Natural gas hydrogen production process key parameter real-time estimation method based on semi-supervised robust hybrid echo state network model

Abstract

The invention discloses a natural gas hydrogen production process key parameter real-time estimation method based on a semi-supervised robust hybrid echo state network model, and the method comprises the steps: firstly designing a semi-supervised robust model structure which is combined with a local echo state network in a probability mode, and then developing a training algorithm based on variational reasoning; according to the algorithm, the local echo state networks can be trained in a unified framework, and important regularization parameters of each echo state network can be adaptively determined. The method can provide real-time estimated values of key parameters in the natural gas hydrogen production process on line in real time. By applying the method, the estimation precision can be effectively improved, and technical support and guarantee are provided for reducing the production cost, enhancing the process operation stability, monitoring the process and making decisions.

Description

Technical Field

[0001] This invention belongs to the field of soft measurement modeling and application in chemical processes, specifically involving a real-time estimation method for key parameters in a natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model. Background Technology

[0002] Hydrogen energy, as a clean energy source with zero carbon emissions and diverse applications, is an ideal energy storage medium for achieving a low-carbon transformation of the energy system. The main hydrogen production technologies fall into four categories: (1) hydrogen production from fossil fuel reforming; (2) hydrogen produced from industrial by-products; (3) hydrogen production from water electrolysis; and (4) hydrogen production from clean energy sources. Currently, over 90% of large-scale industrial hydrogen production worldwide primarily utilizes fossil fuel reforming, and over 50% of these reforming processes use natural gas as the raw material. The natural gas hydrogen production process includes four parts: raw material purification, steam reforming, medium-temperature conversion, and pressure swing adsorption. In this process, CH4, CO, CO2, and H2 are the main process gases. The concentrations of these gases are closely related to product quality and are key parameters in the hydrogen production process. Failure to accurately measure these variables in real time will lead to a decline in system control performance, resulting in significant production losses and even safety hazards. Therefore, accurate real-time measurement of these gas concentrations and reasonable adjustment of system operating parameters are crucial for ensuring efficient, stable, and safe operation of the hydrogen production process.

[0003] Currently, the measurement of key parameters in the hydrogen production process mainly relies on two methods: offline laboratory analysis and online mass spectrometry. Offline laboratory analysis yields relatively accurate results, but its measurement cycle is long, typically several hours or even days, making it unsuitable for measuring key parameters in hydrogen production. Using mass spectrometry to measure these gas concentrations can shorten the measurement cycle and obtain results more quickly, but mass spectrometers suffer from accuracy drift, are expensive, and require frequent calibration and maintenance, significantly increasing the production cost of hydrogen production. Establishing a data-driven model for key parameters is an ideal way to solve these problems. By establishing a mathematical prediction model, using easily measurable parameters (temperature, pressure, flow rate, etc.) as input and the output of the model for the key parameters to be estimated, real-time prediction of key parameters can be achieved, effectively compensating for the shortcomings of laboratory analysis and online mass spectrometry measurement.

[0004] However, the frequent changes in feedstock sources in natural gas-to-hydrogen processes, along with factors such as feed intervals and external disturbances, affect the hydrogen production process, making it difficult to maintain the set steady-state operating point. The collected hydrogen production process data exhibits significant dynamic characteristics. Furthermore, due to noise, sensor malfunctions, storage device failures, and human interference, the collected hydrogen production process data often contains outliers. Additionally, due to the complexity of the reaction, the production process data also exhibits strong nonlinearity. Moreover, because mass spectrometer measurement cycles are long and the number of labeled samples (i.e., samples with known key parameters) is small, traditional supervised modeling methods struggle to obtain accurate model parameters due to overlearning or underlearning.

[0005] Therefore, researching and developing data-driven models for key parameters in the natural gas hydrogen production process that can simultaneously address issues such as the dynamics, outliers, strong nonlinearity, and scarcity of labeled samples in the natural gas hydrogen production process will help the hydrogen production industry to more effectively schedule production, improve production efficiency, and use real-time predictive data to guide production decisions. It can also predict trends in product quality, adjust production parameters in a timely manner, and maintain product quality stability. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a real-time estimation method for key parameters in the natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model. Specifically, to address issues of dynamics and strong nonlinearity, this invention utilizes an echo state network model to capture dynamics, while employing a "divide and conquer" approach by using multiple local echo state network models to simulate nonlinear characteristics. To address the scarcity of labeled samples, a semi-supervised strategy is adopted, combining a probabilistic model with both labeled and unlabeled samples for model training. For outliers, the normal distribution of the random variables in the probabilistic model is changed to a Student's t-distribution, thus exhibiting a long-tail effect and increasing model robustness. Furthermore, a training method based on variational inference is developed to adaptively determine the important regularization parameters for each echo state network.

[0007] The specific technical solution is as follows:

[0008] A real-time estimation method for key parameters in a natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model is characterized by the following steps:

[0009] Step 1: Select key parameters for the natural gas-to-hydrogen process Related auxiliary variables ,in Indicates the number of key parameters. Indicates the number of auxiliary variables;

[0010] Step 2: Collect a labeled sample set that includes both auxiliary variables and key parameters. , compared with an unlabeled sample set containing only auxiliary variables ,in The auxiliary variable representing the l-th labeled sample. This represents the key parameter of the l-th labeled sample. The auxiliary variable representing the u-th unlabeled sample. and These represent the number of labeled and unlabeled samples, respectively. , , ;

[0011] Step 3, for and Perform dimensionless processing to convert the sample variance of the auxiliary variable sample and the key parameter sample to 1;

[0012] Step 4, given the number of local echo state networks The number of hidden layer neurons in each echo state network Leakage rate spectral radius Input scaling factor Input matrix reservoir matrix Maximum number of iterations and convergence threshold Initialize model parameters and model parameters conjugate prior distribution parameters and With posterior distribution parameters ,in, , , , , , , , , , , , , , , In the formula:

[0013] This indicates the probability of belonging to the k-th local model;

[0014] and They represent the auxiliary variables in the k-th local model, respectively. The mean vector and x covariance matrix of the distribution;

[0015] This represents the degrees of freedom of the k-th local model;

[0016] Represents the coefficients of the covariance matrix of the measurement noise in the k-th local model;

[0017] Represents the reservoir state vector and key parameters in the k-th local model. The linear regression coefficients between them;

[0018] express Precision matrix parameters;

[0019] The meanings of conjugate prior distribution parameters and posterior distribution parameters are as follows:

[0020] express The posterior distribution parameters;

[0021] and They represent The prior and posterior distribution parameters;

[0022] Step 5: Construct a labeled sample set and unlabeled sample set and its corresponding set of latent variables , , , The likelihood function, where, and These represent the l-th labeled sample. and the u-th unlabeled sample The corresponding binary implicit variable, and satisfying , , , , , ;

[0023] Step 6: Input the training sample set processed in Step 3, the initial model parameters from Step 4, and the likelihood function constructed in Step 5 into the semi-supervised robust hybrid echo state network model, and learn the various model parameters through variational inference. and optimal posterior distribution ,here This represents the optimal posterior distribution of the corresponding variable;

[0024] Step 7: Collect unknown samples containing only auxiliary variables, eliminate the dimensions of the auxiliary variables as in Step 3, and estimate the key parameters using the model parameters and optimal posterior distribution obtained in Step 6.

[0025] Furthermore, the real-time estimation method for key parameters of the natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model according to claim 1 is characterized in that the labeled sample set constructed in step 5... and unlabeled sample set and the latent variables corresponding to each sample , , , The likelihood function is:

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] ;

[0031] ;

[0032] ;

[0033] in, , Represents the probability density function of Gamma. This represents the probability density function of student t. This represents the Gaussian probability density function. , , This represents the sigmoid activation function. .

[0034] Furthermore, the model parameters in step 6 , , , , and the parameters of the optimal posterior distribution , , , The iterative formula has the following form:

[0035] ;

[0036] ;

[0037] ;

[0038] ;

[0039] ;

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] in, Indicates the expectation. Represents the digamma function. The iterative expression can be solved using the bisection method. , , express An identity matrix of dimension 1 This indicates finding the trace of a matrix. , , , , , , and The calculation method is as follows:

[0045] ;

[0046] ;

[0047] ;

[0048] ;

[0049] ;

[0050] ;

[0051] in:

[0052] ;

[0053] ;

[0054] ;

[0055] ;

[0056] ;

[0057] ;

[0058] in, This represents the Gamma function.

[0059] Furthermore, step 7 specifically includes:

[0060] Given a new auxiliary variable After performing a dimensionless operation on it, its corresponding implicit variables... The posterior distribution is:

[0061] ;

[0062] in All are 0-1 variables, and satisfy the following conditions: ;

[0063] against Prediction models are established for each of the key parameters, and then the first parameter can be calculated. Key parameters The probability distribution is:

[0064] ;

[0065] Therefore, we can obtain the first... The estimated values ​​for the key parameters are:

[0066] .

[0067] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0068] 1. The dynamics are captured by using the echo state network model. The "divide and conquer" approach is used to simulate nonlinear characteristics by employing multiple local echo state network models.

[0069] 2. By combining a probabilistic model and employing a semi-supervised strategy, the model is trained using both labeled and unlabeled samples to address the problem of poor model parameter learning caused by insufficient labeled samples.

[0070] 3. By changing the normal distribution of the random variables in the probabilistic model to a Student's t-distribution, a long-tail effect is introduced, increasing the robustness of the model and thus improving the estimation accuracy of key parameters in the hydrogen production process.

[0071] 4. Develop a training method based on variational inference to adaptively determine the important regularization parameters for each echo state network. Attached Figure Description

[0072] Figure 1 The flowchart shows the real-time estimation method for key parameters in the natural gas hydrogen production process based on a semi-supervised robust hybrid echo state network model of the present invention.

[0073] Figure 2 This is a schematic diagram of a steam reforming process for hydrogen production from natural gas. In the diagram, x1 represents the feed gas flow rate at the pre-reformer inlet, x2 represents the steam flow rate at the pre-reformer inlet, x3 represents the fuel gas flow rate to the reformer, x4 represents the fuel exhaust gas flow rate to the reformer, x5 represents the fuel exhaust gas pressure at the exhaust gas preheater outlet, x6 represents the fuel gas pressure at the reformer outlet, x7 represents the fuel exhaust gas temperature at the exhaust gas preheater outlet, x8 represents the fuel gas temperature at the fuel gas preheater outlet, and x9 represents the process gas temperature at the reformer inlet. 10 Represents the temperature of the fuel gas in the upper left corner of the converter, x 11 Represents the temperature of the fuel gas in the upper right corner of the converter, x 12 Represents the temperature of the mixed gas directly above the converter, x 13 Represents the temperature of the conversion gas at the left outlet of the converter, x 14 Represents the temperature of the conversion gas at the right outlet of the converter, x 15 The temperature of the converted gas at the converter outlet is represented by y, and the concentrations of methane, carbon monoxide, carbon dioxide, and hydrogen are represented by y.

[0074] Figure 3 This is a schematic diagram of the estimation results of key parameters (methane, carbon monoxide, carbon dioxide and hydrogen concentration) in the hydrogen production process according to the present invention. In the diagram, the vertical axis represents the content of key parameters in mole percentage (mol %), the vertical axis represents the test sample number, the solid line represents the actual value of the content of key parameters, and the dashed line represents the estimated value of the content of key parameters.

[0075] Figure 4 This diagram illustrates the estimation results of key parameters (methane, carbon monoxide, carbon dioxide, and hydrogen concentrations) in the hydrogen production process using a Gaussian mixture echo state network model. The vertical axis represents the content of key parameters in mole percentage (mol %), the vertical axis represents the test sample number, the solid line represents the actual value of the key parameter content, and the dashed line represents the estimated value of the key parameter content. Detailed Implementation

[0076] The following detailed embodiments further illustrate the real-time estimation method for key parameters in the natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model. It should be noted that the described embodiments are intended only to enhance understanding of the invention and do not constitute any limitation thereof.

[0077] A real-time estimation method for key parameters in natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model, such as... Figure 1 As shown, the specific steps include the following:

[0078] Step 1: Select key parameters for the natural gas-to-hydrogen process Related auxiliary variables ,in Indicates the number of key parameters. Indicates the number of auxiliary variables;

[0079] This example is based on the steam reforming process of a natural gas-to-hydrogen production process (such as...). Figure 2 The process mechanism analysis (as shown) selected 15 variables with the greatest impact on oxygen content as auxiliary variables, namely:

[0080] Pre-conversion furnace inlet feed gas flow rate (x1), pre-conversion furnace inlet steam flow rate (x2), fuel gas flow rate to the converter (x3), fuel exhaust gas flow rate to the converter (x4), fuel exhaust gas pressure at the exhaust gas preheater outlet (x5), fuel gas pressure at the converter outlet (x6), fuel exhaust gas temperature at the exhaust gas preheater outlet (x7), fuel gas temperature at the fuel gas preheater outlet (x8), process gas temperature at the converter inlet (x9), fuel gas temperature at the upper left of the converter (x1). 10 ), temperature of fuel gas at the upper right of the converter (x) 11 ), the temperature of the mixed gas directly above the converter (x) 12 ), temperature of the conversion gas at the left outlet of the converter (x) 13 ), temperature of the conversion gas at the right outlet of the converter (x) 14 ), temperature of the conversion gas at the converter outlet (x) 15 ).

[0081] Therefore, the auxiliary variable x = [x1,…,x] 15 ],Right now ,d =15.

[0082] And collect the key parameters to be predicted (y1: methane concentration, y2: carbon monoxide concentration, y3: carbon dioxide concentration, y4: hydrogen concentration), i.e. ,q =4.

[0083] Step 2: Collect a labeled sample set that includes both auxiliary variables and key parameters. , compared with an unlabeled sample set containing only auxiliary variables ,in The auxiliary variable representing the l-th labeled sample. This represents the key parameter of the l-th labeled sample. The auxiliary variable representing the u-th unlabeled sample. and These represent the number of labeled and unlabeled samples, respectively. , , ;

[0084] This invention collects 1000 labeled sample sets (denoted as ) from a distributed control system database, which simultaneously contain auxiliary variables and key parameters. ), and a set of 3000 unlabeled samples containing only auxiliary variables (denoted as ), ), as the training dataset, where and These represent the number of labeled samples and the number of unlabeled samples, respectively.

[0085] Step 3, for and Perform dimensionless processing to convert the sample variance of the auxiliary variable sample and the key parameter sample to 1;

[0086] The method for dimensionless measurement is as follows:

[0087] ;

[0088] In the formula:

[0089] ;

[0090] ;

[0091] Let represent the sample standard deviations of the i-th auxiliary variable and the i-th key parameter, respectively. This represents the sampled value of the i-th auxiliary variable in the n-th sample. This represents the sampled value of the i-th key parameter of the n-th sample.

[0092] Step 4, given the number of local echo state networks The number of hidden layer neurons in each echo state network Leakage rate spectral radius Input scaling factor Input matrix reservoir matrix Maximum number of iterations and convergence threshold Initialize model parameters and model parameters The conjugate prior and posterior distribution parameters, and the meaning of the model parameters are as follows:

[0093] This indicates the probability of belonging to the k-th local model;

[0094] and They represent the auxiliary variables in the k-th local model, respectively. The mean vector and x covariance matrix of the distribution;

[0095] This represents the degrees of freedom of the k-th local model;

[0096] Represents the coefficients of the covariance matrix of the measurement noise in the k-th local model;

[0097] Represents the reservoir state vector and key parameters in the k-th local model. The linear regression coefficients between them;

[0098] express Precision matrix parameters;

[0099] In this invention, the conjugate prior and posterior distributions of each model parameter are determined as follows:

[0100] prior distribution and posterior distribution All are Gaussian distributed, i.e. , ,in and The parameters are respectively and Gaussian distribution;

[0101] prior distribution and posterior distribution All are gamma-distributed, that is , ,in and The parameters are respectively and The gamma distribution.

[0102] Therefore, in this step, it is necessary to initialize the model parameters, package , , , , Prior distribution parameters, including and posterior distribution parameters, including , , and .

[0103] In this example, the parameters of the prior distribution are set as follows: , The initial values ​​for the remaining parameters and the posterior distribution parameters are random values.

[0104] Step 5: Construct a labeled sample set and unlabeled sample set and its corresponding set of latent variables , , , The likelihood function, where, and These represent the l-th labeled sample. and the u-th unlabeled sample The corresponding binary implicit variable, and satisfying , , , , , It has the following form:

[0105] ;

[0106] ;

[0107] ;

[0108] ;

[0109] ;

[0110] ;

[0111] ;

[0112] Step 6: Input the training sample set processed in Step 3, the initial model parameters from Step 4, and the likelihood function constructed in Step 5 into the semi-supervised robust hybrid echo state network model, and learn the various model parameters through variational inference. and optimal posterior distribution The specific process includes a variational expectation part and a variational maximization part.

[0113] In the variational expectation part, it is necessary to calculate the latent variables. , , , posterior distribution , , , According to the principle of variational reasoning, we can obtain:

[0114] ;

[0115] in, Indicates and Irrelevant numbers , , , , The following characters also represent constants that are independent of the parameters to be solved, and:

[0116] ;

[0117] therefore:

[0118] ;

[0119] in, .

[0120] Similarly, we can obtain posterior distribution as follows:

[0121] ;

[0122] in:

[0123] ;

[0124] in .

[0125] According to variational reasoning, posterior distribution It can be calculated as:

[0126] ;

[0127] therefore:

[0128] ;

[0129] in, , .

[0130] Similarly, we can obtain posterior distribution :

[0131] ;

[0132] in, , .

[0133] In the variational maximization part, it is necessary to calculate the model parameters. and optimal posterior distribution It still employs the principles of variational reasoning. Specifically, It can be calculated using the following formula:

[0134] ;

[0135] therefore, posterior distribution The parameter update formula is:

[0136] ;

[0137] in, .

[0138] It can be calculated using the following formula:

[0139] ;

[0140] therefore, posterior distribution The parameter update formula is:

[0141] ;

[0142] Model parameters , , , , The iterative formulas are calculated as follows:

[0143] ;

[0144] ;

[0145] ;

[0146] ;

[0147] ;

[0148] By iteratively executing the variational expectation and variational maximization parts, the posterior distribution of the model parameters will converge. In this example, the convergence criterion is that the relative increment of the variational lower bound is lower than a set threshold (10). -7 ).

[0149] Step 7, Online Phase: Collect unknown samples containing only auxiliary variables. Then, eliminate the dimensions of the auxiliary variables as in step 3, and use the optimal posterior distribution of the model parameters obtained in step 6 to estimate the key parameters.

[0150] Specifically Corresponding hidden variables The posterior distribution can be calculated as:

[0151] ;

[0152] in, All are 0-1 variables, and satisfy the following conditions: ;

[0153] Prediction models are established for each of the four key parameters, and then the first one can be calculated. Key parameters The probability distribution is:

[0154] ;

[0155] Therefore, we can obtain the first... The estimated values ​​for the key parameters are:

[0156] ;

[0157] To verify the effectiveness of this invention, 700 additional tagged samples were collected from the distributed control system of the steam reforming process in the natural gas-to-hydrogen production as a validation sample set. Following step 7, models were established and estimated for different key parameters. The estimation results for each key parameter are as follows: Figure 4 As shown. Meanwhile, Figure 3 The estimation results of key parameters by the Gaussian mixture echo state network model are presented. It can be seen that, due to the identification of outliers containing valid information as noise, the Gaussian mixture echo state network model does not predict the true values ​​as accurately as the method provided in this invention.

[0158] The root mean square error (RMSE) is used to quantify the estimation accuracy of this invention and the Gaussian mixture echo state network model, defined as follows:

[0159] ;

[0160] in and These represent the true key parameter and the estimated value of the t-th verification sample for the i-th key parameter, respectively. The estimated RMSE of the method provided in this invention and the Gaussian Mixture Echo State Network model are 0.0745, 0.0785, 0.1191, 0.0795 and 0.0511, 0.0665, 0.1056, 0.0531, respectively. It can be seen that the present invention significantly improves the estimation accuracy of key parameters (methane, carbon monoxide, carbon dioxide, and hydrogen concentrations) compared to the Gaussian Mixture Echo State Network model, with improvements of approximately 31%, 15%, 11%, and 33%, respectively.

[0161] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A real-time estimation method for key parameters in a natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model, characterized in that... Includes the following steps: Step 1: Select key parameters for the natural gas-to-hydrogen process Related auxiliary variables ,in Indicates the number of key parameters. Indicates the number of auxiliary variables; Step 2: Collect a labeled sample set that includes both auxiliary variables and key parameters. , compared with an unlabeled sample set containing only auxiliary variables ,in The auxiliary variable representing the l-th labeled sample. This represents the key parameter of the l-th labeled sample. The auxiliary variable representing the u-th unlabeled sample. and These represent the number of labeled and unlabeled samples, respectively. , , ; Step 3, for and Perform dimensionless processing to convert the sample variances of the auxiliary variable samples and the key parameter samples to 1; Step 4, given the number of local echo state networks The number of hidden layer neurons in each echo state network Leakage rate spectral radius Input scaling factor Input matrix reservoir matrix Maximum number of iterations and convergence threshold , Initialize model parameters and model parameters conjugate prior distribution parameters and With posterior distribution parameters ,in, , , , , , , , , , , , , , , In the formula: This indicates the probability of belonging to the k-th local model; and They represent the auxiliary variables in the k-th local model, respectively. The mean vector and x covariance matrix of the distribution; This represents the degrees of freedom of the k-th local model; Represents the coefficients of the covariance matrix of the measurement noise in the k-th local model; Represents the reservoir state vector and key parameters in the k-th local model. The linear regression coefficients between them; express The precision matrix parameters; The meanings of conjugate prior distribution parameters and posterior distribution parameters are as follows: express The posterior distribution parameters; and They represent The prior and posterior distribution parameters; Step 5: Construct a labeled sample set and unlabeled sample set and its corresponding set of latent variables , , , The likelihood function, where, and These represent the l-th labeled sample. and the u-th unlabeled sample The corresponding binary implicit variable, and satisfying , , , , , ; Step 6: Input the training sample set processed in Step 3, the initial model parameters from Step 4, and the likelihood function constructed in Step 5 into the semi-supervised robust hybrid echo state network model, and learn the various model parameters through variational inference. and optimal posterior distribution ,here This represents the optimal posterior distribution of the corresponding variable; Step 7: Collect unknown samples containing only auxiliary variables, eliminate the dimensions of the auxiliary variables as in Step 3, and estimate the key parameters using the model parameters and optimal posterior distribution obtained in Step 6.

2. The method for real-time estimation of key parameters in the natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model according to claim 1, characterized in that, The labeled sample set constructed in step 5 and unlabeled sample set and the latent variables corresponding to each sample , , , The likelihood function is: ； ； ； ； ； ； ； in, , Represents the probability density function of Gamma. This represents the probability density function of student t. This represents the Gaussian probability density function. , , This represents the sigmoid activation function. .

3. The method for real-time estimation of key parameters in the natural gas-to-hydrogen process based on a semi-supervised robust hybrid echo state network model according to claim 1, characterized in that, The model parameters in step 6 , , , , and the parameters of the optimal posterior distribution , , , The iterative formula has the following form: ； ； ； ； ； ； ； ； ； in, Indicates the expectation. Represents the digamma function. The iterative expression can be solved using the bisection method. , , express An identity matrix of dimension 1 This indicates finding the trace of a matrix. , ,and , , , , and The calculation method is as follows: ； ； ； ； ； ； in: ； ； ； ； ； ； in, This represents the Gamma function.

4. The real-time estimation method for key parameters of natural gas to hydrogen production process based on a semi-supervised robust hybrid echo state network model according to claim 1, wherein step 7 specifically comprises: Given a new auxiliary variable After performing a dimensionless operation on it, its corresponding implicit variables... The posterior distribution is: ； in All are 0-1 variables, and satisfy the following conditions: ; against Prediction models are established for each of the key parameters, and then the first parameter can be calculated. Key parameters The probability distribution is: ； Therefore, we can obtain the first... The estimated values ​​for the key parameters are: 。

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