CN108664706B - Semi-supervised Bayesian Gaussian mixture model-based online estimation method for oxygen content of one-stage furnace in ammonia synthesis process - Google Patents

Abstract

The invention discloses an online estimation method of oxygen content of a first-stage furnace in an ammonia synthesis process based on a semi-supervised Bayesian Gaussian mixture model, which is characterized by firstly designing a new complete Bayesian model structure, making probability of all model parameters and semi-supervised regression learning possible; and then, under a variational reasoning framework, simultaneously mining labeled sample information and unlabeled sample information, and establishing a learning step of model parameters. The method can provide the estimated value of the oxygen content of the first-stage furnace in the ammonia synthesis process in real time on line. By applying the method and the device, the influence of overfitting can be effectively reduced, so that the estimation precision is improved, and technical support and guarantee are provided for reducing the production cost, enhancing the process running stability, monitoring the process and making a decision.

Description

Semi-supervised Bayesian Gaussian mixture model-based online estimation method for oxygen content of one-stage furnace in ammonia synthesis process

Technical Field

The invention belongs to the field of chemical process soft measurement modeling and application, and particularly relates to an online estimation method for the oxygen content of a first-stage furnace in an ammonia synthesis process based on a semi-supervised Bayesian Gaussian mixture model.

Background

Ammonia is a very important basic chemical product, the yield of the ammonia is highest among various chemical products, and the ammonia is industrially used for producing a large amount of urea, soda ash, ammonium nitrogen fertilizer and nitric acid and preparing organic synthetic industrial products such as fibers, plastics, dyes and the like. The raw materials for ammonia synthesis include nitrogen, which is available in large quantities from air, and hydrogen, which needs to be produced by specialized hydrogen production facilities. In most ammonia synthesis processes, a primary reformer (abbreviated as primary furnace) is the main equipment for preparing hydrogen, wherein the chemical reaction (catalyst is nickel) is as follows:

the chemical reaction is endothermic and requires heat to be supplied to the primary furnace. Therefore, the reaction temperature is an important factor for keeping the hydrogen production reaction stably. The conventional heating means of a primary furnace is to burn fuel gas and recovered flue gas in the radiant section. In order to maintain the reaction temperature set by the process, the oxygen content in the first-stage furnace needs to be controlled within a specified range. The oxygen content (in mole percent, mol%) can be determined by a mass analyzer. However, the mass analyzer is not only expensive, has a long measurement period, but also is prone to failure. Losing the measured value of the oxygen content, the closed-loop controller will not work, possibly causing a series of adverse consequences, such as environmental pollution and cost increase caused by the increase of rejection rate, energy consumption and the like, and even leading to potential safety hazards.

The data-driven oxygen content soft measurement model can realize the on-line real-time estimation of the oxygen content to make up for the defects of the mass analyzer. The principle is that a mathematical model is established according to the dependence relationship between the oxygen content and variables (such as temperature, pressure, flow, liquid level and other parameters, also called as auxiliary variables) which are easy to measure in the process at an off-line stage, and then the oxygen content is estimated on line by using the mathematical model, so that the method has the advantages of no measurement lag, low cost, good universality, easy maintenance and the like. However, because the combustion process of the first furnace is very complex, the working conditions are frequently switched, and the production data has the characteristics of uncertainty, multimodality, strong nonlinearity and the like, the traditional soft measurement model (such as a principal component analysis model, a partial least square model, a neural network model, a support vector machine model and the like) is difficult to obtain satisfactory estimation accuracy. On the other hand, because the measurement period of the mass analyzer is long, the number of labeled samples (i.e., samples with known oxygen content) is small, so that the traditional supervised modeling method is difficult to obtain accurate model parameters due to reasons such as "over-learning" or "under-learning". The soft measurement model of oxygen content with poor training cannot necessarily provide satisfactory estimation precision, and manual parameter setting is time-consuming and labor-consuming and has great difficulty.

Therefore, it is necessary and urgent to research and develop a soft oxygen content measurement model capable of simultaneously handling the problems of complex uncertainty, strong nonlinearity, multimodality, rare labeled samples and the like in the process of one stage of the furnace, which is helpful to improve the estimation accuracy of the oxygen content, thereby assisting the ammonia synthesis enterprises to realize the goals of safe production, energy saving, environmental protection, cost reduction and efficiency improvement.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a method for estimating the oxygen content of a first-stage furnace in the ammonia synthesis process on line based on a semi-supervised Bayes Gaussian mixture model, which is characterized in that a probabilistic mathematical model between the oxygen content and auxiliary variables is established in the form of the Bayes Gaussian mixture model, the contribution degree of the mixture model is adaptively distributed according to the working condition switching, the problems of uncertainty, nonlinearity, multimodality and the like are effectively solved, and the problem of low model estimation precision caused by the rare labeled samples is solved by simultaneously utilizing the labeled samples and the unlabeled samples (namely the samples with unknown oxygen content and only known auxiliary variables) through semi-supervised learning. The specific technical scheme is as follows:

a method for estimating the oxygen content of a first-stage furnace in an ammonia synthesis process on line based on a semi-supervised Bayes Gaussian mixture model is characterized by comprising the following steps:

(1) selecting an auxiliary variable associated with the oxygen content y of the primary furnace

Wherein d represents the number of auxiliary variables;

(2) collecting labeled sample sets containing both auxiliary variables and oxygen content

And unlabeled sample set containing only auxiliary variables

Wherein n is_lAnd n_uRespectively representing the number of the labeled samples and the number of the unlabeled samples;

(3) to (X)_l,Y_l) And X_uCarrying out dimensionless treatment, and converting the sample variance of the auxiliary variable sample and the oxygen content sample into unit variance;

(4) initializing model parameters given a truncation level M of the Dirichlet process

Is a conjugate prior distribution parameter of₀、b₀、c₀、d₀、e₀、f₀、β₀、v₀、m₀、W₀And posterior distribution parameters a, b, h_k、l_k、c_k、d_k、e_k、f_k、β_k、v_k、m_k、W_k、ω_k、Ω_kWherein, in the step (A),

and is

α represents a concentration factor of the dirichlet process;

χ_kparameters representing the kth hybrid model coefficients;

μ_kand Λ_kRespectively representing a mean vector and a precision matrix of the distribution of the auxiliary variable x in the kth mixed model;

representing a linear regression coefficient between an auxiliary variable x and an oxygen content y in the kth mixed model;

τ_kto represent

The accuracy matrix parameters of (1);

η_kand representing the precision matrix coefficient of the measurement noise in the k mixed model.

The meaning of the conjugate prior distribution parameter and the posterior distribution parameter is:

(a₀,b₀) And (a, b) a prior distribution parameter and a posterior distribution parameter respectively representing α;

(h_k,l_k) Denotes x_kThe posterior distribution parameters of (1);

(m₀,β₀,W₀,v₀) And (m)_k,β_k,W_k,v_k) Respectively represent (mu)_k,Λ_k) The prior distribution parameter and the posterior distribution parameter;

(c₀,d₀) And (c)_k,d_k) Respectively represent tau_kThe prior distribution parameter and the posterior distribution parameter;

(e₀,f₀) And (e)_k,f_k) Respectively represent η_kThe prior distribution parameter and the posterior distribution parameter;

ω_kand Ω_kTo represent

The posterior distribution parameters of (1);

(5) constructed with labeled samples (X)_l,Y_l) And unlabeled sample X_uAnd its corresponding hidden variable

In which z is_i＝(z_i1,…,z_iM)^TAnd z_j＝(z_j1,…,z_jM)^TRespectively represent the ith labeled sample (x)_i,y_i) And the jth unlabeled sample x_jCorresponding binary hidden variables, and satisfy

(6) Inputting the training sample set processed in the step (3), the initial model parameters in the step (4) and the likelihood function constructed in the step (5) into a semi-supervised Bayes Gaussian mixture model, and learning the optimal posterior distribution q (alpha) and the optimal posterior distribution q (alpha) of each model parameter through variational inference

Where q (-) denotes the optimal posterior distribution of the corresponding variable.

(7) And (4) collecting unknown samples only containing auxiliary variables, eliminating the dimension of the auxiliary variables according to the step (3), and estimating the oxygen content by using the optimal posterior distribution of the model parameters obtained in the step (6).

Further, the labeled sample (X) constructed in the step (5)_l,Y_l) And unlabeled sample X_uAnd its corresponding hidden variable Z_l、Z_uThe likelihood function of (d) is:

wherein χ ═ χ (χ ═ χ)₁,…,χ_M)，μ＝(μ₁,…,μ_M)，Λ＝(Λ₁,…,Λ_M)，

η＝(η₁,…,η_M)，

Represents the mean value of μ_kThe covariance matrix is

The gaussian probability density function of (a) is,

further, the parameters a, b, h of the optimal posterior distribution of the model parameters of step (6)_k，l_k，c_k，d_k，e_k，f_k，β_k，v_k，m_k，W_k，ω_kAnd Ω_kHas the following form:

a＝a₀+M-1

c_k＝c₀+(d+1)/2

where ψ (·) denotes a digamma function, I denotes an identity matrix of the corresponding dimension,

1 is the full 1 column vector, Tr (-) trace of the matrix,

represents the estimation error of the k-th hybrid model,

here, the

Express according to

Distribution calculation of

(iii) a desire; kappa_ikAnd kappa_jkIs calculated in a manner that

Wherein

Further, the step (7) is specifically as follows:

according to the posterior distribution of alpha calculated in the step (6) and the property of the Dirichlet process, each model mixing coefficient pi ═ pi (pi ═ pi)₁,…,π_M) Can be calculated as

q(π)＝Dir(π|φ₁,…,φ_M)

Wherein Dir (π | φ)₁,…,φ_M) The representative parameter is (phi)₁,…,φ_M) Is distributed, and

then, the dimensionless auxiliary variable x can be obtained from the posterior distribution of the model parameters calculated in step (6)_tIs distributed at the edge of

Wherein

The expression parameter is

Student's t distribution. Further, x can be obtained_tCorresponding hidden variable z_t＝(z_t1,…,z_tM) The posterior distribution of

Wherein z is_t1,…,z_tMAre all variable from 0 to 1 and satisfy

The probability distribution of the oxygen content can then be found, thereby obtaining an estimate of the oxygen content.

Further, the oxygen content y_tThe probability distribution of (c) is:

wherein

Thus, an estimate of the oxygen content can be obtained as

Compared with the prior art, the invention has the following beneficial effects:

1. a mathematical model of oxygen content and auxiliary variables is established in a form of a mixed model, so that the problems of multi-mode and strong nonlinearity caused by working condition switching and a complex combustion process can be effectively solved;

2. through semi-supervised learning, labeled samples and unlabeled samples can be simultaneously utilized, and the problem of poor model parameter learning caused by insufficient labeled samples is solved, so that the estimation accuracy of the oxygen content is improved;

3. the method can solve the problems of parameter learning and model selection in one round of training at the same time, and does not need to traverse the number of all candidate mixed models, thereby improving the training efficiency.

Drawings

FIG. 1 is a flow chart of the method for on-line estimation of oxygen content in a first-stage furnace in an ammonia synthesis process based on a semi-supervised Bayesian Gaussian mixture model;

FIG. 2 is a schematic diagram of a process of a first stage furnace apparatus of a certain ammonia synthesis plant;

FIG. 3 is a diagram illustrating the estimation result of oxygen content according to the present invention, wherein the abscissa represents the oxygen content in mol%, the ordinate represents the serial number of the test sample, the solid line represents the true value of oxygen content, and the dotted line represents the estimated value of oxygen content;

fig. 4 is a schematic diagram of an estimation result of a gaussian mixture model on oxygen content, wherein an abscissa represents oxygen content in mol%, an ordinate represents a test sample number, a solid line represents a true value of oxygen content, and a dotted line represents an estimated value of oxygen content;

fig. 5 is a diagram showing the estimation result of the partial least squares model for the oxygen content, in which the abscissa represents the oxygen content in mol%, the ordinate represents the serial number of the test sample, the solid line represents the true value of the oxygen content, and the dotted line represents the estimated value of the oxygen content.

Detailed Description

The method for estimating the oxygen content of the first-stage furnace in the ammonia synthesis process based on the semi-supervised Bayesian Gaussian mixture model is further explained by combining a specific embodiment. It should be noted that the described embodiments are only intended to enhance the understanding of the present invention, and do not have any limiting effect on the present invention.

An online estimation method for oxygen content of a first-stage furnace in an ammonia synthesis process based on a semi-supervised Bayesian Gaussian mixture model is shown in figure 1, and specifically comprises the following steps:

(1) selection of auxiliary variables associated with oxygen content y in a one-stage furnace production plant

Wherein d represents the number of auxiliary variables;

in this embodiment, according to the process mechanism analysis of a first stage furnace device (as shown in fig. 2) in the production process of synthetic ammonia in a certain ICI-AMV process (with a yield of 1000t/d), 13 variables having the greatest influence on the oxygen content are selected as auxiliary variables, which are: fuel gas flow (x) to 03B001₁And the bit number: fr03001.pv), fuel exhaust gas flow (x) to 03B001₂And the bit number: fr03002.pv), pressure of fuel exhaust gas at exit 03E005 (x)₃And the bit number: PC03002.PV), 03B001 outlet fuel gas pressure (x)₄And the bit number: pc03007.pv), temperature of fuel exhaust gas at exit of 03E005 (x)₅And the bit number: ti03001.pv), temperature of fuel gas at exit of 03B002E06 (x)₆And the bit number: ti03009.pv), temperature of process gas at 03B001 inlet (x)₇And the bit number: TR03012.PV), temperature of fuel gas above and to the left of 03B001 (x)₈And the bit number: ti03013.pv), temperature of fuel gas right above 03B001 (x)₉And the bit number: TI03014.PV), temperature (x) of the gas mixture just above 03B001₁₀And the bit number: TR03015.PV), temperature (x) of the switching gas at left outlet of 03B001₁₁And the bit number: TR03016.PV), temperature (x) of the transfer gas at right side outlet of 03B001₁₂And the bit number: TR03017.PV) and 03B001 outlet transfer gas temperature (x)₁₃And the bit number: tr03020. pv). Thus the auxiliary variable x ═ x₁,…,x₁₃]I.e. by

(2) Collecting labeled sample sets containing both auxiliary variables and oxygen content

And unlabeled sample set containing only auxiliary variables

Wherein n is_lAnd n_uRespectively representing the number of the labeled samples and the number of the unlabeled samples;

the invention is distributed control from computerA2000 group of labeled sample sets (noted as auxiliary variables and oxygen content) were collected from the system database

And 5000 groups of unlabeled sample sets (noted as auxiliary variables) containing only auxiliary variables

As a training data set, where n _l2000 and n_u5000 represents the number of labeled and unlabeled swatches, respectively.

(3) To (X)_l,Y_l) And X_uCarrying out dimensionless treatment, and converting the sample variance of the auxiliary variable sample and the oxygen content sample into unit variance;

the dimension removing method comprises the following steps:

in the formula (I), the compound is shown in the specification,

sample standard deviations, x, representing the ith auxiliary variable and oxygen content, respectively_n(l) The sample value of the i auxiliary variable representing the n sample.

(4) Initializing model parameters given a truncation level M of the Dirichlet process

The conjugate prior distribution parameter and the posterior distribution parameter, the meaning of the model parameter is:

α represents a concentration factor of the dirichlet process;

χ_kparameters representing the kth hybrid model coefficients;

μ_kand Λ_kRespectively representing a mean vector and a precision matrix of the distribution of the auxiliary variable x in the kth mixed model;

representing a linear regression coefficient between an auxiliary variable x and an oxygen content y in the kth mixed model;

τ_kto represent

The accuracy matrix parameters of (1);

η_ktable k precision matrix coefficients of the measurement noise in the hybrid model.

In the invention, the conjugate prior distribution and the posterior distribution of each model parameter are determined as follows:

both the a priori distribution p (α) and the a posteriori distribution q (α) of α are gamma distributions, i.e., p (α) ═ Gam (α | a)₀,b₀) Q (α) ═ Gam (α | a, b), where Gam (α | a)₀,b₀) And Gam (α | a, b) each represent a parameter of (a)₀,b₀) Gamma distributions of (a) and (b);

χ_ka priori distribution p (χ)_k) And posterior distribution q (χ)_k) Are all beta distribution, i.e. p (χ)_k)＝Beta(χ_k|1,α)，q(χ_k)＝Beta(χ_k|h_k,l_k) Therein Beta (x)_k1, α) and Beta (χ)_k|h_k,l_k) Respectively representing the parameters (1, alpha) and (h)_k,l_k) Beta distribution of (a);

μ_k,Λ_ka priori distribution p (mu)_k,Λ_k) And posterior distribution q (mu)_k,Λ_k) Are all in a Gaussian-Weisset distribution, i.e.

Wherein

And

respectively represent a parameter of (m)₀,β₀,W₀,v₀) And (m)_k,β_k,W_k,v_k) Gauss-WeishateDistributing;

prior distribution of

And posterior distribution

Are all Gaussian distributed, i.e.

Wherein

Represents a mean vector of 0 and a covariance matrix of

The distribution of the gaussian component of (a) is,

represents the mean vector as ω_kThe covariance matrix is omega_kI denotes the identity matrix of the corresponding dimension;

τ_ka priori distribution p (τ)_k) And a posterior distribution q (tau)_k) Are all gamma distributions, i.e. p (τ)_k)＝Gam(τ_k|c₀,d₀)，q(τ_k)＝Gam(τ_k|c_k,d_k) Wherein Gam (τ)_k|c₀,d₀) And Gam (τ)_k|c_k,d_k) Respectively represent a parameter of (c)₀,d₀) And (c)_k,d_k) The gamma distribution of (1);

η_ka priori distribution p (η)_k) And posterior distribution q (η)_k) Are all gamma distributions, i.e. p (η)_k)＝Gam(η_k|e₀,f₀)，q(η_k)＝Gam(η_k|e_k,f_k) Wherein Gam (η)_k|e₀,f₀) And Gam (. eta.)_k|e_k,f_k) Respectively represent a parameter of (e)₀,f₀) And (e)_k,f_k) The gamma distribution of (1).

Therefore, in this step, it is necessary to initialize a priori distribution parameters, including

And posterior distribution parameters including

In this example, the parameter of the prior distribution is set to a₀＝1，b₀＝1，c₀＝1，d₀＝1，e₀＝1，f₀＝1，β₀＝1，v₀＝1，m₀＝0，W₀I ═ I; parameters a, b, h of the posterior distribution_k，l_k，c_k，d_k，e_k，f_k，β_k，v_k，m_k，W_k，ω_k，Ω_kIs a random value.

(5) Constructed with labeled samples (X)_l,Y_l) And unlabeled sample X_uAnd its corresponding hidden variable

In which z is_i＝(z_i1,…,z_iM)^TAnd z_j＝(z_j1,…,z_jM)^TRespectively represent the ith labeled sample (x)_i,y_i) And the jth unlabeled sample x_jCorresponding binary hidden variables, and satisfy

Has the following form:

(6) inputting the training sample set processed in the step (3), the initial model parameters in the step (4) and the likelihood function constructed in the step (5) into a semi-supervised Bayes Gaussian mixture model, and learning the optimal posterior distribution q (alpha) and the optimal posterior distribution q (alpha) of each model parameter through variational inference

The specific process comprises a variation expectation part and a variation maximization part.

In the expected part of variation, the implicit variable Z needs to be calculated_lAnd Z_uPosterior distribution q (Z)_l) And q (Z)_u). According to the principle of variational reasoning, the method can be obtained

Wherein

Express according to

Distribution calculation of

Is desired, x ═ x₁,…,χ_M)，μ＝(μ₁,…,μ_M)，Λ＝(Λ₁,…,Λ_M)，

η＝(η₁,…,η_M)，

Represents the mean value of μ_kThe covariance matrix is

The gaussian probability density function of (a) is,

and is

Where ψ (·) represents a digamma function. Therefore, the temperature of the molten metal is controlled,

wherein

For simplicity, the constant term in equation (7) is omitted; constant terms are still omitted when posterior distribution of each parameter is calculated subsequently.

By the same token, Z can be obtained_uPosterior distribution q (Z)_u) The following were used:

wherein

Thereby obtaining

Wherein

In the variation maximization part, model parameters need to be calculated

The posterior distribution q (Θ). The principle of variational reasoning is still adopted. Specifically, q (α) is solved by

Therefore, the posterior distribution q (α) of α is represented by the parameter update formula of Gam (α | a, b)

lnq(χ_k) Can be calculated as follows

Therefore, χ_kPosterior distribution q (χ)_k)＝Beta(χ_k|h_k,l_k) The parameter update formula is

lnq(μ_k,Λ_k) Can be calculated as follows

Wherein

The above formula is mu_k,Λ_kPosterior distribution of

The parameter update formula of (1), the trace of the Tr (-) matrix;

can be calculated as follows

Wherein the content of the first and second substances,

1 is the vector of all 1 columns,

represents the estimation error of the k-th hybrid model, and therefore,

posterior distribution of

The parameter update formula is

lnq(τ_k) Can be calculated as follows

Thus, τ_kA posterior distribution q (τ)_k)＝Gam(τ_k|c_k,d_k) The parameter update formula is

lnq(η_k) Can be calculated as follows

Thus η_kA posterior distribution q (η)_k)＝Gam(η_k|e_k,f_k) The parameter update formula is

By iteratively performing the variational expectation portion and the variational maximization portion, the posterior distribution of the model parameters will converge. The criterion for convergence in this example is that the relative increment of the lower bound of the variation is below a set threshold (10)^-7)。

(7) In the on-line phase, unknown samples x containing only auxiliary variables are acquired_tAnd (4) according to the dimension of eliminating the auxiliary variable in the step (3), estimating the oxygen content by using the optimal posterior distribution of the model parameters obtained in the step (6). Specifically, each model mixture coefficient pi ═ according to the posterior distribution of α calculated in step (6) and the properties of the dirichlet process (pi ═ pi₁,…,π_M) Can be calculated as

q(π)＝Dir(π|φ₁,…,φ_M) (25)

Wherein Dir (π | φ)₁,…,φ_M) The representative parameter is (phi)₁,…,φ_M) Is distributed, and

then, the dimensionless auxiliary variable x can be obtained from the posterior distribution of the model parameters calculated in step (6)_tIs distributed at the edge of

Wherein

The expression parameter is

Student's t distribution. Further, x can be obtained_tCorresponding hidden variable z_t＝(z_t1,…,z_tM) The posterior distribution of

Wherein z is_t1,…,z_tMAre all variable from 0 to 1 and satisfy

Finally, the oxygen content y can be obtained_tHas a probability distribution of

Wherein

Therefore, according to equation (29), an estimated value of the oxygen content can be obtained as

In order to verify the effectiveness of the present invention, an additional 4000 sets of labeled samples were collected from the computer distributed control system of the furnace device of the first stage of the ammonia plant as a calibration sample set, and the oxygen content was estimated according to step (7), and the average estimation result is shown in fig. 3. Meanwhile, fig. 4 and 5 show the average estimation results of the oxygen content by the conventional gaussian mixture model and the partial least square model, respectively. In the Gaussian mixture model, the mixing component quantity is set to be 12 through a Bayesian information criterion; in the partial least squares model, the number of principal components is set to 10 by the cross-validation method. It can be seen that the estimated value of oxygen content provided by the partial least squares model deviates significantly from the true value due to the inability to process non-linear objects; the estimation result of the conventional gaussian mixture model, although improved compared to the partial least square model, is still unsatisfactory, especially in the third and fourth operation regions (2500 th and 4000 th samples). In contrast, the method provided by the present invention provides an estimated oxygen level that substantially meets its true value in all operating regions.

The estimation accuracy of the invention and the traditional Gaussian mixture model and partial least square model is quantified by using Root Mean Square Error (RMSE), and is defined as follows

Wherein y is_tAnd

respectively representing the true oxygen content and the estimated value of the t-th check sample. The estimated RMSE of the method provided by the invention, the Gaussian mixture model and the partial least square model are 0.6933, 1.1515 and 1.7143 respectively. Therefore, the estimation accuracy of the oxygen content is obviously improved by the Gauss mixed model and the partial least square model, and the estimation accuracy is improvedThe error of the meter is reduced by about 40% and 60%, respectively.

The above-described embodiments are intended to illustrate rather than to limit the invention, and any modifications and variations of the present invention are within the spirit of the invention and the scope of the claims.

Claims (5)

1. A method for estimating the oxygen content of a first-stage furnace in an ammonia synthesis process on line based on a semi-supervised Bayes Gaussian mixture model is characterized by comprising the following steps:

(1) selecting an auxiliary variable associated with the oxygen content y of the primary furnace

Wherein d represents the number of auxiliary variables;

(2) collecting labeled sample sets containing both auxiliary variables and oxygen content

And unlabeled sample set containing only auxiliary variables

Wherein n is_lAnd n_uRespectively representing the number of the labeled samples and the number of the unlabeled samples;

(3) to (X)_l,Y_l) And X_uCarrying out dimensionless treatment, and converting the sample variance of the auxiliary variable sample and the oxygen content sample into unit variance;

(4) initializing model parameters given a truncation level M of the Dirichlet process

Is a conjugate prior distribution parameter of₀、b₀、c₀、d₀、e₀、f₀、β₀、v₀、m₀、W₀And posterior distribution parameters a, b, h_k、l_k、c_k、d_k、e_k、f_k、β_k、v_k、m_k、W_k、ω_k、Ω_kWherein, in the step (A),

and is

α represents a concentration factor of the dirichlet process;

χ_kparameters representing the kth hybrid model coefficients;

μ_kand Λ_kRespectively representing a mean vector and a precision matrix of the distribution of the auxiliary variable x in the kth mixed model;

representing a linear regression coefficient between an auxiliary variable x and an oxygen content y in the kth mixed model;

τ_kto represent

The accuracy matrix parameters of (1);

η_krepresenting the precision matrix coefficient of the measurement noise in the kth mixed model;

the meaning of the conjugate prior distribution parameter and the posterior distribution parameter is:

(a₀,b₀) And (a, b) a prior distribution parameter and a posterior distribution parameter respectively representing α;

(h_k,l_k) Denotes x_kThe posterior distribution parameters of (1);

(m₀,β₀,W₀,v₀) And (m)_k,β_k,W_k,v_k) Respectively represent (mu)_k,Λ_k) A priori distribution parameters ofChecking distribution parameters;

(c₀,d₀) And (c)_k,d_k) Respectively represent tau_kThe prior distribution parameter and the posterior distribution parameter;

(e₀,f₀) And (e)_k,f_k) Respectively represent η_kThe prior distribution parameter and the posterior distribution parameter;

ω_kand Ω_kTo represent

The posterior distribution parameters of (1);

(5) constructed with labeled samples (X)_l,Y_l) And unlabeled sample X_uAnd its corresponding hidden variable

In which z is_i＝(z_i1,…,z_iM)^TAnd z_j＝(z_j1,…,z_jM)^TRespectively represent the ith labeled sample (x)_i,y_i) And the jth unlabeled sample x_jCorresponding binary hidden variables, and satisfy

(6) Inputting the training sample set processed in the step (3), the initial model parameters in the step (4) and the likelihood function constructed in the step (5) into a semi-supervised Bayes Gaussian mixture model, and learning the optimal posterior distribution q (alpha) and the optimal posterior distribution q (alpha) of each model parameter through variational inference

Where q (-) denotes the optimal posterior distribution of the corresponding variable;

(7) and (4) collecting unknown samples only containing auxiliary variables, eliminating the dimension of the auxiliary variables according to the step (3), and estimating the oxygen content by using the optimal posterior distribution of the model parameters obtained in the step (6).

2. The method for on-line estimation of oxygen content in one-stage furnace in ammonia synthesis process based on semi-supervised Bayesian Gaussian mixture model as recited in claim 1, wherein the labeled sample (X) constructed in the step (5) is_l,Y_l) And unlabeled sample X_uAnd its corresponding hidden variable Z_l、Z_uThe likelihood function of (d) is:

wherein χ ═ χ (χ ═ χ)₁,…,χ_M)，μ＝(μ₁,…,μ_M)，Λ＝(Λ₁,…,Λ_M)，

η＝(η₁,…,η_M)，

Represents the mean value of μ_kThe covariance matrix is

The gaussian probability density function of (a) is,

3. the method for on-line estimation of oxygen content in one-stage furnace in ammonia synthesis process based on semi-supervised Bayesian Gaussian mixture model as recited in claim 1 or 2, wherein the parameters a, b, h of optimal posterior distribution of model parameters in step (6) are_k，l_k，c_k，d_k，e_k，f_k，β_k，v_k，m_k，W_k，ω_kAnd Ω_kHas the following form:

a＝a₀+M-1

c_k＝c₀+(d+1)/2

where ψ (·) denotes a digamma function, I denotes an identity matrix of the corresponding dimension,

1 is the all 1 column vector, Tr (-) is the trace of the matrix,

represents the estimation error of the k-th hybrid model,

here, the

Express according to

Distribution calculation of

(iii) a desire; kappa_ikAnd kappa_jkIs calculated in a manner that

Wherein

4. The method for estimating the oxygen content of the primary furnace in the ammonia synthesis process based on the semi-supervised Bayesian Gaussian mixture model according to claim 1 or 2, wherein the step (7) specifically comprises the following steps:

according to the posterior distribution of alpha calculated in the step (6) and the property of the Dirichlet process, each model mixing coefficient pi ═ pi (pi ═ pi)₁,…,π_M) Can be calculated as

q(π)＝Dir(π|φ₁,…,φ_M)

Wherein Dir (π | φ)₁,…,φ_M) The representative parameter is (phi)₁,…,φ_M) Is distributed, and

then, the dimensionless auxiliary variable x can be obtained from the posterior distribution of the model parameters calculated in step (6)_tIs distributed at the edge of

Wherein

The expression parameter is

Student's t distribution; further, x can be obtained_tCorresponding hidden variable z_t＝(z_t1,…,z_tM) The posterior distribution of

Wherein z is_t1,…,z_tMAre all variable from 0 to 1 and satisfy

The probability distribution of the oxygen content can then be found, thereby obtaining an estimate of the oxygen content.

5. The method for on-line estimation of oxygen content in one-stage furnace in ammonia synthesis process based on semi-supervised Bayesian Gaussian mixture model as recited in claim 4, wherein the oxygen content y is_tThe probability distribution of (c) is:

wherein

Thus, an estimate of the oxygen content can be obtained as

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