSW into the incinerator via the feeder. After drying, combustion 1, combustion 2, and the burnout grate, the combustible components in the MSW are completely burned. The required combustion air is injected from below the grate and into the middle of the furnace by the primary and secondary fans. The ash produced by the combustion finally falls from the end of the burnout grate to the slag remover, and after water cooling, is sent to the slag pool. To ensure that the DXN contained in the raw MSW and generated during incineration can be completely decomposed under the high-temperature combustion conditions in the furnace, the furnace combustion process must strictly control the flue gas temperature to above 850°C, the high-temperature flue gas residence time in the furnace to exceed 2 seconds, and ensure sufficient flue gas turbulence, among other process requirements.

[0010] 3) Waste Heat Boiler Stage: High-temperature flue gas (above 850°C) generated in the furnace is drawn into the waste heat boiler system by an induced draft fan. It then passes through the superheater, evaporator, and economizer. Heat exchange between the high-temperature flue gas and the liquid water in the boiler drum produces high-temperature steam, which in turn cools the high-temperature flue gas to below 200°C at the waste heat boiler outlet (i.e., flue gas G1). From the perspective of the DXN formation mechanism, when the high-temperature flue gas is cooled by the waste heat boiler, the chemical reactions that lead to DXN formation include high-temperature gas-phase synthesis reactions (800°C to 500°C), precursor synthesis (450°C to 200°C), and de novo synthesis (350°C to 250°C), but there is currently no consensus on this.

[0011] 4) Steam power generation stage: The high-temperature steam generated by the waste heat boiler is used to drive the steam turbine generator, converting mechanical energy into pure electrical energy, achieving self-sufficiency in plant-level electricity consumption and supplying surplus electricity to the grid, realizing resource utilization and obtaining economic benefits.

[0012] 5) Flue Gas Purification: Flue gas purification in the MSWI process primarily involves denitrification (NOx), desulfurization (HCl, HF, SO2, etc.), heavy metal removal (Pb, Hg, Cd, etc.), dioxin (DXN) adsorption, and dust removal (particulate matter), ultimately achieving compliance with incineration flue gas pollutant emission standards. The most widely used technology is the use of an activated carbon injection system to adsorb DXN from incineration flue gas. The adsorbed DXN is then concentrated in the fly ash.

[0013] 6) Flue Gas Emission Stage: After cooling and purification, the incineration flue gas (flue gas G2) containing trace amounts of DXN is drawn by an induced draft fan and discharged through the chimney into the atmosphere. The continuous, long-term operation of the MSWI process results in a large amount of DXN adhering to the particles on the chimney wall (known as the memory effect). The conditions under which this release is possible remain a research challenge.

[0014] At present, the research on DXN soft measurement detection for MSWI process mainly focuses on the DXN concentration detection in the emission stage (i.e. flue gas G3). The focus of this paper is to construct a soft measurement model for DXN emission at G3 flue gas. Summary of the Invention

[0015] Here we discuss the basic definition of TS fuzzy reasoning and describe the process of constructing BDT for regression.

[0016] TS fuzzy reasoning was first proposed by Takagi and Sugeno and is widely used in modeling, control and parameter identification.

[0017] For a dataset with M input features The complex industrial system, the output y is a continuous value, the modeling data set is recorded as N is the number of modeling data.

[0018] The basic definition of TS fuzzy reasoning is as follows:

[0019] For M input features x=[x1...x m ...x M ]∈R 1×M , K IF-THEN fuzzy rules are used to describe the local linear relationship, where the kth fuzzy rule is expressed as:

[0020]

[0021] Where R k The meaning is: when x1 is and... and x m for and... and x M for Time φ k =g k (x1,...,x M ); and Represent the input features x1, x m and x M The fuzzy set specified by the membership function; φ k represents the output of the kth fuzzy rule, g k (x1,...,x M ) is specifically expressed as follows:

[0022] g k (x1,...,x M )=ω1x1+ω2x2+...+ω M x M (2)

[0023] Where ω1, ω2 and ω m The 1st, 2nd and mth input features x1, x2 and x m The corresponding weight.

[0024] Therefore, based on K fuzzy rules TS fuzzy inference system f T-S (x) represents the following:

[0025]

[0026] Where, Representing fuzzy sets The fuzzy operation between them usually uses t-norms, s-norms or Cartesian product.

[0027] This paper uses the CART algorithm in binary decision tree (BDT) for regression modeling. BDT consists of a feature set (clear set) The top-down recursive segmentation dataset is constructed, and its structure is as follows Figure 2 shown.

[0028] In order to realize the top-down recursive process, the crisp set theory is applied in all non-leaf nodes. Assume that the BDT model consists of T node nodes. Therefore, the number of non-leaf nodes is T node / 2-1, and the membership function of the clear set is expressed as The t-th membership function is expressed as follows:

[0029]

[0030] Where μ CS (x i ) represents the input x i The clear membership function of δ t is the splitting node of the t-th membership function, which is determined by minimizing the mean square error (MSE). The calculation process is as follows:

[0031]

[0032] Where, Ω is the loss value; f MSE (D Left ) and f MSE (D Right ) represent the left subset D Left and right subset D Right MSE of y Left and y Right Respectively represent D Left and D Right The truth vector in ; and Respectively represent D Left and D Right The mean of the target values ​​is calculated as follows:

[0033]

[0034] Where, and D Left and D Right The number of samples in y Left,i and y Right,i y Left and y Right The i-th true value of .

[0035] Therefore, the BDT model can be expressed as:

[0036]

[0037] Where, Indicates the tth leaf The mean of leaf nodes.

[0038] DXN emission concentration modeling based on integrated T–S fuzzy regression tree

[0039] First, the structure of the TSFRT model of DXN emission concentration is introduced; then, the learning algorithm of the TSFRT model is provided; finally, the EnTSFRT model of DXN emission concentration is proposed.

[0040] 4.1 Construction of TSFRT model for DXN emission concentration

[0041] The DXN emission concentration TSFRT model includes a screening layer (clear set) and a fuzzy reasoning layer (fuzzy set), where the screening layer is used for feature screening and the fuzzy reasoning layer is used for TS fuzzy reasoning. Figure 3 shown.

[0042] In the filtering layer, the training dataset As input. First, traverse each feature value in the data set D and calculate its MSE value using formula (5). Then, the clear set is obtained by minimizing the MSE. The first membership in Therefore, the dataset D is divided into two left and right subsets as follows:

[0043]

[0044] Where, Indicates when When the left subset D Left Belongs to N Left ×M real number space, Indicates when Right subset D Right Belongs to N Right ×M real number space.

[0045] In addition, clear set The first element in (δ1=x i,m ) is determined by formula (4) and is expressed as follows:

[0046]

[0047] Repeat the above process, the DXN emission concentration TSFRT model has T node / 2-1 internal nodes. Therefore, Tnode / 2 subsets In addition, the t leaf clear set Expressed as

[0048]

[0049] The simplified form is

[0050]

[0051] Therefore, the t leaf clear set The input of TS fuzzy reasoning is expressed as follows:

[0052]

[0053] Where, is the training data of TS fuzzy inference, that is, the tth leaf leaf nodes; Indicates the tth leaf clear set Input features of y i is the i-th true value; Indicates the tth leaf The number of samples in a leaf node; t is the number of sample features.

[0054] In the fuzzy inference layer, K fuzzy rules are defined to represent the local linear relationship between the input features and the target, which is expressed as follows:

[0055]

[0056] Where R k Means: If δ1 is And... and x t for Time k =g k (x1,...,x t ).

[0057] The simplified form is

[0058]

[0059] Where R k Means: If for And... and for Time k =g k (x1,...,x t ); For the tthleaf clear set Features in for The membership function of express for degree of affiliation.

[0060] Gaussian function is used as membership function It is expressed as follows:

[0061]

[0062] Where c t,k and σ t,k Respectively The center and width.

[0063] Therefore, the k-th fuzzy rule for the t-th input feature is calculated as follows:

[0064]

[0065] Wherein, k represents the product output of the kth fuzzy rule, Represents the Cartesian product.

[0066] Based on formula (3), the output of the Cartesian product is Normalize and calculate the weight of the antecedent part as follows:

[0067]

[0068] Where, is the kth weight of the antecedent part.

[0069] Therefore, the fuzzy rule output obtained by combining the antecedent and the consequent is expressed as

[0070]

[0071] Where, is the output of the consequent of the i-th fuzzy rule.

[0072] Finally, x is calculated by linear combination of fuzzy rules i The predicted values ​​of DXN emission concentrations are as follows:

[0073]

[0074] Where, For input x i The predicted output.

[0075] Therefore, the TSFRT model of DXN emission concentration is simplified as follows:

[0076]

[0077] Where, f TSFRT (·) represents the DXN emission concentration TSFRT model; θ leaf is the minimum number of samples for the hyperparameter; ω is the consequent weight matrix; c and σ are the center and width of the membership function respectively; X is the input data; K is the number of fuzzy rules.

[0078] In most cases, prior knowledge and prefuzzification are often used to set the parameters of the fuzzy system. However, this increases the modeling burden and hinders the rapid construction of a soft-sensing model for DXN emission concentrations during the MSWI process. To address this issue, an update strategy is employed to determine the parameters of the TS fuzzy inference.

[0079] Parameter Update Learning Algorithm for TSFRT Model of DXN Emission Concentration

[0080] 4.2.1 Parameter Identification of T-S Precursor

[0081] For the DXN emission concentration TSFRT model f TSFRT (·), first define the training square error as follows:

[0082]

[0083] Where E represents the square error of all samples; X, K and θ leaf f TSFRT (·) is the input; ω, c and σ represent the parameters that need to be further identified in the modeling process.

[0084] As shown in formula (15), the parameter of the antecedent part is the center c t and width σ t In order to achieve the expected performance, these parameters are confirmed based on the training data D and updated using the gradient descent (GD) method.

[0085] 1) Sample-by-sample update

[0086] The sample-by-sample update strategy of the center c and width σ is expressed as follows:

[0087] c i+1 =c i -η c ▽c i (E i ) (twenty three)

[0088] σ i+1 =σ i -η b ▽σi (E i ) (twenty four)

[0089] Where c i+1 Update the matrix for the center of the i+1th sample, σ i+1 Update the matrix for the width of the i+1th sample, η c and η b denote the learning rates of the center and width respectively, ▽c i (E i ) and ▽σ i (E i ) represent the gradient of the center and width of the i-th sample, and the gradient of the center and width of the t-th input feature of the i-th sample ▽c i,t (E i ) and ▽σ i,t (E i ) is calculated as follows:

[0090]

[0091] Where, E i is the square error of the i-th sample; is the i-th predicted value; φ i The fuzzy rule output obtained by the combination of the antecedent and the consequent; k is the product output of the kth fuzzy rule; g i (x1,...,x t ) represents the fuzzy rule consequent output of the i-th sample; μ k (x t ) represents the kth fuzzy rule pair x t The membership degree of c i,t and σ i,t are the center and width of the t-th input feature of the ith sample respectively; e i represents the error of the i-th sample, which is expressed as follows:

[0092]

[0093] Therefore, the model is expressed as the DXN emission concentration TSFRT-I model.

[0094] 2) Batch sample update

[0095] The batch sample update strategy is based on batch GD (BGD), which can effectively reduce the training time of the TSFRT-I model of DXN emission concentration. The batch identified in Expressed as

[0096]

[0097] Where n batch is the number of samples in a batch, For the tth leaf The number of samples in a leaf node.

[0098] The center matrix c and width matrix σ in the batch The process of updating once can be expressed as follows:

[0099]

[0100] Where, and Respectively represent the center and width in the batch The BGD is calculated from a single sample.

[0101] Therefore, the model is expressed as the DXN emission concentration TSFRT-Ⅱ model.

[0102] Parameter identification of TS post-processing part

[0103] Three different methods are provided to determine the weight of TS consequents.

[0104] 1) Update sample by sample

[0105] In the TSFRT-I model of DXN emission concentration, the GD method is used to identify the center and width. Similarly, GD is used to update the consequent weights, which are expressed as follows:

[0106] ω i+1 =ω i -η w ▽ω i (E i )(31)

[0107] Where η w is the learning rate of the consequent weight; ▽ω i (E i ) represents the gradient of the posterior weight of the i-th sample, and the posterior weight of the t-th feature of the i-th sample ▽ω i,t (E i ) is calculated as follows:

[0108]

[0109] 2) Least Squares Update

[0110] Generally speaking, the least squares method is used to express the linear relationship between input and output, and (19) can be reformulated as follows:

[0111]

[0112] Where,

[0113] Given an input matrix X * And the output vector y, the weight of the TS posterior part is calculated as follows:

[0114] ω=((X * ) T X * ) -1 (X * ) T y(34)

[0115] Where, ω is t×1; X * Depend on indivual Composition, its size is (X * ) T Represents X * The transpose of .

[0116] The premise of using the least squares method to update the weights is that the antecedent part k Already obtained. Input matrix X * The i-th vector of is The i-th element of vector y is y i , the recursive calculation is as follows:

[0117]

[0118] In the formula, the initial value of ω0 is given randomly; S0 can be initialized to S0 = αI, where α is any positive number and I is the unit matrix.

[0119] The weight ω in the results section i The size of is the main difference between the sample-by-sample and least squares update methods. The weight ω of the sample-by-sample update is i The size is equal to the rule set The number of ω i The size range is [1, +∞], and the specific value is determined by the number of fuzzy rules. The least squares update has a fixed weight size ω i From formula (33), we can see that the weight ω i The size and number of fuzzy rules K are determined by the input matrix X * Therefore, the fuzzy rules updated sample by sample are the hyperparameters of the predefined DXN emission concentration TSFRT model through expert knowledge or adaptive adjustment, and the fuzzy rules updated by least squares are no longer the hyperparameters of the DXN emission concentration TSFRT model, but the coefficient matrix S i .

[0120] 3) Weight initialization based on prior knowledge

[0121] The weights are initialized by formula (5) to further utilize the prior knowledge of the screening layer. The scheme is as follows Figure 4 shown.

[0122] According to (5), (8) and (9), the MSE loss function is reformulated as follows:

[0123]

[0124] In addition, the loss value Ω is obtained as t<<(T / 2)-1 t , and then initialize the weights of the subsequent normalization parts as follows:

[0125]

[0126] Therefore, the t leaf The input of TS fuzzy inference is expressed as follows:

[0127]

[0128] Where, Denotes the initial weight ω0. Then, the final weight is obtained by recursively calculating formulas (34) and (35).

[0129] It should be noted that for the DXN emission concentration TSFRT model, various parameter update strategies are provided in the antecedent part, including sample-by-sample and BGD strategies. The consequent part uses sample-by-sample update, least squares update, and prior knowledge to initialize weights. Therefore, there are five types of DXN emission concentration TSFRT models with different antecedent and consequent identification methods, as shown below:

[0130] ●TSFRT-Ⅰ: The antecedent is updated sample by sample, the consequent is updated sample by sample, and the parameters are randomly initialized.

[0131] ●TSFRT-Ⅱ: The antecedent part is GBD update, the posterior part is least square update, and the number of samples in a batch is n batch Equal to t leaf The number of samples in a leaf node Parameters are initialized randomly.

[0132] ●TSFRT-Ⅲ: This method is the same as the TSFRT-Ⅱ model, but the consequent weights are initialized by prior knowledge.

[0133] ●TSFRT-Ⅳ: This method is the same as the TSFRT-Ⅱ model, but the number of samples in a batch is n batch Equal to t leaf The number of samples in a leaf node

[0134] ●TSFRT-V: This method is the same as the TSFRT-IV model, except that the consequent weights are initialized by prior knowledge.

[0135] The above five types of DXN emission concentration TSFRT models only differ in the updating method, which can be selected according to needs.

[0136] 4.3 Integrated TSFRT of DXN Emission Concentration (EnTSFRT)

[0137] Here, we propose an integrated modeling method for DXN emission concentration based on the TSFRT-Ⅲ model as the base learner, namely the DXN emission concentration EnTSFRT model. Figure 5 shown.

[0138] exist Figure 5 The structure of the DXN emission concentration EnTSFRT is the same as that of the normal parallel ensemble method. However, the difference between this structure and the random forest (RF) is that the bootstrap and random subspace methods are not used in EnTSFRT, and the pseudo-inverse method is used for the parallel ensemble output.

[0139] The modeling process of DXN emission concentration EnTSFRT is as follows:

[0140] First, by giving the input X∈R N×M , N and M are the number of samples and the number of features respectively. The output of the TSFRT-Ⅲ model for DXN emission concentration is converted to Represented as a j ∈R N×1 Therefore, the J DXN emission concentration TSFRT-Ⅲ model The output can be expressed as a matrix A∈R N×J .

[0141] Next, the pseudo-inverse is calculated by employing the following optimal problem to estimate the weights that minimize the training error.

[0142]

[0143] Where, is the weighted square sum constraint term, λ is the constraint term coefficient given arbitrarily in (0,1); y is the sample output.

[0144] The above optimal results are obtained by using the Moore-Penrose inverse matrix to calculate the weight matrix, as follows:

[0145] When the number J of DXN emission concentration TSFRT-Ⅲ models is greater than the number of samples N, the weight for

[0146]

[0147] When the number J of DXN emission concentration TSFRT-Ⅲ models is less than the number of samples N, the weight for

[0148]

[0149] Finally, the output of the EnTSFRT model for DXN emission concentration is

[0150] BRIEF DESCRIPTION OF THE DRAWINGS

[0151] Figure 1 Municipal solid waste incineration process flow chart

[0152] Figure 2 BDT structure diagram

[0153] Figure 3 TSFRT structure diagram

[0154] Figure 4 Weight initialization scheme based on prior knowledge

[0155] Figure 5 EnTSFRT structure diagram

[0156] Figure 6 Fitting curves of different methods DETAILED DESCRIPTION

[0157] This paper uses actual DXN data from a MSWI power plant for industrial validation. The DXN data originates from a Beijing MSWI incineration power plant. The data covers 141 sets of DXN emission concentration samples from 2009 to 2020. The true DXN value is the converted concentration after two hours of flue gas sampling and testing. After removing missing and abnormal variables during process data collection, the process variable is 116-dimensional. The mean of the process data within the current DXN true value sampling period is used as the input feature.

[0158] In this paper, we select root mean square error (RMSE), mean absolute error MAE and coefficient of determination (R 2 ) There are three evaluation indicators to compare the performance of different soft sensing methods, which are calculated as follows:

[0159]

[0160] Where y i represents the i-th true value, represents the i-th predicted value, represents the average output value, and N represents the number of samples.

[0161] The TSFRT-Ⅰ, TSFRT-Ⅱ, TSFRT-Ⅲ, TSFRT-Ⅳ, TSFRT-Ⅴ and EnTSFRT models for DXN emission concentration were compared with the TS fuzzy neural network (FNN), BDT and RF models.

[0162] During training, t-norms were used for all DXN emission concentration TSFRT and EnTSFRT models. Generally, the fuzzy inference process in FDT is highly dependent on initial conditions, particularly the initial values ​​of the center and width. In this application, the initial random number generation method for the DXN emission concentration TSFRT-I, TSFRT-II, TSFRT-III, TSFRT-IV, TSFRT-V, EnTSFRT, and FNN models was fixed, and the corresponding hyperparameters are shown in Table 1.

[0163] Table 1 Detailed information of hyperparameters

[0164]

[0165]

[0166] The experimental results are shown in Table 2 and Figure 6 shown.

[0167] Table 2 Statistical results of different methods

[0168]

[0169] As shown in Table 2 and Figure 6 The results show that: (1) the proposed DXN emission concentration TSFRT model can effectively reduce the overfitting of BDT in the training set and then improve the accuracy in the test set; (2) DXN emission concentration TSFRT-I has the longest training time among all DXN emission concentration TSFRT methods, while the training time of other DXN emission concentration TSFRT methods is lower than that of BDT methods; (3) complex machine learning methods, such as FNN, RF, and EnTSFRT, outperform single learners on the DXN dataset. Among them, the DXN emission concentration EnTSFRT model performs the best, with far fewer fuzzy rules and shorter training time compared to the FNN method.

[0170] The results show that the EnTSFRT model of DXN emission concentration proposed in this application has significant advantages and practical application potential compared with existing methods.

[0171] This paper proposes a novel EnTSFRT model for soft sensing of DXN emission concentrations during MSWI processes. This model features a top-down structure, employing a growth process for feature selection. Each leaf node performs TS fuzzy inference, updates antecedent and consequent parameters using multiple update strategies, and employs a pseudo-inverse model integration mechanism to improve model generalization performance. The proposed method significantly outperforms other methods on real-world datasets.

Claims

1. A soft-sensing method for dioxin emissions from MSWI processes based on an integrated TS fuzzy regression tree is characterized by: For a dataset with M input features The complex industrial system, the output y is a continuous value, the modeling data set is recorded as N is the number of modeling data; The basic definition of TS fuzzy reasoning is as follows: For M input features x=[x1...x m ...x M ]∈R 1×M , K IF-THEN fuzzy rules are used to describe the local linear relationship, where the kth fuzzy rule is expressed as: Where R k The meaning is: when x1 is and... and x m for and... and x M for Time φ k =g k (x1,...,x M ); and Represent the input features x1, x m and x M The fuzzy set specified by the membership function; φ k represents the output of the kth fuzzy rule, g k (x1,...,x M ) is specifically expressed as follows: g k (x1,...,x M )=ω1x1+ω2x2+...+ω M x M (2) Where ω1, ω2 and ω M The 1st, 2nd and Mth input features x1, x2 and x M The corresponding weight; Therefore, based on K fuzzy rules TS fuzzy inference system f T-S (x) represents the following: Where, Representing fuzzy sets The fuzzy operation between them uses t-norms, s-norms or Cartesian product; The CART algorithm in the binary decision tree (BDT) is used for regression modeling; BDT consists of a feature set Top-down recursive segmentation dataset construction; To implement the top-down recursive process, crisp set theory is applied in all non-leaf nodes; Assume that the BDT model consists of T node nodes; Therefore, the number of non-leaf nodes is T node / 2-1, and the membership function of the clear set is expressed as The t-th membership function is expressed as follows: Where μ CS (x i ) represents the input x i The clear membership function of δ t is the splitting node of the t-th membership function, which is determined by minimizing the mean square error. The calculation process is as follows: Where, Ω is the loss value; f MSE (D Left ) and f MSE (D Right ) represent the left subset D Left and right subset D Right MSE of y Left and y Right Respectively represent D Left and D Right The truth vector in ; and Respectively represent D Left and D Right The mean of the target values ​​is calculated as follows: Where, and D Left and D Right The number of samples in y Left,i and y Right,i y Left and y Right The i-th true value of ; Therefore, the BDT model is expressed as: Where, Indicates the tth leaf The mean of leaf nodes; DXN emission concentration modeling based on integrated T–S fuzzy regression tree First, the structure of the TSFRT model for DXN emission concentration is introduced; then, the learning algorithm of the TSFRT model is provided; finally, the EnTSFRT model for DXN emission concentration is proposed; The TSFRT model of DXN emission concentration includes a screening layer, i.e., a clear set, and a fuzzy reasoning layer, i.e., a fuzzy set. The screening layer is used for feature screening, and the fuzzy reasoning layer is used for TS fuzzy reasoning. In the filtering layer, the training dataset As input, first, traverse each feature value in the data set D and calculate its MSE value using formula (5); then, obtain the clear set by minimizing MSE. The first membership in Therefore, the dataset D is divided into two left and right subsets as follows: Where, Indicates when When the left subset D Left Belongs to N Left ×M real number space, Indicates when Right subset D Right Belongs to N Right ×M real number space; In addition, clear set The first element in δ1=x i,m Determined by formula (4), it is expressed as follows: Repeat the above process, the DXN emission concentration TSFRT model has T node / 2-1 internal nodes; therefore, T node / 2 subsets In addition, the t leaf clear set Expressed as The simplified form is Therefore, the t leaf clear set The input of TS fuzzy reasoning is expressed as follows: Where, is the training data of TS fuzzy inference, that is, the tth leaf leaf nodes; Indicates the tth leaf clear set Input features of y i is the i-th true value; Indicates the tth leaf The number of samples in a leaf node; For the tth leaf The number of sample features in the leaf nodes; in the fuzzy inference layer, K fuzzy rules are defined to represent the local linear relationship between the input features and the target, which is expressed as follows: Where, Means: If δ1 is And... and for hour The simplified form is Where, Means: If for And... and for hour For the tth leaf clear set Features in for The membership function of express for degree of affiliation; Gaussian function is used as membership function It is expressed as follows: Where, and Respectively the center and width of Therefore, the k-th fuzzy rule for the t-th input feature is calculated as follows: Wherein, k represents the product output of the kth fuzzy rule, represents the Cartesian product; Based on formula (3), the output of the Cartesian product is Normalize and calculate the weight of the antecedent part as follows: Where, is the kth weight of the antecedent part; Therefore, the fuzzy rule output obtained by combining the antecedent and the consequent is expressed as Where, is the output of the consequent of the i-th fuzzy rule; Finally, x is calculated by linear combination of fuzzy rules i The predicted values ​​of DXN emission concentrations are as follows: Where, For input x i The predicted output of Therefore, the TSFRT model of DXN emission concentration is simplified as follows: Where, f TSFRT (·) represents the DXN emission concentration TSFRT model; θ leaf is the minimum number of samples for the hyperparameter; ω is the consequent weight matrix; c and σ are the center and width of the membership function respectively; X is the input data; K is the number of fuzzy rules; Parameter Update Learning Algorithm for TSFRT Model of DXN Emission Concentration Parameter identification of TS antecedent For the DXN emission concentration TSFRT model f TSFRT (·), first define the training square error as follows: Where E represents the square error of all samples; X, K and θ leaf f TSFRT (·) is the input; ω, c and σ represent the parameters that need to be further identified in the modeling process; As shown in formula (15), the parameters of the antecedent part are centered and width In order to achieve the expected performance, these parameters are confirmed based on the training data D and updated using the gradient descent method; 1) Sample-by-sample update The sample-by-sample update strategy of the center c and width σ is expressed as follows: c i+1 =c i -η c ▽c i (AND i ) (23) s i+1 =s i -or b ▽s i (E i ) (24) Where c i+1 Update the matrix for the center of the i+1th sample, σ i+1 Update the matrix for the width of the i+1th sample, η c and η b denote the learning rates of the center and width respectively, ▽c i (E i ) and ▽σ i (E i ) represent the gradient of the center and width of the i-th sample, respectively. The gradient of the center and width of the input features and The calculation method is as follows: Where, E i is the square error of the i-th sample; is the i-th predicted value; φ i The fuzzy rule output obtained by the combination of the antecedent and the consequent; k is the product output of the kth fuzzy rule; represents the fuzzy rule consequent output of the i-th sample; represents the kth fuzzy rule pair The degree of membership; and are the first The center and width of the input features; e i represents the error of the i-th sample, which is expressed as follows: Therefore, the model is expressed as the DXN emission concentration TSFRT-I model; 2) Batch sample update The batch sample update strategy is based on batch to effectively reduce the training time of the TSFRT-I model of DXN emission concentration; from the training dataset The batch identified in Expressed as Where n batch is the number of samples in a batch, For the tth leaf The number of samples in a leaf node; The center matrix c and width matrix σ in the batch The process of updating once is as follows: Where, and Respectively represent the center and width in the batch The BGD is calculated from a single sample; Therefore, the model is expressed as the DXN emission concentration TSFRT-Ⅱ model; Parameter identification of TS post-processing part Three different methods are provided to determine the weight of the TS consequent; 1) Update sample by sample In the TSFRT-I model of DXN emission concentration, the GD method is used to identify the center and width; similarly, GD is used to update the consequent weights, which are expressed as follows: oh i+1 =ω i -or w ▽ω i (E i ) (31) Where η w is the learning rate of the consequent weight; ▽ω i (E i ) represents the gradient of the weight of the posterior of the i-th sample, and the i-th sample The consequent weight of the feature ▽ω i,t (E i ) is calculated as follows: 2) Least Squares Update Generally speaking, the least squares method is used to express the linear relationship between input and output, and (19) can be reformulated as follows: Where, Given an input matrix X * And the output vector y, the weight of the TS posterior part is calculated as follows: ω=((X * ) T X * ) -1 (X * ) T y(34) In the formula, the size of ω is X * Depend on indivual Composition, its size is (X * ) T Represents X * The transpose of The premise of using the least squares method to update the weights is that the antecedent part k Already obtained; input matrix X * The i-th vector of is The i-th element of vector y is y i , the recursive calculation is as follows: In the formula, the initial value of ω0 is randomly given; S0 is initialized to S0 = αI, where α is an arbitrary positive number and I is the unit matrix; The weight ω in the results section i The size of is the main difference between the sample-by-sample and least squares update methods; the weight ω of the sample-by-sample update i The size is equal to the rule set The number of ω i The size interval is [1, +∞], and the specific value is determined by the number of fuzzy rules; the least squares update has a fixed weight size ω i ; According to formula (33), the weight ω i The size and number of fuzzy rules K are determined by the input matrix X * Definition; Therefore, the fuzzy rules updated sample by sample are the hyperparameters of the predefined DXN emission concentration TSFRT model through expert knowledge or adaptive adjustment, and the fuzzy rules updated by least squares are no longer the hyperparameters of the DXN emission concentration TSFRT model, but the coefficient matrix S i ; 3) Weight initialization based on prior knowledge The weights are initialized by formula (5) to further utilize the prior knowledge of the screening layer; According to (5), (8) and (9), the MSE loss function is reformulated as follows: In addition, the loss value Ω is obtained as t<<(T / 2)-1 t , and then initialize the weights of the subsequent normalization parts as follows: Therefore, the t leaf The input of TS fuzzy inference is expressed as follows: Where, Denotes the initial weight ω0; then, the final weight is obtained by recursively calculating formulas (34) and (35); It should be noted that for the DXN emission concentration TSFRT model, multiple parameter update strategies, including sample-by-sample and BGD strategies, are provided in the antecedent part. The consequent part uses sample-by-sample update, least squares update, and prior knowledge to initialize the weight strategy. Therefore, there are a total of five types of DXN emission concentration TSFRT models with different antecedent and consequent identification methods, as shown below: TSFRT-I: The antecedent and the consequent are updated sample by sample, and the parameters are randomly initialized. ●TSFRT-Ⅱ: The antecedent part is GBD update, the posterior part is least square update, and the number of samples in a batch is n batch Equal to t leaf The number of samples in a leaf node Parameters are randomly initialized; TSFRT-III: This method is the same as the TSFRT-II model, but the consequent weights are initialized using prior knowledge. ●TSFRT-Ⅳ: This method is the same as the TSFRT-Ⅱ model, but the number of samples in a batch is n batch Equal to t leaf The number of samples in a leaf node TSFRT-V: This method is the same as the TSFRT-IV model, except that the consequent weights are initialized using prior knowledge. The above five types of DXN emission concentration TSFRT models differ only in the updating method, which can be selected arbitrarily according to the needs; Here, an integrated modeling method for DXN emission concentration based on the TSFRT-III model as the base learner is proposed, namely the EnTSFRT model of DXN emission concentration; The modeling process of DXN emission concentration EnTSFRT is as follows: First, by giving the input X∈R N×M , N and M are the number of samples and the number of features respectively; the output of the TSFRT-Ⅲ model of DXN emission concentration is Represented as a j ∈R N×1 Therefore, the J DXN emission concentration TSFRT-Ⅲ model The output is represented as a matrix A∈R N×J ; Next, the pseudo-inverse is calculated to estimate the weights that minimize the training error by using the following optimal problem; Where, is the weighted square sum constraint term, λ is any given constraint term coefficient in (0,1); y is the sample output; The above optimal results are obtained by using the Moore-Penrose inverse matrix to calculate the weight matrix, as follows: When the number J of DXN emission concentration TSFRT-Ⅲ models is greater than the number of samples N, the weight for When the number J of DXN emission concentration TSFRT-Ⅲ models is less than the number of samples N, the weight for Finally, the output of the EnTSFRT model for DXN emission concentration is

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