CN108171002B - Polypropylene melt index prediction method based on semi-supervised hybrid model - Google Patents

Polypropylene melt index prediction method based on semi-supervised hybrid model Download PDF

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CN108171002B
CN108171002B CN201711236164.4A CN201711236164A CN108171002B CN 108171002 B CN108171002 B CN 108171002B CN 201711236164 A CN201711236164 A CN 201711236164A CN 108171002 B CN108171002 B CN 108171002B
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邵伟明
宋执环
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Zhejiang University ZJU
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Abstract

The invention discloses a polypropylene melt index prediction method based on a semi-supervised mixed model, which comprises the steps of firstly, separately processing an auxiliary variable and a melt index, and explicitly considering the dependency relationship between the melt index and the auxiliary variable to establish a probabilistic mathematical model; and then, simultaneously mining labeled sample information and unlabeled sample information, and performing automatic model parameter learning and model selection by using an expectation maximization algorithm and a Bayesian information criterion. The method can provide the predicted value of the melt index on line in real time and evaluate the reliability of the melt index. By applying the method and the device, the accuracy of model parameter learning and model selection can be improved, all parameters do not need to be manually set, the prediction precision of the melt index can be effectively improved, and technical support and guarantee are provided for improving the product quality, reducing the cost, monitoring the process and making a decision.

Description

Polypropylene melt index prediction method based on semi-supervised hybrid model
Technical Field
The invention belongs to the field of process system soft measurement modeling and application, and particularly relates to a polypropylene melt index prediction method based on a semi-supervised hybrid model.
Background
The polypropylene resin has the advantages of small specific gravity, no toxicity, no odor, easy processing, high impact strength, good distortion resistance, good electrical insulation property and the like, and is widely applied to various industrial fields of national economy. Melt index is an important measure of the quality of polypropylene products and is usually measured by laboratory test analysis for a period of 2-4 hours. Such a large measurement lag can significantly reduce the dynamics and stability of a closed-loop control system, and the card edge control is more irrelevant, so that the polypropylene production process has strong fluctuation and more product waste, thereby not only increasing the production cost of enterprises, but also aggravating the environmental pollution.
The soft measurement model of the melt index can realize the online real-time prediction of the melt index. The current soft measurement modeling methods of melt index can be summarized into two main categories. The first is to build a mechanism model according to the mechanism of polymerization reaction, and the other is to build a data-driven model by using production process data. The accurate mechanism model has good dynamic property, high prediction precision and wide application range, however, the mechanism of the polypropylene production process is very complex and not completely clear, and the melt index mechanism model usually needs a large number of strong assumed conditions and is generally more suitable for steady-state design. In comparison, the data-driven model directly adopts the process data to establish the model, and can reflect the real-time performance of the actual production condition, so that the method is more suitable for the online prediction of the melt index.
There are two major difficulties in establishing a data-driven melt index soft measurement model. First, due to market demands, the polypropylene production process has many operating conditions, and the product usually has a plurality of grades (up to tens of grades), so that the polypropylene production process data shows strong non-linearity, non-gaussian property and other characteristics. A single global model (e.g., partial least squares model, neural network model, support vector machine model, etc.) has difficulty providing satisfactory prediction accuracy across all brands. Second, current melt index prediction methods are typically supervised learning methods, i.e. relying only on labeled samples with melt index assay values. However, because the melt index assay period is long, the number of labeled samples is generally small, so that accurate model parameters of the melt index soft measurement model are difficult to obtain due to 'over learning' or 'under learning' and the like. A soft measurement model with poor training cannot provide satisfactory prediction precision necessarily, and manual parameter setting is time-consuming and labor-consuming and has high difficulty. On the other hand, unlabeled samples containing only auxiliary variables exist in large quantity, but the existing melt index soft measurement model cannot mine important process information contained in the unlabeled samples. Furthermore, after the soft measurement model provides the melt index prediction value, it is desirable to know how reliable the prediction value is, i.e., to evaluate the prediction accuracy of the melt index. Unfortunately, most current melt index soft measurement models do not have this functionality. Therefore, it is necessary and urgent to research and develop a melt index soft measurement model having the functions of multi-grade processing, label-free sample information mining, prediction accuracy evaluation and the like, which is helpful to improve the prediction accuracy of the melt index and help the production enterprises to realize the aims of energy conservation, environmental protection, cost reduction and efficiency improvement.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a polypropylene melt index prediction method based on a semi-supervised mixed model, which establishes the nonlinear relation between an auxiliary variable and a melt index in the form of the mixed model, can adaptively respond to grade switching, effectively solves the problems of strong nonlinearity and non-Gaussian caused by a plurality of product grades, and simultaneously excavates the information contained in a labeled sample and a non-labeled sample through semi-supervised learning so as to ensure that the model training is more reliable. The specific technical scheme is as follows:
a polypropylene melt index prediction method based on a semi-supervised mixing model is characterized by comprising the following steps:
(1) selecting an auxiliary variable x ∈ R associated with the polypropylene melt index ymWherein m represents the number of auxiliary variables;
(2) collecting a labeled sample set containing both auxiliary variables and melt indexAnd unlabeled sample set containing only auxiliary variables
Figure BDA0001489021850000022
Wherein n is1And n2Respectively representing the number of the labeled samples and the number of the unlabeled samples;
(3) to [ X ]L,YL]And XUPerforming dimensionless processing, and converting the sample variance of the auxiliary variable sample and the melt index sample into unit variance;
(4) giving the number K of Gaussian components of the semi-supervised hybrid model, and randomly initializing model parameters
Figure BDA0001489021850000023
Wherein alpha iskIs the k-th Gauss componentThe prior probability of (a) being,
Figure BDA0001489021850000024
and
Figure BDA0001489021850000025
mean and covariance matrices, ω, of the edge probability density of x in the kth Gaussian component, respectivelykAnd ωkAre the regression coefficients of x and y in the kth gaussian component respectively,a measurement noise variance representing the melt index y in the kth gaussian component;
(5) inputting the labeled sample set, the unlabeled sample set and the initial model parameters in the step (4) after the processing in the step (3) into a semi-supervised hybrid model, and maximizing a semi-supervised objective function L (theta)K) Learning model parameters ΘK；
(6) Traverse K ═ Kmin,Kmin+1,…,KmaxRepeating the steps (4) and (5), calculating the optimal number of Gaussian components by using a Bayesian information criterion, and recording the number as KoptAnd from
Figure BDA0001489021850000027
To select a corresponding set of model parameters
Figure BDA0001489021850000028
(7) Collecting unknown samples only containing auxiliary variables, eliminating the dimension of the auxiliary variables according to the step (3), and utilizing the optimal Gaussian component number K obtained in the step (6)optAnd corresponding model parameter set
Figure BDA0001489021850000029
The melt index of polypropylene is predicted and a confidence interval for the prediction is provided.
Further, the objective function L (Θ) of the semi-supervised hybrid model constructed in the step (5) isK) Comprises the following steps:
Figure BDA00014890218500000210
wherein, Pk(yi|xi) Given x in the k-th Gaussian componentiConditional probability density of melt index y, Pk(xi) And Pk(xj) X in the k-th Gauss componentiAnd xjEdge probability density of RikIs [ x ]i,yi]Degree of membership to the kth Gaussian component, RjkIs xjFor the membership degree of the kth Gaussian component, the calculation formula is as follows:
Figure BDA0001489021850000031
Figure BDA0001489021850000032
Figure BDA0001489021850000034
wherein N (·; μ, Σ) represents a normal distribution having a mean μ and a covariance matrix Σ;
Figure BDA0001489021850000035
and
Figure BDA0001489021850000036
a mean vector and a covariance matrix representing the joint probability densities of x and y, respectively, wherein,
Figure BDA0001489021850000037
Figure BDA0001489021850000038
further, the model parameters
Figure BDA0001489021850000039
Has the following form:
Figure BDA00014890218500000310
Figure BDA00014890218500000311
Figure BDA00014890218500000312
Figure BDA00014890218500000314
in the formula 1 is a full 1 column vector.
Further, in the step (5), the optimal number of Gaussian components KoptThe calculation formula of (a) is as follows:
wherein, BIC (K) represents the value of Bayesian information criterion when the number of Gaussian components is K, and the calculation formula is as follows:
Figure BDA0001489021850000042
compared with the prior art, the invention has the following beneficial effects:
1. the nonlinear relation between the auxiliary variable and the melt index is established in a form of a mixed model, so that the grade switching can be adaptively responded, and the problems of strong nonlinearity and non-Gaussian caused by a plurality of product grades are effectively solved;
2. the information contained in the labeled sample and the unlabeled sample is mined simultaneously through semi-supervised learning, so that the model training is more reliable;
3. all model parameters can be adaptively learned without manual intervention and additional verification data sets, so that the time and the energy for model development are greatly saved;
4. besides providing the predicted value of the melt index, the invention can also provide the confidence interval of the predicted value, which is used for judging the prediction precision, classifying abnormal samples, ensuring the reliability of model updating and the like.
Drawings
FIG. 1 is a flow chart of a semi-supervised mixing model based polypropylene melt index prediction method of the present invention;
FIG. 2 is a schematic diagram of a process of a Spheripol-II liquid-phase bulk-process polypropylene production apparatus of a petrochemical enterprise;
FIG. 3 is a diagram illustrating the online prediction result and confidence interval of the melt index of polypropylene based on a semi-supervised mixing model;
FIG. 4 is a diagram illustrating the prediction result of partial least squares model on melt index.
Detailed Description
The method for predicting the melt index of polypropylene based on semi-supervised mixing model of the present invention is further illustrated below with reference to specific examples. 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.
A polypropylene melt index prediction method based on a semi-supervised mixing model is shown in figure 1, and specifically comprises the following steps:
(1) selecting an auxiliary variable x ∈ R associated with the polypropylene melt index ymWherein m represents the number of auxiliary variables;
in this embodiment, according to the mechanism analysis of the Spheripol-II liquid-phase bulk-process polypropylene production process (as shown in FIG. 2) of a petrochemical company, 8 variables having the greatest influence on the melt index are selected as auxiliary variables, and are respectively the hydrogen/propylene concentration ratio (x) in the reactor R2011) Catalyst/propylene concentration ratio (x) in reactor R2012) Heat of reaction (x) of reactor R2013) Reaction Density (x) of reactor R2014) Hydrogen/propylene concentration ratio (x) in reactor R2025) Catalyst/propylene concentration ratio (x) in reactor R2026) Heat of reaction (x) of reactor R2027) And the reaction density (x) of reactor R2028) Thus the auxiliary variable x ═ x1,x2,…,x8]I.e. x ∈ Rm,m＝8。
(2) Collecting a labeled sample set containing both auxiliary variables and melt index
Figure BDA0001489021850000051
And unlabeled sample set containing only auxiliary variablesWherein n is1And n2Respectively representing the number of the labeled samples and the number of the unlabeled samples;
the invention collects a set of labeled sample sets 200 (noted as auxiliary variables and melt index) from a decentralized computer control system database) And unlabeled exemplar set 200 (denoted as the set of auxiliary variables) containing only auxiliary variables
Figure BDA0001489021850000054
) As a training data set, where n1200 and n2The number of labeled and unlabeled swatches is 200.
(3) To [ X ]L,YL]And XUPerforming dimensionless processing, and converting the sample variance of the auxiliary variable sample and the melt index sample into unit variance;
the dimension removing method comprises the following steps:
Figure BDA0001489021850000055
in the formula (I), the compound is shown in the specification,
Figure BDA0001489021850000056
sample standard deviations, x, for the ith auxiliary variable and melt index, respectivelyn(l) The sample value of the i auxiliary variable representing the n sample.
(4) Giving the number K of Gaussian components of the semi-supervised hybrid model, and randomly initializing model parameters
Figure BDA0001489021850000057
Wherein alpha iskIs the prior probability of the kth gaussian component,
Figure BDA0001489021850000058
and
Figure BDA0001489021850000059
mean and covariance matrices, ω, of the edge probability density of x in the kth Gaussian component, respectivelykAnd ωkAre the regression coefficients of x and y in the kth gaussian component respectively,
Figure BDA00014890218500000510
a measurement noise variance representing the melt index y in the kth gaussian component;
(5) inputting the labeled sample set, the unlabeled sample set and the initial model parameters in the step (4) after the processing in the step (3) into a semi-supervised hybrid model, and maximizing a semi-supervised objective function L (theta)K) Learning model parameters ΘK(ii) a I.e. the prior probability of each gaussian component, the edge probability density of x, the functional relationship of x and y, and the noise variance, given the number K of gaussian components of the mixture model. The specific process is as follows:
edge probability density P of auxiliary variable x in K-th (K-1, 2, …, K) gaussian componentk(x) And the functional relationship y of x and y is fk(x) Is defined as
Figure BDA0001489021850000061
Wherein N (·; mu, sigma) represents the normal distribution with mean value mu and covariance matrix sigma;
the conditional probability density P of the melt index y can be obtained by performing linear Gaussian operation on the formula (2)k(y | x) and the joint probability density P of x and yk(x, y) as shown in the following formula:
Figure BDA0001489021850000062
according to the definition of covariance and the addition rule of probability, the global probability density P (x) of x, and the global joint probability density P (x, y) of x and y can be obtained in the following form:
Figure BDA0001489021850000063
in the formula (I), the compound is shown in the specification,
Figure BDA0001489021850000064
and
Figure BDA0001489021850000065
a mean vector and a covariance matrix representing the joint probability densities of x and y, respectively; alpha is alphakRepresenting the prior probability of the kth Gaussian component in the mixed model;
in the semi-supervised hybrid model, the parameters to be estimated given the number K of Gaussian components comprise
Figure BDA0001489021850000066
Because modeling is carried out by a probabilistic learning method, EM algorithm can be adopted to learn the model parameter thetaK. In step E, the posterior probability of each hidden variable is calculated according to the following formula:
in the formula ziAnd zjRespectively representing discrete hidden variables corresponding to the ith labeled sample and the jth unlabeled sample;
in step M, the log-likelihood function of the complete data is first determined as shown in the following equation:
Figure BDA0001489021850000071
wherein
Figure BDA0001489021850000072
Andrepresenting sets of hidden variables corresponding to labeled exemplars and unlabeled exemplars, respectively. Introducing Lagrange multiplier beta, combining constraint
Figure BDA0001489021850000074
Constructing the following Lagrangian function;
Figure BDA0001489021850000075
will be provided with
Figure BDA0001489021850000076
For alphakDerivation and beta elimination to obtain alphakThe learning formula of (2) is shown as follows:
Figure BDA0001489021850000077
mixing L (theta)K) Are paired respectivelyAnd
Figure BDA0001489021850000079
derived by derivation
Figure BDA00014890218500000710
Figure BDA00014890218500000711
Figure BDA00014890218500000712
Figure BDA00014890218500000713
In the formula
Figure BDA00014890218500000714
Figure BDA00014890218500000714
1 is a full 1 column vector.
The above-mentioned derivative is set to zero to obtain
Figure BDA00014890218500000715
And
Figure BDA00014890218500000716
of learning formulae, i.e.
Figure BDA00014890218500000717
Figure BDA00014890218500000718
Figure BDA0001489021850000081
Figure BDA0001489021850000082
(6) Traverse K ═ Kmin,Kmin+1,…,KmaxRepeating the steps (4) and (5), calculating the optimal number of Gaussian components by using a Bayesian information criterion, and recording the number as KoptAnd fromTo select a corresponding set of model parameters
Figure BDA0001489021850000085
Wherein, bic (K) represents the value of the bayesian information criterion when the number of gaussian components is K, and the calculation formula is as follows:
Figure BDA0001489021850000086
(7) acquiring unknown samples x containing only auxiliary variablesqEliminating the dimension of the auxiliary variable according to the step (3), and utilizing the optimal Gaussian component number K obtained in the step (6)optAnd corresponding model parameter set
Figure BDA0001489021850000087
Predicting the melt index of the polypropylene and providing a confidence interval of the predicted value, wherein the confidence interval is as follows:
the melt index y is calculated as followsqConditional probability density P (y) ofq|xq)：
Figure BDA0001489021850000088
In the formula
Figure BDA0001489021850000089
zqRepresentation and unknown sample xqThe corresponding hidden variables.
According to formula (19) for the melt index yqIs predicted to be
Figure BDA00014890218500000810
Further, the variance of the predicted value is calculated as follows
Figure BDA00014890218500000811
According to the formulae (20) and (21), the melt index yqThe confidence interval (three times standard deviation) of (A) is calculated as
Figure BDA0001489021850000091
To verify the effectiveness of the present invention, additional sets of labeled samples 148 are collected from the petrochemical company computer decentralized control system as a verification sample set, and the melt index is predicted according to step (7), with the prediction results and corresponding confidence intervals as shown in fig. 3. Meanwhile, fig. 4 shows the prediction result of the conventional partial least squares model on the melt index. The prediction accuracy of the present invention and the partial least squares model is quantified using Root Mean Square Error (RMSE), as defined below
Wherein y istAnd
Figure BDA0001489021850000093
respectively representing the assay value and the predicted value of the t-th check sample. The predicted RMSE of the method and the partial least square model provided by the invention are 0.3233 and 0.4097 respectively. Therefore, compared with the traditional partial least square model, the method has the advantages that the prediction precision of the melt index is obviously improved, and the prediction error is reduced by 21%. In addition, FIG. 3 shows that the confidence interval provided by the present invention provides a suitable width to cover the true distribution of melt indices that conventional methods do not.
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 (4)

1. A polypropylene melt index prediction method based on a semi-supervised mixing model is characterized by comprising the following steps:
(1) selecting an auxiliary variable x ∈ R associated with the polypropylene melt index ymWherein m represents the number of auxiliary variables;
(2) collecting a labeled sample set containing both auxiliary variables and melt index
Figure FDA0002177281980000011
And unlabeled sample set containing only auxiliary variables
Figure FDA0002177281980000012
Wherein n is1And n2Respectively representing the number of the labeled samples and the number of the unlabeled samples;
(3) to [ X ]L,YL]And XUPerforming dimensionless processing, and converting the sample variance of the auxiliary variable sample and the melt index sample into unit variance;
(4) giving the number K of Gaussian components of the semi-supervised hybrid model, and randomly initializing model parameters
Figure FDA0002177281980000013
Wherein alpha iskIs the prior probability of the kth gaussian component,
Figure FDA0002177281980000014
and
Figure FDA0002177281980000015
mean and covariance matrices, ω, of the edge probability density of x in the kth Gaussian component, respectivelykAnd ωkAre the regression coefficients of x and y in the kth gaussian component respectively,
Figure FDA0002177281980000016
a measurement noise variance representing the melt index y in the kth gaussian component;
(5) inputting the labeled sample set and the unlabeled sample set processed in the step (3) and the initialization model parameters in the step (4) into a semi-supervised hybrid model, and maximizing a semi-supervised objective function L (theta)K) Learning model parameters ΘK；
(6) Traverse K ═ Kmin,Kmin+1,…,KmaxRepeating the steps (4) and (5), calculating the optimal number of Gaussian components by using a Bayesian information criterion, and recording the number as KoptAnd from
Figure FDA0002177281980000017
To select a corresponding set of model parameters
Figure FDA0002177281980000018
(7) Collecting unknown samples only containing auxiliary variables, eliminating the dimension of the auxiliary variables according to the step (3), and utilizing the optimal Gaussian component number K obtained in the step (6)optAnd corresponding model parameter set
Figure FDA0002177281980000019
The melt index of polypropylene is predicted and a confidence interval for the prediction is provided.
2. The method for predicting melt index of polypropylene based on semi-supervised mixing model as recited in claim 1, wherein the objective function L (Θ) of the semi-supervised mixing model constructed in the step (5)K) Comprises the following steps:
Figure FDA00021772819800000110
wherein, Pk(yi|xi) Given x in the k-th Gaussian componentiMelt index yiConditional probability density of (1), Pk(xi) And Pk(xj) X in the k-th Gauss componentiAnd xjEdge probability density of RikIs [ x ]i,yi]Degree of membership to the kth Gaussian component, RjkIs xjFor the membership degree of the kth Gaussian component, the calculation formula is as follows:
Figure FDA0002177281980000021
Figure FDA0002177281980000022
Figure FDA0002177281980000023
Figure FDA0002177281980000024
Figure FDA0002177281980000025
wherein N (·; mu, sigma) represents a normal distribution with a mean value mu and a covariance matrix sigma;
Figure FDA0002177281980000026
anda mean vector and a covariance matrix representing the joint probability densities of x and y, respectively, wherein,
Figure FDA0002177281980000028
3. the semi-supervised hybrid model-based polypropylene melt index prediction method as recited in claim 2, wherein the model parameters
Figure FDA00021772819800000210
Has the following form:
Figure FDA00021772819800000212
Figure FDA00021772819800000213
Figure FDA00021772819800000214
in the formula
Figure FDA00021772819800000216
1 is a full 1 column vector.
4. The semi-supervised mixing model-based polypropylene melt index prediction method as recited in claim 3, wherein in the step (6), the optimal number K of Gaussian componentsoptThe calculation formula of (a) is as follows:
Figure FDA00021772819800000217
wherein, bic (K) represents the value of the bayesian information criterion when the number of gaussian components is K, and the calculation formula is as follows:
Figure FDA0002177281980000031
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