Adaptive Soft Sensor Based On Incremental Gaussian Mixture Regression

In modern industrial processes,the measuring of key quality-related indexes and parameters is one of the crucial factors to effectively monitor process-operating status and achieve process stability control.Due to measurement environment limitations in practical industrial processes introduced by complicated physical/chemical reactions and material energy transfer,it is hard for traditional hardware sensor devices to directly measure key indexes that closely relate to product quality,such as measurement delay and high maintenance cost.In recent years,soft sensors have been widely accepted for online estimating quality variables in industrial processes,which establish a mathematical model using auxiliary variables(process variables)that are easily measured as input and key product indexes(quality variables)as output.Nevertheless,the data characteristics in practical industrial processes become more and more complicated,new methods are supposed to deal with these modeling problems further.Based on Gaussian mixture regression(GMR),an adaptive soft sensor modeling scheme is proposed for time-varying,non-Gaussian and limited labeled-data industrial processes in this paper.The whole studies are mainly divided into three parts as follows:1)Aiming at non-Gaussian and time-varying industrial processes,an incremental Gaussian mixture regression(IGMR)is proposed to tackle the common adaptive model updating issues.The IGMR first combines incremental learning with the GMR,which merges real-time new data blocks with historical model incrementally.The model fusion incorporates symmetrical Kullback-Leibler(KL)divergence for judging the similarity of Gaussian components,which shows the effectiveness in tackling the model inefficiency.Besides,since the historical data are not saved,the rate of model updating is accelerated effectively.2)Dedicating to solve the industrial process of insufficient labeled data and time-varying property,a semi-supervised incremental Gaussian mixture regression(S~2IGMR)is proposed.In S~2IGMR,both labeled and unlabeled data are fully utilized for learning the model.Component parameters(probability density function,regression relationship coefficients,mixture coefficients)are learned via expectation–maximization(EM)algorithm,and model selection is based on Bayesian information criterion(BIC),so the S~2IGMR achieves higher prediction accuracy.3)In order to handle automatic Gaussian components selection,singular covariances and overfitting easily occurring in parameter learning processes,a semi-supervised variational incremental Gaussian mixture regression model(S~2VIGMR)is proposed.In S~2VIGMR,all parameters are treated as stochastic variables with prior distributions.Meanwhile,the variational Bayesian expectation maximization(VBEM)algorithm is developed to drive the contributions of ineffective components to zero,which could effectively alleviate the difficulties of overfitting and model selection,and therefore,can improve the model performance.