Water quality monitoring method based on feature enhancement type width learning system

Abstract

The present application proposes a water quality monitoring method based on a feature enhancement type width learning system. In view of the problems of low precision and time-consuming training of traditional models caused by the nonlinearity, time sequence and non-Gaussian characteristics of sewage treatment data. The method first preprocesses the sewage historical data, then constructs three groups of features of time sequence, non-Gaussian and enhanced nodes. The time sequence group extracts the time sequence features by using the time window and the cycle structure, the non-Gaussian group mines the non-Gaussian information through the over-complete independent component analysis (OICA), and the enhanced group expands the features through the nonlinear mapping. The output of the combined feature group is a matrix, and the output weight is determined by combining the ridge regression pseudo-inverse to complete the model training. During online monitoring, the new data is preprocessed and input into the trained model, and the COD, BOD5, SS and TN four indexes are output simultaneously. Experiments show that the model has high training efficiency and high prediction accuracy, and can meet the real-time monitoring needs of sewage treatment plants.

Description

Technical Field

[0001] This invention belongs to the field of data-driven intelligent optimization technology and proposes a water quality monitoring method based on Feature Enhancement Broad Learning System (FAEBLS). This method combines a breadth learning model (BLS) from the field of artificial intelligence to address the need for multi-indicator collaborative monitoring of four indicators in the wastewater treatment process: Chemical Oxygen Demand (COD), Five-day Biochemical Oxygen Demand (BOD5), Suspended Solids (SS), and Total Nitrogen (TN). Multiple water quality indicators are simultaneously analyzed and predicted as multiple outputs of the artificial intelligence model. Background Technology

[0002] In recent years, data-driven water quality monitoring models have played a crucial role in pollution control and environmental monitoring. However, the complexity of wastewater data processes presents numerous challenges to related water quality monitoring modeling research. First, because wastewater treatment processes involve various physical, biological, and chemical nonlinear reactions, wastewater data often exhibits significant nonlinear characteristics. Second, this process is subject to dynamic and continuous diurnal variations and seasonal fluctuations in influent flow, resulting in temporal characteristics in the collected wastewater data, meaning that data collected at different times are correlated. Finally, wastewater data also exhibits strong non-Gaussianity, manifesting as skewed, heavy-tailed, or multimodal distributions, typically caused by the superposition of multiple pollution sources, abnormal discharge events, and dynamic environmental changes. The temporal and non-Gaussian characteristics of wastewater data not only reflect the complex dynamic processes of wastewater treatment systems but also place higher demands on data-driven water quality monitoring modeling.

[0003] Currently, deep neural networks, with their powerful data interpretation capabilities, can deeply mine complex process characteristics contained in data, and have gradually become the preferred method for water quality monitoring and modeling of wastewater and other complex industrial processes. However, the training efficiency of complex models is low, and the computational cost is too high. Furthermore, when pre-trained deep neural networks are applied to wastewater treatment sites, their accuracy is affected by harsh environments, leading to a significant time expenditure for retraining. This fails to meet the dual requirements of high accuracy and low time overhead for water quality monitoring models in actual wastewater treatment plants. To address the shortcomings of deep neural networks, Chen and Liu et al. developed a single-layer incremental neural network called the Broad Learning System (BLS). [1-2]The basic idea of ​​BLS is to map data into linear feature nodes and nonlinear enhancement nodes, and then merge them into a single layer directly connected to the output layer. Unlike deep neural networks that improve model accuracy by stacking network layers, BLS expands the network horizontally by increasing the number of nodes, requiring only the training of the connection weights between this layer and the output layer. This flattened network structure allows BLS to avoid complex training processes, significantly reducing the time cost of network training. Furthermore, a trained BLS can be rapidly updated by adding nodes to improve model accuracy without retraining the network, and its general approximation ability and nonlinear feature extraction capability are theoretically guaranteed. As the above analysis shows, BLS has a flexible structure, and many researchers have improved the feature node layers of wide learning systems to extract different features from the data in order to obtain better network performance.

[0004] A breadth learning system is a single-layer incremental neural network with a flattened structure. The classic BLS network structure diagram is attached. Figure 1 As shown, this network has only one hidden layer containing two parts: a feature node group and a booster node group. The flat network structure, combined with simple matrix operations, greatly reduces the network training time. However, in BLS, each feature node group and booster node group is independent, and the network lacks the ability to learn complex features of sewage data, such as temporal and non-Gaussian features. Summary of the Invention

[0005] To address the limitation in model accuracy caused by the inability of current BLS-based water quality monitoring methods to simultaneously extract the temporal and non-Gaussian features of wastewater process variables, this invention proposes a Feature-Augmented Broad Learning System (FAEBLS) method for water quality monitoring. To enhance BLS's ability to extract temporal features, FAEBLS utilizes time windows to serialize input data, ensuring that the input of each feature node group has temporal correlation. Simultaneously, the feature node groups of BLS are cascaded, making the output at the current time step dependent on the output at the previous time step and the input at the current time step, thereby enhancing BLS's ability to extract temporal features. This study's wastewater quality indicator monitoring mainly consists of two parts: "model training" and "online monitoring." The monitoring method involves inputting 13 process variables (the specific meanings of which are shown in Table 1) S... I S S X I X BH X BA X P S O S NO S NH S ND S ALK X NDX S The data is input into the FAEBLS network in vector form and then output in one go the four water quality indicators to be monitored: Chemical Oxygen Demand (COD), Five-day Biochemical Oxygen Demand (BOD5), Suspended Solids (SS), and Total Nitrogen (TN). The specific process is shown in the attached figure. Figure 2 As shown.

[0006] Model training

[0007] Step 1: Obtain the training set X based on historical wastewater data train and label dataset Y train .

[0008] Step 2: Calculate X train The mean and standard deviation of the data are normalized.

[0009] Step 3: Construct the temporal feature set of the feature-enhanced width learning system. The preprocessed training data is scanned sequentially through a time window to obtain the input data X. P The data is then fed into the FAEBLS time-series feature set based on a loop structure, and the training data X is extracted according to the following formula. train The temporal characteristics are obtained, and the temporal characteristic node Z is obtained simultaneously. P .

[0010] The main processing methods are as follows:

[0011] First, assume the input data is , Indicates the number of samples. This represents the number of process variables in the wastewater data. The data is serialized using a time window to ensure that the input data for each feature group is time-dependent. The serialized data is represented as follows: in, This represents the length of the sequence after the time window is divided, and also the number of temporal feature groups. X t This represents the input data for each temporal feature group, where the serialized X1 is input into the first feature node group and passed through a linear activation function. After mapping, the output is represented as:

[0012] (1)

[0013] in, These are the weights and bias matrices randomly initialized according to the corresponding input dimensions, which are simplified and expressed in the subsequent formula derivation as follows: , among them All are subscripts, for easy distinction. and Use uppercase and lowercase letters, in the following text , , This is true for all of them. For the first set of input data With weights and bias matrices The activation function that performs a linear mapping after computation. and , Equivalent to a linear activation function, the general formula for a linear activation function is: The operations involved are all matrix and vector operations. When x is a single variable, it can be written as y = kx + b. When there are multiple input variables, it can be written as y = w1x1 + w2x2 + ... + b. When the multiple variables are converted into a matrix, X consists of multiple input variables stored in order, and W represents their corresponding weights. This is the biased column vector. The corresponding feature group is the first time-series feature group, and so on. This corresponds to the second time series feature group. This corresponds to the third time-series feature group. The corresponding number is FAEBLS uses a set of temporal feature nodes. To capture the correlation between adjacent time steps, FAEBLS introduces a recurrent neural network-like concept between the feature node groups, specifically the output Z from the previous time step. p-1 Integrate into the next moment X p In the input mapping. Therefore, the first FAEBLS The feature node group should not only include the current X p Feature extraction should also start from X at previous time steps. p-1 Temporal features are extracted from them, and the output of each feature node group can be expressed as shown in formula (2).

[0014] (2)

[0015] In formula (2), U is the forgetting factor, which usually takes values ​​between (0,1) and is used to attenuate historical information that is far away, similar to the forgetting gate in a recurrent neural network. The ⊙ symbol is the element-wise multiplication symbol, which is the core operation for realizing the element-wise attenuation of historical features by the forgetting factor. Through the rules of dimension matching and element-wise multiplication, it allows historical time-series features to be selectively transmitted according to the actual degree of influence, which not only preserves the key temporal correlation, but also avoids the interference of invalid historical information, providing important support for the accurate output of the subsequent linear activation function. Specifically, when p≥2, formula (2) will form a multi-layer nested structure when expanded. Taking n=3 as an example, the process can be expressed as follows:

[0016]

[0017] (3)

[0018]

[0019] By analogy with the p-th feature node group, we get a summation form from t = 1 to p:

[0020] (4)

[0021] As can be seen from formula (4), the output of each feature node group can be derived from the current time X. t Perform feature extraction, X t This is a single set of wastewater process variable data generated based on time window serialization processing. The subscript "t" represents the "indicator of the time-series feature group" (t = 1, 2, ..., p, where p is the total number of time-series feature groups, i.e., the sequence length corresponding to the time window). The original data is truncated using a time window, dividing the process variable data over p consecutive time points into p groups, each group being X. t (t from 1 to p), thus giving the data a temporal correlation. It can also be from X1 to X... t-1 The data at each time step is used for feature learning, and samples that are far apart at time steps are attenuated by a forgetting factor U to ensure that local temporal features are enhanced while extracting global temporal features. The output of the p-fold cyclic feature node group can be represented as:

[0022] (5)

[0023] Step 4: Constructing the non-Gaussian feature set of the feature-enhanced wide learning system. Since wastewater data does not follow a Gaussian distribution, relying solely on a time-series feature set based on a cyclic structure can only extract the time-series features of the input data, failing to uncover the non-Gaussian information. Independent component analysis (ICA) is a statistical method for separating multidimensional signals, aiming to decompose multiple mixed signals into independent components with non-Gaussian features. Unlike traditional linear transformation methods, ICA, based on the assumption of signal independence, can better extract the non-Gaussian features hidden in the data. The overcomplete independent component analysis algorithm OICA, as an improved algorithm of ICA, can separate more independent components than the data source dimension, thus better capturing the non-Gaussian feature information hidden in complex data. Therefore, FAEBLS, while establishing the time-series feature node set, simultaneously establishes a non-Gaussian feature node set based on OICA to improve the model's ability to learn complex process features in wastewater data. The core idea of ​​the OICA algorithm is to transform the training data X of d-dimensional process variables through a mixing matrix... Representing independent elements α (k>d) with higher-order features in k dimensions to extract non-Gaussian information can be expressed by formula (6):

[0024] (6)

[0025] In the formula, It is the transpose of X. Let i represent an independent component matrix, where the i-th independent component is... Here, n represents the number of samples. In the OICA algorithm, it is typically assumed that each independent component is a statistically independent element with significant non-Gaussianity. Each independent component captures higher-order statistical features of the original data X. Therefore, by estimating the mixture matrix D, the independent components α can be indirectly obtained, thereby extracting non-Gaussian features from the data. Specifically, obtaining the mixture matrix... The process can be divided into three main steps: 1) estimating the subspace W; 2) estimating the atoms in the subspace W. 3) Use these atoms to calculate the mixing matrix. The specific process is as follows.

[0026] First, the subspace W spanned by the atoms needs to be estimated using the generalized covariance matrix of x. The generalized covariance matrix is ​​defined as the Hessian matrix of the cumulant generating function (GCF) at some non-zero vector. It is well known that the Taylor expansion coefficients of the GCF, i.e., the cumulants of each order, provide crucial information about the distribution of the random variable. Therefore, this step aims to capture higher-order statistical information from the data x. For the input x, its cumulant generating function can be expressed as:

[0027] (7)

[0028] In the formula, Represents the mathematical expectation. It is a randomly generated non-zero vector. Specifically, the generalized covariance matrix (the Hessian matrix of the cumulant generating function CGF, i.e., the second-order partial derivative) can be expressed as:

[0029] (8)

[0030] (9)

[0031] In the formula Indicated as to Find the gradient. That is, variables The first-order gradient, which reflects the statistical anomalies of wastewater process variables, is essentially the logarithmic gradient of the data probability distribution, used to characterize the rate of change of the local probability density of data x in the direction of parameter v. To find the second gradient of the expression following the sign, we obtain x and... through formula (6). Substituting the relationships into formulas (8) and (9), we can obtain the generalized covariance matrix of the independent components α. The generalized covariance matrix of x The relationship between them:

[0032] (10)

[0033] (11)

[0034] In the formula, Furthermore, since the independent components are independent of each other, and It is a diagonal matrix, therefore, It can also be written as:

[0035] (12)

[0036] In the above formula, It is to characterize data In parameters The rate of change of local probability density in the direction is used to obtain parameters. The first-order gradient, due to x and The relationship has been defined by formula (6), and is derived from... as well as Formula (11) can be obtained. R is the set of real numbers. Represented as a The set of real numbers matrix, The number of independent components. Represents the number of variables in the wastewater process. The value is determined by the actual monitored variables. It is usually greater than 3, with no fixed range but must meet the requirement of the number of features. It is used to define the dimension of the observation data. If d is too small (such as d=1 or 2), the model will be unable to capture the complete pattern of water quality changes due to insufficient variable information. The range of values ​​must be greater than , is used to define the decomposition dimension of non-Gaussian features. This is the core difference between overcomplete independent component analysis (OICA) and traditional ICA. Traditional ICA requires k=d (number of independent components = number of observed variables), while OICA performs over-decomposition by k > d, which can separate more hidden non-Gaussian features. (12) In the formula Represents independent components α i The generalized variance. It can be seen from formulas (10)-(12) that... The information of both the mixing matrix D and the information of the mixing matrix D can be obtained from Representation. Therefore, through a sufficient number of generalized covariance matrices... The spanned subspace W can be used to estimate the corresponding atoms. Specifically, through R d Select a sufficient number of vectors v1, v2, ..., v s Thus, the same number of matrices are obtained. (s is a variable, determined by the quantity). The subspace W consists of these s matrices. Zhang Cheng can be represented as a set of matrices:

[0037] (13)

[0038] Based on the above process, an approximate estimate of the subspace W can be obtained. From formula (12), it can be seen that the atoms... It is the generalized covariance matrix These atoms are also components of [the material], therefore they are also made of [the material]. The elements of the subspace W spanned by atoms. Clearly, atoms... W is a rank-1 matrix. Therefore, an atom can be estimated from the subspace W using a semidefinite programming (SDP) algorithm. The optimization process of SDP is summarized below:

[0039] (14)

[0040] In the formula, It is the optimal solution of a semidefinite programming problem under strict constraints, which directly satisfies the strict requirements. constraint set In the middle, maximize the inner product This is a standard semidefinite programming (SDP) problem. The solution strictly satisfies all constraints. The "*" in the upper right corner represents the optimal solution that makes the objective function reach its extreme value (maximum value in this article). It is a standard symbol in optimization theory. The only difference is in the way it is expressed. The core is to restrict the range of values ​​of B to ensure the rationality of the solution. It is a randomly generated matrix of rank 1, representing the direction of atomic contraction. and It is the set of constraints for the SDP optimization problem, and the two are equivalent. In this set, This indicates that B must belong to the previously estimated subspace W (given by the generalized covariance matrix). Zhang Cheng), to ensure that B is correlated with the statistical characteristics of wastewater data; This indicates that the trace (sum of diagonal elements) of matrix B equals 1. It's a normalization constraint method to prevent the value of B from being too large or too small, ensuring that different atoms ( The uniformity of the scale facilitates the subsequent splicing of the hybrid matrix D. This means that B is a positive semi-definite matrix (all eigenvalues ​​are non-negative), because B is essentially a vector outer product (…). The outer product of any vector is a positive semi-definite matrix. This constraint ensures that the physical meaning of B conforms to the characteristics of the outer product of vectors. Add constraints to this set It represents the basis vectors of the orthogonal complement of B and subspace W. orthogonal ( (representing the inner product), F j It is the orthogonal complement of subspace W, equivalent to (because Zhang Cheng W's orthogonal complement, and Orthogonal means belonging to W). The constraint B∈W is equivalent to B being orthogonal to the orthogonal complement of W, where m represents the dimension of the entire space spanned by the subspace W and its orthogonal complement. Other constraints ( )and Completely identical, therefore and These are two representations of the same set of constraints, namely... ( Equivalent to Equivalence is to ensure that B is a normalized positive semi-definite matrix in the subspace W.

[0041] Furthermore, due to the limited sample size and noise, the subspace W obtained above is only an approximate representation. To accommodate this incomplete or inaccurate subspace estimation, OICA introduces a relaxation factor in SDP. The modified optimization process is as follows:

[0042] (15)

[0043] In the formula, It is an approximate optimal solution with a penalty term, achieved by introducing a penalty term. Strict constraints will be imposed. Transforming it into a soft constraint allows for temporary deviations from the null space constraint during the optimization process, ultimately forcing the solution to converge through a penalty term. This is an approximate optimization problem with regularization. Here, μ > 0 is the regularization parameter. In overcomplete independent component analysis (OICA) scenarios like the non-Gaussian feature extraction of wastewater in this paper, the typical empirical range of μ is: [1,10000], needs to be screened through cross-validation to make The steps to estimate μ with the highest accuracy are as follows:

[0044] (1) Validation set: A portion of the data from the samples in subspace W is used as the validation set (not participating in the validation process). (Optimization)

[0045] (2) Define a candidate μ set: Select several candidate values ​​within the empirical range (e.g.) = 1, 10, 100, 1000, 10000).

[0046] (3) Calculate the evaluation index corresponding to each μ: For each candidate μ, optimize to obtain and calculate With real atoms Similarity (here, similarity can be...) and The larger the value of the inner product (the product of the two elements), the higher the similarity.

[0047] (4) Select the optimal μ: Select the μ that makes the highest similarity as the final value.

[0048] At this point, the formula can be solved using a fast iterative shrinkage threshold algorithm and the principal component maximization principle to obtain an estimate of a single optimal atom. In the above process, an optimal atom is first estimated using the SDP algorithm with a relaxation factor. To estimate all atoms, FAEBLS uses a novel semi-adaptive shrinkage algorithm. Specifically, FAEBLS first iteratively adjusts the shrinkage direction G in SDP to initially estimate l atoms. Then, cluster analysis is performed on these l atoms to select k optimal atoms (here, k has the same meaning as k above, referring to the number of independent components, which is also the final number of atoms needed; the value logic is completely consistent with the previous text, requiring l>k here). Next, within the existing clustering results, the similarity matrix between each atom is calculated to select the atom with the lowest similarity to other atoms (in practice, each atom is expanded into a column vector for calculation). After all selections are completed, if the total number of selected atoms... If the number is less than the preset number k, a semi-adaptive shrinkage algorithm is used to estimate the remaining atoms, and the process is repeated. The adaptive shrinkage algorithm works as follows: first, the currently selected optimal atom is removed from the subspace W, the constraint set is updated, and then SDP is used again to estimate the remaining optimal atoms.

[0049] Based on the above steps, FAEBLS can extract k atoms. In the initial assumptions of OICA, the mixing matrix D is represented as... Therefore, by obtaining d i Thus, the mixture matrix D can be constructed. Clearly, the matrix... It is a symmetric matrix with rank 1 and only one non-zero eigenvalue. After performing eigenvalue decomposition on this matrix, its non-zero eigenvalue and its corresponding eigenvector d can be directly obtained. iTherefore, by performing eigenvalue decomposition on each atom and extracting its largest eigenvector, the column vectors d of the mixture matrix D can be obtained. i .

[0050] Therefore, the preprocessed training data is then input into the non-Gaussian feature node group of FAEBLS, and X is calculated using formulas (6) to (15). train The mixture matrix D. After obtaining the mixture matrix D, the non-Gaussian features G of the data. N The extraction process is as follows:

[0051] (16)

[0052] In the formula, This represents the non-Gaussian features extracted from the data. It is about finding the pseudo-inverse of matrix D. It is the standard notation for the generalized inverse in matrix theory, also known as the "pseudo-inverse," and its core function is to extract information from observed data. Separate independent components (i.e., non-Gaussian features) from the data. Thus, training data X is obtained based on formula (16). train The independent element representation is used to obtain the non-Gaussian feature node G. N .and It is a regularization parameter. It is an identity matrix (a matrix with 1s on the main diagonal), and the core function of both is to ensure and Same dimensions, and solve Singular value phenomena may occur (matrix is ​​not invertible).

[0053] Step 5: Construct the enhancement node group of the feature-enhanced width learning system. Map the complex process features of water quality data obtained from the FAEBLS temporal feature group with cyclic structure and the non-Gaussian feature group to the enhancement node group for nonlinear expansion to extract the nonlinear features of the wastewater data. Specifically, the j-th enhancement window H j The calculation formula is as follows:

[0054] (17)

[0055] in, The meaning is the original features and The intermediate features after nonlinear mapping serve as the basis for subsequent generation of enhanced nodes. Provides a foundation. Enhances the window's W... hj and β hjIt consists of orthogonally normalized random weights and biases between temporal feature nodes, non-Gaussian feature nodes, and the j-th group of enhancement nodes. This is achieved through... This transformation yields the total output H of the enhanced node group. J Here In the transformation This refers to the number of transformations of the enhanced node group, i.e., the generation. Group Enhancement Nodes The number of iterations for feature enhancement is a scalar. The formula is as follows:

[0056] (18)

[0057] Based on formulas (17) and (18), the outputs of the temporal and non-Gaussian feature node groups are mapped through a nonlinear activation function to obtain the enhanced node group H. J This allows for further extraction of the nonlinear features of wastewater data. In summary, the FAEBLS network constructed based on temporal feature node groups, non-Gaussian feature node groups, and enhancement node groups can be represented as:

[0058] (19)

[0059] The outputs of each feature node group obtained in steps 3 to 5 are merged to form matrix A, which is then connected to the output layer. The connection weights from the feature node group to the output layer are then calculated according to formulas (20) and (22). To simplify the representation, let Introducing equation (19), the output layer of FAEBLS can be further expressed as: .in, These are the connection weights from the hidden layer to the output layer, which are composed of temporal feature node groups, non-Gaussian feature node groups, and enhancement node groups. Considering that A might be an ill-conditioned matrix, direct solution may present problems. Therefore, this problem is transformed into an objective function optimization problem:

[0060] (20)

[0061] in, These are labels used during training. In essence, they are the actual values ​​of water quality indicators to be monitored during the wastewater treatment process, such as biochemical oxygen demand (BOD5) and chemical oxygen demand (COD) at the effluent end.

[0062] and The norm regularization constraint is a specific type, and λ² represents the coefficient of the regularization constraint term. Its main function is to suppress model overfitting by adding a weight penalty term to the loss function. This constrains the size of the model parameters to avoid overfitting the model to noise in the training data. The value needs to be adjusted on the training set through "cross-validation". The core principle is to find the value that minimizes the error on the validation set. Value, set within a range of approximately 10 -6 Up to 10 -1 Since the noise level of wastewater treatment data is moderate, excessive regularization is usually unnecessary. Let This transforms the problem into a standard L2 norm regularization problem. Simultaneously, to prevent overfitting, λ2 is applied to the weights W. J The square of the factorial imposes additional constraints. The output layer weights W are quickly computed using the Moore-Penrose generalized inverse matrix. J Approximate solution:

[0063] (twenty one)

[0064] In the formula It is the identity matrix. It is to let Guarantee and Dimension matching. Theoretically, if the regularization constraint term approaches zero infinitely, the inverse problem transforms into a least squares problem, which can be expressed by the following formula:

[0065] (twenty two)

[0066] As shown in Equation (22), unlike traditional deep neural networks that optimize network parameters layer by layer through iteration and gradient updates, the FAEBLS model uses ridge regression to calculate the pseudo-inverse to determine the network connection weights in one step. Since the pseudo-inverse calculation has low time complexity, FAEBLS only needs to calculate the network parameters once for a given combination of node parameters, significantly improving the model's learning speed and effectively avoiding problems such as gradient vanishing, gradient exploding, and slow convergence. This advantage makes FAEBLS highly efficient in network training and online updates. The network structure diagram of FAEBLS is attached. Figure 3 As shown.

[0067] Step 6: Obtain the output weights of each feature group. Then, the model training is complete. The training output of the model can be obtained through formula (19).

[0068] Online monitoring

[0069] For the newly collected online process data X Online This data contains 13 process variables S. I S S X I XBH X BA X P S O S NO S NH S ND S ALK X ND X S The specific meanings of these 13 process variables are shown in Table 1. Using the FAEBLS model that has already been trained during the network training phase, the online monitoring result Y is calculated. Online The specific process for the online application phase is as follows.

[0070] Step 7: Process the newly collected online process data X Online To perform data preprocessing operations.

[0071] Step 8: Transfer the preprocessed online data X Online The input is fed into the FAEBLS model established during the training phase, and the output of the temporal feature groups is obtained respectively. Non-Gaussian characteristic group Output and enhanced node group output .

[0072] Step 9: Proceed as described in Step 5, and merge the outputs of each feature node group to form matrix A. Online Then, based on the weights from the feature layer to the output layer obtained during the training phase... According to the expression Calculate the final output Y Online The output includes four indicators that need to be monitored: COD, BOD5, SS, and TN. The simultaneous output of these four indicators allows for rapid monitoring. Attached Figure Description

[0073] Figure 1 This is a structural diagram of the BLS network in this study;

[0074] Figure 2 Modeling flowchart based on FAEBLS water quality monitoring model;

[0075] Figure 3 The FAEBLS network structure diagram proposed in this study;

[0076] Figure 4 BSM1-based wastewater treatment process flow diagram;

[0077] Figure 5 Changes in inflow rate under different weather conditions using the BSM1 simulation platform; Detailed Implementation

[0078] (1) Introduction to the wastewater treatment process based on BSM1

[0079] In 2001, the International Water Quality Association (IWA) and the European Union's scientific and technological cooperation jointly developed the Benchmark Simulation Model No. 1 (BSM1) platform for wastewater treatment processes. Based on a mathematical model of activated sludge reactions, this platform constructs wastewater influent load data applicable to different weather conditions, thus providing a standardized experimental environment for simulation studies of wastewater treatment processes. Specifically, when designing influent load data, BSM1 considers the fluctuations in influent flow under different weather conditions, providing wastewater flow simulations for three weather environments: sunny days, rainy days, and heavy rain days.

[0080] Furthermore, in terms of the specific structure of the treatment process, the BSM1 platform includes one primary sedimentation tank, five activated sludge reaction tanks (including two anaerobic tanks and three aeration tanks), and one secondary sedimentation tank. The anaerobic tank in the activated sludge biological reaction tank is 1000 m³, the aeration tank is 1333 m³, and the secondary sedimentation tank has 10 layers, each 4 m deep, with a total volume of 6000 m³. During the wastewater treatment process, the BSM1 simulation platform employs both internal and external recirculation reaction methods, with a daily wastewater treatment capacity of 20,000 m³ / d, and a chemical oxygen demand (COD) of 300 mg / L for biodegradable wastewater. (Appendix) Figure 4 The wastewater treatment process based on BSM1 is shown in detail.

[0081] (2) Experimental data acquisition and preprocessing

[0082] Based on three types of inflow rate data, BSM1 can simulate the operational status of the system over 14 days under three weather conditions: sunny, rainy, and heavy rain. The data sampling period for each weather condition is 15 minutes, with 96 samples taken per day, totaling 1344 samples over 14 days. The specific inflow rate changes under each weather condition are attached. Figure 5 As shown in the figure, the influent flow rate was the same for the first 7 days under various weather conditions. Rainy days added approximately four days of rainfall from the eighth day on top of the sunny days, while heavy rain days introduced two additional heavy rain events on top of the rainy days, causing a surge in process data and significant changes in process water quality parameters. The wastewater treatment process was in an unstable operating state overall. Therefore, this section's experiment selected 1152 sets of samples from the first 12 days as training data and 192 sets from the last two days as test data. Following initial screening using the BSM1 guidelines, 13 process variables were selected to validate the performance of the method and model; the specific variables are shown in Table 1.

[0083] Table 1

[0084]

[0085] The specific application of the FAEBLS model proposed in this study to actual wastewater treatment processes is described below:

[0086] A. FAEBLS network preprocessing and training phases in wastewater treatment:

[0087] Step 1: First, preprocess the sampling data collected from the wastewater treatment plant.

[0088] Step 2: Construct the temporal feature set of the feature-enhanced width learning system. The preprocessed training data is scanned sequentially through a time window to obtain the input data X. P The data is then fed into the FAEBLS-based temporal feature set, and the temporal features of the training data are extracted according to the following formula, thus obtaining the temporal feature node Z. P .

[0089] Step 3: At the same time, the preprocessed training data is input into the non-Gaussian feature node group of FAEBLS, and the mixing matrix D is added.

[0090] Step 4: After obtaining the mixture matrix D, obtain the independent element representations of the training data, thereby obtaining the non-Gaussian feature nodes G. N .

[0091] Step 5: Map the outputs of the temporal and non-Gaussian feature node groups through a non-linear activation function to obtain the enhanced node group H. J This is to further extract the nonlinear characteristics of wastewater data.

[0092] Step 6: Combine the outputs of each feature node group obtained in steps 3 to 5 to form matrix A, and connect it to the output layer to calculate the connection weights from the feature node group to the output layer. .

[0093] Step 7: After obtaining the output weights of each feature group, calculate the training parameters of the model. Output training. After confirming that four output quantities—COD, BOD5, SS, and TN—could be output, the necessary network parameters for establishing the FAEBLS model were determined through 50 repeated experiments. This model was then applied to the online wastewater treatment process.

[0094] B. Online application training phase for wastewater treatment processes based on FAEBLS networks:

[0095] Step 8: First, preprocess the sampling data collected from the BSM1 simulation platform.

[0096] Step 9: In the online monitoring phase, the wastewater treatment output results obtained using one-heat coding were used. To comprehensively evaluate the predictive performance of the water quality monitoring model in the wastewater treatment process, the root mean square error (RMSE) and mean absolute error (MAE) were selected as key indicators to measure the deviation between the model's predicted values ​​and the actual values. RMSE, calculated using squared error, is more sensitive to larger errors and can highlight the impact of outliers, making it suitable for strict monitoring and control of key pollutant indicators. MAE, calculated using absolute error, can more stably reflect the overall error level and is suitable for measuring the model's overall predictive ability and robustness. The calculation formulas for the above indicators are shown below:

[0097] (25)

[0098] (26)

[0099] Where n is the size of the dataset samples, Y represents the predicted value, and Y represents the actual value. Furthermore, the training time cost of the water quality monitoring model is also an important evaluation indicator, directly related to the model's feasibility and real-time performance in practical applications. Therefore, the BSM1 experiment uses RMSE as the main error indicator to evaluate the model's ability to reduce large errors. Simultaneously, MAE is used to measure the overall error level to ensure the model maintains good robustness under different water quality conditions. Through a comprehensive evaluation of RMSE, MAE, and training time cost, the predictive performance of the FAEBLS model in wastewater treatment water quality monitoring can be analyzed more accurately. Table 2 shows the evaluation indicators of the four outputs after the entire monitoring process is completed. Overall, the FAEBLS network shows excellent prediction results for the four indicators SS, TN, COD, and BOD5 in wastewater, demonstrating certain effectiveness in water quality monitoring.

[0100] Table 2

[0101]

[0102] Step 10: Furthermore, to evaluate the superiority of the FAEBLS network in water quality monitoring, the evaluation index of one output, BOD5, was selected for evaluation on rainy and sunny days and compared with the BOD5 evaluation index of other networks. BOD5 is a key parameter for evaluating wastewater treatment effectiveness. This parameter provides insight into the concentration of biodegradable organic pollutants present in wastewater. However, current BOD5 evaluation mainly relies on time-consuming manual laboratory testing. Therefore, this study's BSM1 experiment focused on effluent BOD5 to validate the effectiveness of the proposed model. Table 3 shows the RMSE and MAE values ​​of the six models, as well as the average training time cost. This data was used for the analysis of FAEBLS's water quality monitoring results.

[0103] Among them, the SAE model, as a representative of traditional deep neural networks, can extract the nonlinear features of wastewater data, but it fails to effectively model its complex temporal dependencies and non-Gaussian distribution characteristics, resulting in limited prediction accuracy. In all three test environments, SAE had the highest RMSE and MAE, indicating not only a large prediction bias but also a high overall error level. Compared to FAEBLS, RBLS, OBLS, and LSTM models, which can capture richer features, SAE's performance disadvantage is particularly obvious. Although the prediction accuracy difference between BLS and SAE is not significant, BLS, with its flat structure and fewer parameters, significantly reduces model training time, making it more computationally efficient in practical applications. This further highlights the limitations of SAE in modeling complex wastewater data. The traditional deep neural network LSTM, similar to SAE, significantly improves prediction accuracy by simultaneously capturing nonlinear features and temporal dependencies, outperforming SAE in all test environments. Furthermore, as shown in Table 3, the RMSE and MAE results indicate that RBLS significantly outperforms BLS. This is mainly due to RBLS's improvement over BLS. Its feature extraction group is optimized from a static structure to a cyclic structure, enabling it to effectively capture the temporal features of wastewater data, thereby improving the prediction accuracy of water quality indicators. It is worth noting that the comparison between RBLS and BLS shows that BLS, which relies solely on static extraction, has limitations when handling complex wastewater data. The comparison between SAE and LSTM further verifies the crucial role of temporal features in improving the model's predictive ability. Similarly, under sunny and rainy conditions, OBLS's prediction accuracy is better than BLS, and FAEBLS's performance is also significantly better than RBLS. This is because, compared to BLS and RBLS, OBLS and FAEBLS additionally utilize the OICA method to extract non-Gaussian features from wastewater data, thus significantly improving the prediction accuracy and reliability of water quality monitoring models under complex environments.

[0104] Table 3

[0105]

[0106] References

[0107] [1] Chen C L P, Liu Z. Broad learning system: An effective and efficient incremental learning system without the need for deep architecture[J]. IEEE transactions on neural networks and learning systems, 2017, 29(1):10-24.

[0108] [2] Gong X, Zhang T, Chen C L P, et al. Research review for broad learning system: Algorithms, theory, and applications[J]. IEEE Transactions on Cybernetics, 2021, 52(9): 8922-8950.

[0109] [3] J. Zhou, X. Wang, C. Yang, and W. Xiong, “A novel soft sensor modeling approach based on difference-LSTM for complex industrial process,” IEEE Trans. Ind. Informat., vol. 18, no. 5, pp. 2955–2964, May 2022.

[0110] [4] P. Chang, S. Zhang, and Z. Wang, “Soft sensor of the key effluent index in the municipal wastewater treatment process based on transformer,” IEEE Trans. Ind. Inform., vol. 20, no. 3, pp. 4021–4028, Mar. 2024.

Claims

1. A water quality monitoring method based on a feature-enhanced width learning system, which mainly divides wastewater quality index monitoring into two parts: "model training" and "online monitoring," is characterized by... The monitoring method involves using 13 process variables S I S S X I X BH X BA X P S O S NO S NH S ND S ALK X ND X S The data is input into the FAEBLS network in vector form and then output in one go the four water quality indicators that need to be monitored: chemical oxygen demand (COD), five-day biochemical oxygen demand (BOD5), suspended solids (SS), and total nitrogen (TN). Model training Step 1: Obtain the training set X based on historical wastewater data train and label dataset Y train ; Step 2: Calculate X train The mean and standard deviation of the data were normalized. Step 3: Construct the temporal feature set of the feature-enhanced width learning system. The preprocessed training data is scanned sequentially through a time window to obtain the input data X. P The data is then fed into the FAEBLS time-series feature set based on a loop structure, and the training data X is extracted according to the following formula. train The temporal characteristics are obtained, and the temporal characteristic node Z is obtained simultaneously. P ; The main processing methods are as follows: First, assume the input data is , Indicates the number of samples. This represents the number of process variables in the wastewater data; the data is serialized using a time window to ensure that the input data for each feature group is time-dependent. The serialized data is represented as follows: in, X represents the sequence length after the time window is divided, and is also the number of temporal feature groups; t This represents the input data for each temporal feature group, where the serialized X1 is input into the first feature node group and passed through a linear activation function. After mapping, the output is represented as: （1） in, These are the weights and bias matrices randomly initialized according to the corresponding input dimensions, which are simplified and expressed in the subsequent formula derivation as follows: , among them All are subscripts, for easy distinction. and Use uppercase and lowercase letters, in the following text , , This is true for all of them; For the first set of input data With weights and bias matrices The activation function that performs a linear mapping after computation. and , Equivalent to a linear activation function, the general formula for a linear activation function is: The operations involved are all matrix and vector operations. When x is a single variable, it can be written as y = kx + b. When there are multiple input variables, it can be written as y = w1x1 + w2x2 + ... + b. When the multiple variables are converted into a matrix, X consists of multiple input variables stored in order, and W represents their corresponding weights. It is then the biased column vector; The corresponding feature group is the first time-series feature group, and so on. This corresponds to the second time series feature group. This corresponds to the third time-series feature group. The corresponding number is FAEBLS uses a set of temporal feature groups. To capture the correlation between adjacent time steps, it introduces a recurrent neural network-like concept between the feature node groups, that is, the output Z of the previous time step. p-1 Integrate into the next moment X p In the input mapping; therefore, the first FAEBLS The feature node group should not only include the current X p Feature extraction should also start from X at previous time steps. p-1 Temporal features are extracted from them, and the output of each feature node group is represented as shown in formula (2). （2） In formula (2), U is the forgetting factor, which usually takes values ​​between (0,1) and is used to attenuate historical information from distant locations, similar to the forgetting gate in a recurrent neural network; the ⊙ symbol is the element-wise multiplication symbol. When p≥2, formula (2) will form a multi-layered nested structure when expanded; when n=3, it is expressed as follows: ； （3） ； By analogy with the p-th feature node group, we get a summation form from t = 1 to p: （4） As can be seen from formula (4), the output of each feature node group not only changes from the current time X t Perform feature extraction, X t This is a single set of wastewater process variable data generated based on time window serialization processing. The subscript "t" represents the "indicator of the time-series feature group", t = 1, 2, ..., p, where p is the total number of time-series feature groups, i.e., the sequence length corresponding to the time window. The original data is truncated using a time window, dividing the process variable data over p consecutive time points into p groups, each group being X. t (t from 1 to p), thus giving the data temporal correlation; also from X1 to X t-1 The data at time t is used for feature learning, and samples that are far apart at time t are decayed using a forgetting factor U. The output of the p-fold recurrent feature node group is represented as: （5） Step 4: While establishing the temporal feature node group, a non-Gaussian feature node group based on OICA was established in parallel; the core idea of ​​the OICA algorithm is to transform the training data X of the d-dimensional process variable through a mixing matrix. The independent elements α with higher-order features in k dimensions, k>d, are represented to extract non-Gaussian information. This process is expressed by formula (6): （6） In the formula, It is the transpose of X; Let i represent an independent component matrix, where the i-th independent component is... where n is the number of samples; By estimating the mixture matrix D, the independent components α are indirectly obtained, thereby extracting non-Gaussian features from the data; whereby obtaining the mixture matrix... The process consists of three steps: 1) estimating the subspace W; 2) estimating the atoms in the subspace W. 3) Use these atoms to calculate the mixing matrix; the specific process is as follows; First, the subspace W spanned by the atoms needs to be estimated using the generalized covariance matrix of x; the generalized covariance matrix is ​​defined as the Hessian matrix of the cumulant generating function GCF at some non-zero vector; for the input x, its cumulant generating function is expressed as: （7） In the formula, Represents the mathematical expectation. It is a randomly generated non-zero vector; the generalized covariance matrix is ​​represented as: （8） （9） In the formula Indicated as to Find the gradient. That is, variables The first-order gradient of x is the logarithmic gradient of the data probability distribution, used to characterize the rate of change of the local probability density of data x along the direction of parameter v. To find the second gradient of the expression following the sign, we obtain x and... through formula (6). Substituting the relationships into formulas (8) and (9), we obtain the generalized covariance matrix of the independent components α. The generalized covariance matrix of x The relationship between them: （10） （11） In the formula, Furthermore, since the independent components are independent of each other, and It is a diagonal matrix, therefore, It was also written as: （12） In the above formula, It is to characterize data In parameters The rate of change of local probability density in the direction is used to obtain parameters. The first-order gradient, due to x and The relationship has been defined by formula (6), and is derived from... as well as Formula (11) can be obtained; R is the set of real numbers. Represented as a The set of real numbers matrix, The number of independent components. Represents the number of variables in the wastewater process. The value is determined by the actual monitored variables. Greater than 3 The range of values ​​must be greater than , used to define the decomposition dimension of non-Gaussian features, in formula (12) Represents independent components α i Generalized variance; through R d Select vectors v1, v2, ..., v s Thus, the same number of matrices are obtained. The subspace W consists of these s matrices. Zhang Cheng, represented as a set of matrices: （13） A semidefinite programming (SDP) algorithm is used to estimate an atom from the subspace W; the optimization process of SDP is summarized below: （14） In the formula, It is the optimal solution of a semidefinite programming problem under strict constraints, which directly satisfies the strict requirements. constraint set In the middle, maximize the inner product This is a standard semidefinite programming (SDP) problem. The solution strictly satisfies all constraints. The "*" in the upper right corner represents the optimal solution that maximizes the objective function. It is a randomly generated matrix of rank 1, representing the direction of atomic contraction. and It is the set of constraints for the SDP optimization problem, and the two are equivalent; In this set, This indicates that B must belong to the previously estimated subspace W, determined by the generalized covariance matrix. Zhang Cheng; This means that the trace of matrix B, i.e., the sum of its diagonal elements, is equal to 1. This means that B is a positive semi-definite matrix and all its eigenvalues ​​are non-negative; Add constraints to this set It represents the basis vectors of the orthogonal complement of B and subspace W. Orthogonal F represents the inner product. j It is the orthogonal complement of subspace W, equivalent to The constraint B∈W is equivalent to B being orthogonal to the orthogonal complement of W, where m represents the dimension of the entire space spanned by the subspace W and its orthogonal complement; the remaining constraints and Completely identical, therefore and These are two representations of the same set of constraints, namely... Right now Equivalent to ; The subspace W obtained above is only an approximate representation; to accommodate this incomplete or inaccurate subspace estimation, OICA introduces a relaxation factor in SDP; the modified optimization process is as follows: （15） In the formula, It is an approximate optimal solution with a penalty term, achieved by introducing a penalty term. Strict constraints will be imposed. Transforming it into a soft constraint allows for temporary deviations from the null space constraint during the optimization process, ultimately forcing the solution to converge through a penalty term. This is a regularized approximate optimization problem; where μ > 0 is the regularization parameter, and in overcomplete independent component analysis (OICA) scenarios like the non-Gaussian feature extraction of wastewater in this paper, the range of μ is: [1,10000], needs to be screened through cross-validation to make The steps to estimate μ with the highest accuracy are as follows: (1) Divide the validation set: Divide a portion of the data from the samples in the subspace W as the validation set; (2) Define a candidate μ set: select several candidate values ​​within the empirical range; (3) Calculate the evaluation index corresponding to each μ: For each candidate μ, optimize to obtain and calculate With real atoms The similarity is , the similarity is and The larger the value of the inner product, the higher the similarity. (4) Select the optimal μ: Select the μ that maximizes the similarity as the final value; At this point, the fast iterative shrinkage threshold algorithm and the principal component maximization principle are used to solve the formula, thereby obtaining an estimate of a single optimal atom. In the above process, the SDP algorithm with a relaxation factor is first used to estimate an optimal atom. To estimate all atoms, FAEBLS uses a new semi-adaptive shrinkage algorithm. Specifically, FAEBLS first iteratively adjusts the shrinkage direction G in SDP to initially estimate l atoms. Then, cluster analysis is performed on these l atoms to select k optimal atoms (here, k has the same meaning as k above, both referring to the number of independent components, which is also the final number of atoms required, and the value logic is completely consistent with the previous text, requiring l>k here). Next, within the existing clustering results, the similarity matrix between each atom is calculated to select the one with the lowest similarity to other atoms. After all selections are completed, if the total number of selected atoms is... If the number is less than the preset number k, a semi-adaptive shrinkage algorithm is used to estimate the remaining atoms, and the process is repeated. The adaptive shrinkage algorithm works as follows: first, the currently selected optimal atom is removed from the subspace W, the constraint set is updated, and then SDP is used again to estimate the remaining optimal atom. Based on the above steps, FAEBLS can extract k atoms. In the initial assumptions of OICA, the mixing matrix D is represented as... Therefore, by obtaining d i Thus, the mixture matrix D can be constructed; obviously, the matrix It is a symmetric matrix with rank 1 and only one non-zero eigenvalue; by performing eigenvalue decomposition on this matrix, its non-zero eigenvalue and its corresponding eigenvector d can be directly obtained. i Therefore, by performing eigenvalue decomposition on each atom and extracting its largest eigenvector, the column vectors d of the mixture matrix D can be obtained. i ; Therefore, the preprocessed training data is then input into the non-Gaussian feature node group of FAEBLS, and X is calculated using formulas (6) to (15). train The mixture matrix D; after obtaining the mixture matrix D, the non-Gaussian features G of the data. N The extraction process is as follows: （16） In the formula, This represents the non-Gaussian features extracted from the data; It is about finding the pseudo-inverse of matrix D. It is the standard notation for the generalized inverse in matrix theory, also known as the "pseudo-inverse," and is used to obtain training data X based on formula (16). train The independent element representation is used to obtain the non-Gaussian feature node G. N ;and It is a regularization parameter. It is an identity matrix, that is, a matrix whose main diagonal elements are all 1s; Step 5: Construct the augmentation node group of the feature-enhanced width learning system. Specifically, the j-th augmentation window H j The calculation formula is as follows: （17） in, The meaning is the original features and The intermediate features after nonlinear mapping serve as the basis for subsequent generation of enhanced nodes. Provides a foundation; enhances the window's W. hj and β hj It consists of orthogonally normalized random weights and biases between temporal feature nodes, non-Gaussian feature nodes, and the j-th group of enhancement nodes; through... This transformation yields the total output H of the enhanced node group. J ; here In the transformation This refers to the number of transformations of the enhanced node group, i.e., the generation. Group Enhancement Nodes The number of iterations for feature enhancement is a scalar; the formula is as follows: （18） Based on formulas (17) and (18), the outputs of the temporal and non-Gaussian feature node groups are mapped through a nonlinear activation function to obtain the enhanced node group H. J This is to further extract the nonlinear features of wastewater data. In summary, the FAEBLS network constructed based on temporal feature node groups, non-Gaussian feature node groups, and enhancement node groups is represented as follows: （19） The outputs of each feature node group obtained in steps 3 to 5 are merged to form matrix A, which is then connected to the output layer. The connection weights from the feature node group to the output layer are then calculated according to formulas (20) and (22). To simplify the representation, let Introducing equation (19), the output layer of FAEBLS is further expressed as: ;in, These are the connection weights from the hidden layer to the output layer, which are composed of the temporal feature node group, the non-Gaussian feature node group, and the enhancement node group. Considering that A may be an ill-conditioned matrix, direct solution may have problems; therefore, the problem is transformed into an objective function optimization problem: （20） in, These are labels used during training. In essence, they are the actual values ​​of water quality indicators to be monitored during the wastewater treatment process, including biochemical oxygen demand (BOD5) and chemical oxygen demand (COD) at the effluent end. and The norm regularization constraint represents a specific type, and λ² represents the coefficient of the regularization constraint term. Its main function is to suppress model overfitting by adding a weight penalty term to the loss function. This constrains the size of the model parameters to avoid overfitting the model to noise in the training data. The value needs to be adjusted on the training set through "cross-validation". The core principle is to find the value that minimizes the error on the validation set. Value, set range 10 -6 Up to 10 -1 ;make This transforms the problem into a standard L2 norm regularization problem; the output layer weights W are calculated using the Moore-Penrose generalized inverse matrix. J Approximate solution: （21） In the formula It is the identity matrix. It is to let Guarantee and Dimension matching; theoretically, if the regularization constraint term approaches zero infinitely, the inverse problem transforms into a least squares problem, expressed by the following formula: （22） Step 6: Obtain the output weights of each feature group. Then, the model training is complete; the training output of the model is obtained through formula (19); Online monitoring For the newly collected online process data X Online This data contains 13 process variables S. I S S X I X BH X BA X P S O S NO S NH S ND S ALK X ND X S The specific meanings of these 13 process variables are shown in Table 1. Using the FAEBLS model that has already been trained during the network training phase, the online monitoring result Y is calculated. Online The specific process for the online application phase is as follows; Step 7: Process the newly collected online process data X Online Perform data preprocessing operations; Step 8: Transfer the preprocessed online data X Online The input is fed into the FAEBLS model established during the training phase, and the output of the temporal feature groups is obtained respectively. Non-Gaussian characteristic group Output and enhanced node group output ; Step 9: Proceed as described in Step 5, and merge the outputs of each feature node group to form matrix A. Online Then, based on the weights from the feature layer to the output layer obtained during the training phase... According to the expression Calculate the final output Y Online The output includes four indicators that need to be monitored: COD, BOD5, SS, and TN.

Images