Ore grinding particle size robust soft measurement method based on double-Gaussian mixture distribution

Abstract

The invention discloses an ore grinding particle size robust soft measurement method based on double Gaussian mixture distribution. The method comprises the following steps: constructing a random configuration network as a basic prediction model; secondly, explicitly establishing a noise model weighted and mixed by a small variance Gaussian component and a large variance Gaussian component, and respectively fitting conventional measurement noise and process abnormal noise; under a Bayesian framework, an expectation maximization algorithm is adopted to jointly iteratively optimize network output weight and noise model parameters, and adaptive learning of the parameters and intelligent weighting of samples are achieved. According to the method, noise decomposition parameters including the normal working condition weight and the abnormal variance ratio can be output, and interpretable decision support is provided for ore grinding operation. Experiments show that the method effectively improves the prediction precision, robustness and interpretability of the model in the mixed noise environment, and provides reliable technical support for optimization control of the ore grinding process.

Description

Technical Field

[0001] This invention belongs to the field of intelligent mineral processing technology, specifically relating to a robust soft measurement method for grinding particle size based on a dual Gaussian mixed distribution. Background Technology

[0002] Grinding is a crucial step in mineral processing. Grinding particle size, defined as the proportion of particles smaller than 0.074 mm, directly impacts subsequent sorting efficiency, concentrate grade, and metal recovery rate. Accurate and real-time prediction of grinding particle size is essential for stable production, reduced energy consumption, and improved economic efficiency. However, the grinding process in actual production is complex, exhibiting strong nonlinearity, multivariate coupling, and large time delays, making accurate modeling based on physical mechanisms extremely difficult. Data-driven soft measurement techniques offer an effective solution, with stochastic configuration networks demonstrating significant advantages in modeling complex industrial processes due to their randomized learning, incremental construction, and universal approximation capabilities.

[0003] However, in the specific scenario of grinding, data quality faces severe challenges: on the one hand, sensor measurement noise and environmental electromagnetic interference result in Gaussian background noise in the data; on the other hand, sudden changes in ore hardness, sudden equipment failures (such as liner detachment or steel ball breakage), valve jamming, and even human error can introduce outliers with large amplitudes and abnormal distributions, forming mixed noise interference with both impulse noise and heavy-tailed noise. Traditional stochastically configured networks rely on the least squares criterion for parameter estimation, which is highly sensitive to outliers, severely impairing model accuracy and generalization ability. Existing robust improvement methods, such as models based on a single Laplace distribution or kernel density estimation, can suppress the impact of outliers to some extent, but still have significant limitations: they usually assume that the noise follows a single heavy-tailed distribution, making it impossible to explicitly distinguish and quantify the different types of conventional measurement noise and sudden abnormal interference during grinding; the models lack interpretability, making it difficult to clearly reveal the process status to process engineers; at the same time, some methods have high computational complexity, making it difficult to meet the needs of real-time monitoring and rapid response in the grinding process.

[0004] Therefore, developing a soft measurement method that can accurately match the mixed noise characteristics of the grinding process and has superior robustness, good interpretability and efficient computing power has become an urgent technical problem to be solved in order to achieve refined and intelligent operation of the grinding process. Summary of the Invention

[0005] The purpose of this invention is to propose a robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution, targeting the characteristics of grinding process data. The core of this method lies in constructing a dual-component Gaussian mixture model to explicitly separate and quantify normal measurement noise and abnormal process noise in the grinding process. On this basis, the expectation-maximization (EM) algorithm is used to achieve joint Bayesian learning of network parameters and noise parameters, and reliability weights are adaptively assigned to each training sample. This significantly improves the robustness and interpretability of the model to abnormal operating conditions while ensuring prediction accuracy.

[0006] Technical Solution: To solve the above-mentioned technical problems and achieve the above-mentioned objectives, this invention proposes a robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution. This method includes the following steps:

[0007] Step 1: Collect historical data of the grinding process and construct a training dataset. , where the input vector These represent the ore feed rate, respectively. Mill inlet water flow and overflow concentration of grading equipment Output scalar Represents grinding particle size. This represents the total amount of data in the dataset.

[0008] Step 2: Based on the training dataset, construct a randomized network as the basic prediction model. The randomized network adds hidden nodes sequentially through an incremental supervision mechanism. The network output of each node is represented as follows:

[0009] ;

[0010] in, For having The network prediction output with a random configuration of hidden nodes. For the input vector, For activation function, The first The input weights, biases, and output weights of each hidden node. To randomly configure the output weights of the network, it has The output matrix of the hidden layer of each hidden node Defined as:

[0011] ;

[0012] If it has If the randomly configured network of hidden nodes does not meet the termination condition, a new hidden node is generated, and its hidden layer output is... Represented as:

[0013] ;

[0014] Step 3, calculate the prediction error of the randomly configured network. ;in, Let n be the predicted output for the nth sample; a double Gaussian mixture noise model describing the prediction error of the grinding process is established for the prediction error of the randomly configured network, and its probability density function is:

[0015] ;

[0016] in, express Follow the mean The variance is Gaussian distribution, The mixing weights are for noise under normal operating conditions. This represents the normal noise variance. is the amplification factor of the abnormal noise variance relative to the normal noise, and ;

[0017] Step 4: Within the Bayesian framework, configure the output weights of the randomly assigned network. Introduce a Gaussian prior distribution;

[0018] ;

[0019] in, It is the identity matrix. Output weights for the network variance Indicates output weights The covariance matrix;

[0020] Step 5: Use the expectation-maximization algorithm to jointly estimate the parameter set. Optimal parameters For new grinding operating conditions input Its grinding particle size prediction value is .

[0021] Furthermore, in step 3, the process of constructing the dual Gaussian mixture noise model is as follows:

[0022] Introducing binary latent variables ,satisfy and Define its prior distribution: , In a given Under these conditions, observation data The conditional probability follows a Gaussian distribution when , Follow the mean The variance is Gaussian distribution, when , Follow the mean The variance is The Gaussian distribution, that is:

[0023] , ;

[0024] By analyzing latent variables Marginalization is used to obtain observation data The marginal probability distribution is the aforementioned bi-Gaussian mixture model:

[0025] ;

[0026] To reduce computational complexity, the complete dataset is used. The logarithm of the posterior distribution is used to calculate the log-posterior probability of the complete data. Conditional expectation:

[0027] ;

[0028] in, It is a constant. Indicates the first The parameter set of the next iteration, superscript Indicates the first The next iteration.

[0029] Furthermore, in step 5, the iterative optimization steps of the expectation maximization algorithm are as follows:

[0030] Step 4-1, the parameter set to be estimated Perform initialization. , =0;

[0031] Step 4-2, Estimating based on current parameters Calculate latent variables Characterization Samples Belongs to the Posterior expectation of each Gaussian component , ;

[0032] Step 4-3 updates the parameters by maximizing the expected value of the log-posterior of the complete data. The update of the output weights is achieved by solving a weighted least squares problem. In the formula, This is the diagonal weight matrix assigned to each training sample based on the current noise model;

[0033] Step 4-4: Iterate through steps 4-2 and 4-3 until the parameters converge and the following parameter convergence condition is met to obtain the optimal parameters. Once the model is trained, it is used for robust prediction of grinding particle size.

[0034] ;

[0035] in, It is a preset positive number.

[0036] Furthermore, in step 4-2, the latent variables are calculated. posterior expectation The formula is as follows:

[0037] ;

[0038] ;

[0039] in, , Indicates the first The error of each sample originates from the probability of normal noise components. Indicates the first The error of each sample originates from the probability of abnormal noise components.

[0040] Furthermore, in steps 4-3, the first The noise model parameters are updated once. The formula, by maximizing the expectation By subtracting the corresponding part from the equation and setting the partial derivatives to zero, we obtain:

[0041] ;

[0042] ;

[0043] .

[0044] Furthermore, in step 4-3, the weight matrix The nth diagonal element The calculation formula is:

[0045] ;

[0046] This weight is used to adaptively reduce the impact of samples identified as anomalous on the model when updating network weights. The anomalous samples are... Samples that are greater than the preset threshold.

[0047] Furthermore, in step 4-3, the prior variance is updated. By maximizing expectation Zhongyu Related items And setting the partial derivatives to zero, we get:

[0048] .

[0049] Furthermore, in step 2, when constructing the randomly configured network, an incremental supervision mechanism is used to add hidden nodes; in constructing the... When there are hidden nodes, multiple sets of candidate parameters are randomly generated from a preset interval. ,parameter The following inequality constraints must be satisfied:

[0050] ;

[0051] in, and For preset scalar, , , , Indicates having The network SCN residual vector is randomly configured for each node until the maximum number of hidden nodes is reached or the residual satisfies the termination condition, at which point the generation of new nodes stops.

[0052] Furthermore, this invention proposes a robust neural network modeling system for grinding processes based on dual Gaussian mixed noise, comprising a processor and a memory, wherein the memory stores a computer program, and the processor executes the program to implement the steps of any of the grinding particle size robust soft measurement methods based on dual Gaussian mixed distributions described in the present invention.

[0053] Furthermore, the present invention proposes a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of any of the methods for robust soft measurement of grinding particle size based on a dual Gaussian mixture distribution.

[0054] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0055] This invention provides a robust neural network modeling method for dual Gaussian mixture noise in the grinding process. First, by constructing a dual-component Gaussian mixture noise model, this invention explicitly distinguishes between conventional Gaussian measurement noise and impulse noise caused by abnormal operating conditions in the grinding process at the probability distribution level, achieving accurate matching of complex noise structures and thus enhancing the model's anti-interference capability at its root. Second, based on a joint learning mechanism of expectation-maximization algorithm and Bayesian framework, it can adaptively assign weights reflecting the reliability of each training sample, automatically weakening the negative impact of abnormal samples during model training, eliminating the need for subjective thresholding for data cleaning, and greatly improving the model's robustness and generalization performance. Furthermore, the noise model parameters output by this invention have clear physical meaning, intuitively quantifying the operational health and abnormal risks of the grinding process, providing process engineers with interpretable evidence for diagnosing process status and locating the root causes of anomalies, strongly supporting production decisions. In addition, the computational core of this method has an efficient analytical solution, meeting the stringent real-time requirements of the grinding process while ensuring prediction accuracy. Attached Figure Description

[0056] To more clearly illustrate the technical solution of this application, the drawings used in the embodiments will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0057] Figure 1 Here is a flowchart of the EM algorithm iteration process;

[0058] Figure 2 Overall algorithm block diagram of the DG-RSC method for grinding process described in this invention;

[0059] Figure 3 This is a schematic diagram of the grinding process;

[0060] Figure 4 Comparison of actual and sampled values ​​of hematite grinding particle size;

[0061] Figure 5 The DG-RSC algorithm predicts the results based on grinding data of a certain hematite ore. Detailed Implementation

[0062] See Figure 1 This invention proposes a robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution. The method includes the following steps:

[0063] Step 1: Collect historical data of the grinding process and construct a training dataset. , where the input vector These represent the ore feed rate, respectively. Mill inlet water flow and overflow concentration of grading equipment Output scalar Represents grinding particle size. This represents the total amount of data in the dataset.

[0064] Step 2: Based on the training dataset, construct a randomized network as the basic prediction model. The randomized network adds hidden nodes sequentially through an incremental supervision mechanism. The network output of each node is represented as follows:

[0065] ;

[0066] in, For having The network prediction output with a random configuration of hidden nodes. For the input vector, For activation function, The first The input weights, biases, and output weights of each hidden node. To randomly configure the output weights of the network, it has The output matrix of the hidden layer of each hidden node Defined as:

[0067] ;

[0068] If it has If the randomly configured network of hidden nodes does not meet the termination condition, a new hidden node is generated, and its hidden layer output is... Represented as:

[0069] ;

[0070] Step 3, calculate the prediction error of the randomly configured network. ;in, Let n be the predicted output for the nth sample; a double Gaussian mixture noise model describing the prediction error of the grinding process is established for the prediction error of the randomly configured network, and its probability density function is:

[0071] ;

[0072] in, express Follow the mean The variance is Gaussian distribution, The mixing weights are for noise under normal operating conditions. This represents the normal noise variance. is the amplification factor of the abnormal noise variance relative to the normal noise, and ;

[0073] Step 4: Within the Bayesian framework, configure the output weights of the randomly assigned network. Introduce a Gaussian prior distribution;

[0074] ;

[0075] in, It is the identity matrix. Output weights for the network variance Indicates output weights The covariance matrix;

[0076] Step 5: Use the expectation-maximization algorithm to jointly estimate the parameter set. Optimal parameters For new grinding operating conditions input Its grinding particle size prediction value is .

[0077] Furthermore, in step 3, the process of constructing the dual Gaussian mixture noise model is as follows:

[0078] Introducing binary latent variables ,satisfy and Define its prior distribution: , In a given Under these conditions, observation data The conditional probability follows a Gaussian distribution when , Follow the mean The variance is Gaussian distribution, when , Follow the mean The variance is The Gaussian distribution, that is:

[0079] , ;

[0080] By analyzing latent variables Marginalization is used to obtain observation data The marginal probability distribution is the aforementioned bi-Gaussian mixture model:

[0081] ;

[0082] To reduce computational complexity, the complete dataset is used. The logarithm of the posterior distribution is used to calculate the log-posterior probability of the complete data. Conditional expectation:

[0083] ;

[0084] in, It is a constant. Indicates the first The parameter set of the next iteration, superscript Indicates the first The next iteration.

[0085] Furthermore, in step 5, the iterative optimization steps of the expectation maximization algorithm are as follows:

[0086] Step 4-1, the parameter set to be estimated Perform initialization. , =0;

[0087] Step 4-2, Estimating based on current parameters Calculate latent variables Characterization Samples Belongs to the Posterior expectation of each Gaussian component , ;

[0088] Step 4-3 updates the parameters by maximizing the expected value of the log-posterior of the complete data. The update of the output weights is achieved by solving a weighted least squares problem. In the formula, This is the diagonal weight matrix assigned to each training sample based on the current noise model;

[0089] Step 4-4: Iterate through steps 4-2 and 4-3 until the parameters converge and the following parameter convergence condition is met to obtain the optimal parameters. Once the model is trained, it is used for robust prediction of grinding particle size.

[0090] ;

[0091] in, It is a preset positive number.

[0092] Furthermore, in step 4-2, the latent variables are calculated. posterior expectation The formula is as follows:

[0093] ;

[0094] ;

[0095] in, , Indicates the first The error of each sample originates from the probability of normal noise components. Indicates the first The error of each sample originates from the probability of abnormal noise components.

[0096] Furthermore, in steps 4-3, the first The noise model parameters are updated once. The formula, by maximizing the expectation By subtracting the corresponding part from the equation and setting the partial derivatives to zero, we obtain:

[0097] ;

[0098] ;

[0099] .

[0100] Furthermore, in step 4-3, the weight matrix The nth diagonal element The calculation formula is:

[0101] ;

[0102] This weight is used to adaptively reduce the impact of samples identified as anomalous on the model when updating network weights. The anomalous samples are... Samples that are greater than the preset threshold.

[0103] Furthermore, in step 4-3, the prior variance is updated. By maximizing expectation Zhongyu Related items And setting the partial derivatives to zero, we get:

[0104] .

[0105] Furthermore, in step 2, when constructing the randomly configured network, an incremental supervision mechanism is used to add hidden nodes; in constructing the... When there are hidden nodes, multiple sets of candidate parameters are randomly generated from a preset interval. ,parameter The following inequality constraints must be satisfied:

[0106] ;

[0107] in, and For preset scalar, , , , Indicates having The network SCN residual vector is randomly configured for each node until the maximum number of hidden nodes is reached or the residual satisfies the termination condition, at which point the generation of new nodes stops.

[0108] Furthermore, this invention proposes a robust neural network modeling system for grinding processes based on dual Gaussian mixed noise, comprising a processor and a memory, wherein the memory stores a computer program, and the processor executes the program to implement the steps of any of the grinding particle size robust soft measurement methods based on dual Gaussian mixed distributions described in the present invention.

[0109] Furthermore, the present invention proposes a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of any of the methods for robust soft measurement of grinding particle size based on a dual Gaussian mixture distribution.

[0110] Example

[0111] In this embodiment, the grinding process of a hematite beneficiation plant is used as the application object to verify the effectiveness and superiority of the method described in this invention (hereinafter referred to as DG-RSC). Figure 3 As shown, this grinding process adopts a closed-loop process of ball mill-spiral classifier. After the raw ore is crushed by the ball mill, it is classified by particle size by the spiral classifier. The overflow (fine particles) is sent to subsequent separation operations, while the return sand (coarse particles) is returned to the ball mill for regrinding. The specific implementation steps are as follows:

[0112] Algorithm input: These represent the ore feed rate, respectively. Mill inlet water flow and overflow concentration of grading equipment The algorithm's output includes: This represents the grinding particle size.

[0113] Step 1: Data Acquisition and Preprocessing. From the grinding hardware-in-the-loop simulation platform of the concentrator, 2000 sets of key variable data closely related to grinding particle size during continuous production were collected. Input data: Feed rate. Reflects system processing load and mill inlet feedwater flow rate Used to adjust grinding concentration and overflow concentration of classifying equipment. It is a comprehensive indirect indicator of grinding product particle size; prediction target: grinding particle size. .

[0114] To eliminate dimensional differences and accelerate model convergence, the input data was standardized, and the output data was linearly normalized to the [0, 1] interval. Subsequently, all data were randomly divided into a training set (1200 sets), a validation set (400 sets), and a test set (400 sets) in a 6:2:2 ratio.

[0115] Step 2: Abnormal Operating Condition Simulation and Noise Injection. To simulate abnormal data caused by sudden changes in ore hardness and occasional equipment failures in actual production, the output data (grinding particle size values) of the training set is artificially contaminated: 10% of the training samples (i.e., 120 sample points) are randomly selected as outliers. A random perturbation uniformly distributed in the interval [-0.4, 0.4] is superimposed on the particle size values ​​of these outliers. The amplitude of this perturbation is significantly greater than the normal measurement noise, in order to simulate severe abnormal operating conditions.

[0116] Step 3: Based on the processed data, construct the DG-RSC-Grinding model described in this invention. Specific parameters are determined based on validation set performance optimization. The network component is randomly configured with the following parameters: maximum number of nodes. Random configuration range parameters Maximum number of configurations Prediction error tolerance Double Gaussian noise model section: Double Gaussian noise model section Normal noise initial variance Initial value of abnormal noise variance amplification factor .

[0117] Step 4: Update the posterior probability distribution and expectation: Calculate the loss cost function using the formula given in E-step;

[0118] Step 5: Update model parameter estimates: Update SCN model parameters and prior variance Noise model parameters ;

[0119] Step 6: Increase the value of q by 1 until the parameters of the ARX model to be identified converge.

[0120] A total of 2000 sets of key variable data closely related to grinding particle size were collected during the continuous production process. Input data: feed rate. Reflects system processing load and mill inlet feedwater flow rate Used to adjust the concentration of grinding and classifying equipment overflow. It is a comprehensive indirect indicator of grinding product particle size; prediction target: grinding particle size. .

[0121] Set simulation parameters:

[0122] To verify the effectiveness of the method of the present invention in dealing with problems such as noise and outliers in the collected data, it is assumed that the output data contains 10% outliers and mixed Gaussian noise.

[0123] Simulation verification:

[0124] To demonstrate the superiority of the method in this invention, four existing representative methods were selected as benchmarks for comparison: Lap-RSC, a robust randomized network based on the Laplace distribution; RSC-KDE, a robust randomized network based on kernel density estimation; SCN, a standard randomized network (using least squares estimation); and RVFL, a randomized vector function linking network. All comparison methods used the same training, validation, and test sets and underwent independent parameter tuning. Each method was trained independently 50 times to eliminate randomness, and the average and standard deviation of the performance metrics were used as the final evaluation. Evaluation metrics included: Root Mean Square Error (RMSE), used to measure the absolute deviation between the predicted and actual values. The coefficient of determination (R²) measures the model's ability to explain the variance of the data. Running time (t) measures the computational efficiency of the algorithm. Specific simulation results are shown in Table 1. Figure 4 To visualize the comparison results.

[0125]

[0126] Summary of simulation results:

[0127] In the comprehensive performance evaluation, the DG-RSC algorithm consistently ranked first. Error analysis showed that its RMSE value was the lowest among all methods, indicating that the algorithm's predicted results had the smallest deviation from the actual grinding particle size values, demonstrating higher prediction accuracy. In terms of fitting ability, its R² value also reached the highest level, indicating that the model has the strongest explanatory power for the variation in grinding particle size data and possesses the best goodness of fit. Furthermore, the algorithm also performed well in terms of runtime, achieving a good balance between high accuracy and high stability while maintaining computational efficiency, thus realizing an effective synergy in overall performance.

[0128] This invention provides a robust neural network modeling method for grinding processes using dual Gaussian mixed noise. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

[0129] In this embodiment of the application, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of any of the methods described above.

[0130] In this embodiment of the application, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of any of the methods described above.

[0131] The various embodiments in this specification are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0132] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention described herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not invented herein. The specification and embodiments are to be considered exemplary only.

[0133] The above specific embodiments further illustrate the purpose, technical solution and beneficial effects of this application. It should be understood that the above are only specific embodiments of this application and are not intended to limit the scope of protection of this application. Any modifications, equivalent substitutions, improvements, etc., made on the basis of the technical solution of this application should be included within the scope of protection of this application.

Claims

1. A robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution, characterized in that, The method includes the following steps: Step 1: Collect historical data of the grinding process and construct a training dataset. , where the input vector They represent the amount of ore fed. Mill inlet water flow rate and overflow concentration of grading equipment Output scalar Represents grinding particle size. This represents the total amount of data in the dataset. Step 2: Based on the training dataset, construct a Randomized Network (SCN) as the basic prediction model. The SCN adds hidden nodes sequentially through an incremental supervision mechanism. The network output of each node is represented as follows: ； in, For having The network prediction output with a random configuration of hidden nodes. For the input vector, For activation function, The first The input weights, biases, and output weights of each hidden node. To randomly configure the output weights of the network, it has The output matrix of the hidden layer of each hidden node Defined as: ； If it has If the randomly configured network of hidden nodes does not meet the termination condition, a new hidden node is generated, and its hidden layer output is... Represented as: ； Step 3, calculate the prediction error of the randomly configured network. ;in, Let n be the predicted output for the nth sample; a double Gaussian mixture noise model describing the prediction error of the grinding process is established for the prediction error of the randomly configured network, and its probability density function is: ； in, express Follow the mean The variance is Gaussian distribution, The mixing weights are for noise under normal operating conditions. This represents the normal noise variance. is the amplification factor of the abnormal noise variance relative to the normal noise, and ; Step 4: Within the Bayesian framework, configure the output weights of the randomly assigned network. Introduce a Gaussian prior distribution; ； in, It is the identity matrix. Output weights for the network variance Indicates output weights The covariance matrix; Step 5: Use the expectation-maximization algorithm to jointly estimate the parameter set. Optimal parameters For new grinding operating conditions input Its grinding particle size prediction value is .

2. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 1, characterized in that, In step 3, the process of constructing the dual Gaussian mixture noise model is as follows: Introducing binary latent variables ,satisfy and Define its prior distribution: , In a given Under these conditions, observation data The conditional probability follows a Gaussian distribution when , Follow the mean The variance is Gaussian distribution, when , Follow the mean The variance is The Gaussian distribution, that is: ， ； By analyzing latent variables Marginalization is used to obtain observation data The marginal probability distribution is the bi-Gaussian mixture model: ； Get the complete dataset The logarithm of the posterior distribution is used to calculate the log-posterior probability of the complete data. Conditional expectation: ； in, It is a constant. Indicates the first The parameter set for the next iteration, superscript Indicates the first The next iteration.

3. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 2, characterized in that, In step 5, the iterative optimization steps of the expectation maximization algorithm are as follows: Step 4-1, the parameter set to be estimated Perform initialization. , =0; Step 4-2, Estimating based on current parameters Calculate latent variables Characterizing the first The sample belongs to the first Posterior expectation of each Gaussian component , ; Step 4-3 updates the parameters by maximizing the expected value of the log-posterior of the complete data. The update of the output weights is achieved by solving a weighted least squares problem. In the formula, This is the diagonal weight matrix assigned to each training sample based on the current noise model; Step 4-4: Iterate through steps 4-2 and 4-3 until the following parameter convergence condition is met to obtain the optimal parameters. Model training complete, optimal parameters Robust prediction of grinding particle size; ； in, It is a preset positive number.

4. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 3, characterized in that, In step 4-2, the latent variables are calculated. posterior expectation The formula is as follows: ； ； in, , Indicates the first The error of each sample originates from the probability of normal noise components. Indicates the first The error of each sample originates from the probability of abnormal noise components.

5. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 4, characterized in that, In step 4-3, the first The noise model parameters are updated once. The formula, by maximizing the expectation By taking the corresponding part of the equation and setting the partial derivatives to zero, we obtain: ； ； 。 6. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 5, characterized in that, In step 4-3, the weight matrix The nth diagonal element The calculation formula is: ； This weight is used to adaptively reduce the impact of samples identified as anomalous on the model when updating network weights. The anomalous samples are... Samples that are greater than the preset threshold.

7. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 4, characterized in that, In step 4-3, the prior variance is updated. By maximizing expectation Zhongyu Related items And setting the partial derivatives to zero, we get: 。 8. The robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution according to claim 1, characterized in that, In step 2, an incremental supervision mechanism is used to add hidden nodes when constructing the random configuration network; in constructing the... When there are hidden nodes, multiple sets of candidate parameters are randomly generated from a preset interval. ,parameter The following inequality constraints must be satisfied: ； in, and For preset scalar, , , , Indicates having The network SCN residual vector is randomly configured for each node until the maximum number of hidden nodes is reached or the residual satisfies the termination condition, at which point the generation of new nodes stops.

9. A robust neural network modeling system for grinding processes with dual Gaussian mixed noise, characterized in that, It includes a processor and a memory, the memory storing a computer program, and the processor executing the program to implement the steps of the robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution as described in any one of claims 1-8.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the robust soft measurement method for grinding particle size based on a dual Gaussian mixture distribution as described in any one of claims 1 to 8.

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