Microsoft Word - 211.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 61, 2017 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš Copyright © 2017, AIDIC Servizi S.r.l. ISBN 978-88-95608-51-8; ISSN 2283-9216 Soft Sensor Development in Fermentation Processes Using Recursive Gaussian Mixture Regression Based on Model Performance Assessment Yuhan Ding, Yong Su, Congli Mei* Jiangsu University, Zhenjiang, 212013, Jiangsu, China clmei@ujs.edu.cn The soft sensor modeling method based on moving window method (MW) and Gaussian mixture regression (GMR) has been developed in fermentation processes. However, the MW method always results in low computational efficiency because of frequency updating. GMR has the multi-model structure. In order to reduce calibration frequency of recursive GMR modeling methods, a recursive GMR soft sensor based on model performance assessment (MPA) is developed. According to the results of the model performance assessment, the model updating is selectively activated. The developed model was investigated to estimate biomass concentration in an industrial Erythromycin fermentation process. Compared with the GMR model, the prediction accuracy is improved obviously. 1. Introduction Accurate online estimation and prediction of key product quality variables are vital for continuously characterizing the dynamic behavior of chemical processes. For the past decades, soft sensors have been widely used to handle these problems, which provide frequent estimations of key process variables through those that are easy to be measured online (Jin et al., 2015 and Liu et al., 2010 and Luttmann et al., 2012). The most popular soft sensor methods are partial least squares (PLS) (Sharmin et al., 2006 and Wang et al., 2015), artificial neural networks (ANN) (Cui et al., 2012 and Sun et al., 2014), and support vector machines (SVM) (Liu et al., 2010 and Kaneko et al., 2014). Usually, soft sensors are constructed based on process measurements easy to measure online. From the measurement point of view, the performance of these traditional soft sensors may decrease even fail due to the actual working conditions, the processing of the nonlinear characteristics and the influence of the external disturbance of the system, which restricts above- mentioned soft sensors in practical cases. The recursive PLS algorithm (RPLS) is firstly proposed for the on-line calibration of model parameters. Subsequently, the RPLS algorithm is modified and extended, and it is applied to the modeling and quality prediction of chemical processes. The online calibration method of LS-SVM model (Li et al., 2009) is also applied to a chemical reaction process, hydro-isomerization process of C8-aromatics. Recently, a relative new machine learning method, i.e., Gaussian mixture regression (GMR), has been developed, and began to be applied in soft sensor modeling (Yuan et al., 2014). GMR updates the model usually by using moving window (MW) algorithm (Kadlec et al., 2011). However, the problem is that it does not have a policy to update the model simply by using data accumulation. In this paper, MPA-GMR modeling method based on model performance evaluation is proposed to solve these problems. The method eliminates the influence of the old samples to the new samples by the sliding window method, selects the model based on the model performance evaluation and adapts the model’s confidence limits along with the changes in process characteristics and model performance evaluation results. DOI: 10.3303/CET1761304 Please cite this article as: Ding Y., Su Y., Mei C., 2017, Soft sensor development in fermentation processes using recursive gaussian mixture regression based on model performance assessment, Chemical Engineering Transactions, 61, 1837-1842 DOI:10.3303/CET1761304 1837 2. Principle of GMR modeling based on model performance assessment 2.1 Principle of GMR modelling Assume X represents the space of the explanatory variables and Y is the space of the response variables. x is the input of training data ( x ∈ X ) and y is the ideal output data ( y ∈ Y ). For the given x and y , the joint probability density is given as (Sung, 2004) 1 ( , ) ( , ; , ) K jXY j j j f x y x yπ φ μ = =   (1) Where, subsequently, the mean jμ and covariance jΣ can be divided into the input and output parts like the following jx j jy μ μ μ   =     ， jXX jXY j jYX jYY   =          Eq. (1) shows that the relationship between the explanatory variables and the prediction value can be described by Gaussian mixture models (GMMs). Where ( , ; , )jjx yφ μ  denotes the probability density function of the multivariate GMM. The parameters of this model include the number of the mixture components K , the prior jπ , the mean value jμ , and the variance of each Gaussian component j , which are represented as 1 2( , , , )kθ θ θ θ=  with ( , , )j j j jθ π μ=  and the constraint 1 1 k j j π = = ( 0 1jπ≤ ≤ ). Similarly, the marginal probability density can be given as (H. G. Sung, 2004) , 1 ( ) ( , ) ( ; , ) K X X Y j jX jX j f x f x y dy xπ φ μ = = =   (2) The global GMR function can be deduced by combining Eq. (1) and (2) 2 1 ( , ) ( ) ( ) ( ; ( ), ) ( ) K XY Y X j j j jX f x y f y x w x y m x f x φ σ = = =  (3) with the mixing weight 1 ( ; , ) ( ) ( ; , ) j jX jX j K j jX jX j x w x x π φ μ π φ μ = =    (4) The mean and the variance of the conditional distribution can be acquired in closed form by 1( ) ( )jXj jX jX jYX m x xμ μ−= + −  (5) 12 jXj jYY jYX jXYσ −= −    (6) The prediction given a new input can be obtained by computing the expectation over the conditional distribution ( )Y Xf y x (Sung, 2004) 1 ( ) ( ) ( ) K Y X j j j E f y x w x m x =   =   (7) 2.2 Moving window algorithm for GMR The strategy of the local GPR models adaptation proposed is the moving window approach (Kadlec et al., 2011). As the new complete input–output samples are acquired, the window slides along the samples belonging to the local model such that the oldest sample is discarded and the new one is added to the 1838 window. In that way, not only is the current process behavior tracked, but also computational and memory requirements are constrained by the fixed window size. At some time instant t during online operation, the samples in a window of size N of the kth local model can be written as: [ ] [ ]( ) 1 1{ , , , , , }tk t N t t N tD x x y y+ − + −=   (8) with an appropriate regularized covariance matrix 2( )t t jK K Iσ= + . The new complete input-output sample 1 1{ , }t tx y+ + ，GPR model can be simply updated by removing first column and first row of covariance matrix due to marginalization property of the Gaussian process (Rasmussen, 2004) 1, 1temp t K K − −  =     (9) The new observation is added to the model by adding an appropriate row and column to the matrix tempK : 1 1 2 1 1 temp t t T t t j K K K K k σ + + + +   =   +     (10) where [ ]1 1( , )t i tiK K x x+ += ， ( )ti kx D∈ and 1 1 1( , )t t tk k x x+ + += .Then, the inverse of the matrix 1tK + is calculated and used for prediction. However, each model update requires the inverse of the N N× regularized covariance matrix 1tK + which is computationally and memory demanding. This inverse can be efficiently calculated using the matrix inversion formula with the previous inverse 1tK − . Generally, if the sample is removed from the window and a new sample added, the inverse of the new regularized covariance matrix can be efficiently calculated by the following expressions (Liu et al., 2011) ( ) 1 1 , ,1 1 , 1 , ( ) T t ti i i i temp t i i t i i K K K K K − − − −− − − − −         = −          (11) 1 1 1 2 1 0 1 0 0 1 T temp t T t K K αα α σ α − − + +    − = +    −      (12) where 2 2 1 1 1 1 1 T t t j t temp tk k K kσ σ − + + + += + −  and 1 1temp tK kα − +=  . 2.3 Online calibration model Firstly, Initialize the prior kπ , mean kμ , covariance k and meanwhile the sample is standardized. Assume ( ) k tπ , ( ) k tμ , and ( ) k t are the prior, mean, and covariance of the t step, respectively. When new samples [ ]1 1t tx y+ + are available, a new sample is added to the training sample set, and the most used samples are removed from the training samples. Modeling sample length and the number of components remain unchanged during online operation, then recursive update equations can be used in order to update the mean and covariance matrix of k-th Gaussian component and the related mixing coefficient. These are obtained from Zivkovic (2004) by ignoring Dirichlet prior on a number of components: ( 1) ( ) ( ) 1( ( ) ) t t t k k j t kw xπ π β π + += + − (13) 1( 1) ( ) ( ) 1( ) ( ) ( )j tt t tk k t kt k w x xμ μ β μ π ++ += + − (14) 1839 1( 1) ( ) ( ) ( ) ( ) 1 1( ) ( ) (( )( ) )j tt t t t T tk k t k t k kt k w x x xβ μ μ π ++ + += + − − −   (15) where 1tx + is the sample acquired at time instant t + 1. The influence of the samples is controlled by parameter β which is usually set to a value of 1/T where T can be understood as the number of already acquired samples that are used in the mixture update. In the process of correction of Gauss parameters, the model performance evaluation criteria and confidence limits are needed to be set up. Generally, the model performance can be measured by the prediction error. The root mean square error (RMSE) is commonly used a measure, which can be calculated as (Mu et al., 2006) 2 1 1 ˆ( ) SN i i is RMSE y y N = = − (16) where sN denotes the number of samples in the test set, iy and ˆiy are the actual and predicted measurements respectively. 2.4 Online implementation of MPA-GMR The implementation steps of MPA-GMR modeling method are as follows: Step1. Determine the length of training sample N and Gaussian component K , initialize the Gaussian mixture model parameters: the prior, mean and covariance. Step2. The sample is standardized, constituting the training sample set [X, Y]. Step3. According to the GMR model, to estimate the prediction error of the modeling sample; according to eq.(18) to calculate the standard deviation, and regarded it as the confidence limit of forecast. Step4. The sliding window moves one step forward to collect a new sample [ ]new newX Y . According to equation (4) calculating the mixing weight. The prior, the mean and the covariance of the training sample set are derived by the eq.(13), (14) (15) to take recursive correction. Step5. Calculate the predicted measurements and RMSE using eq.(7) and eq.(16), respectively. Step6. If eRMSE σ> , turn to step2, otherwise turn to step4. 3. Industrial case study In this work, the industrial data were collected from a practical erythromycin fermentation process in Zhenjiang medicine company, P. R. China (Liu et al., 2010). In the process, 15 process variables can be measured online: time, pH, dissolved oxygen, dextrin flow, propanol flow, soybean oil flow, water flow, air flow, soybean oil volume, propanol volume, dextrin volume, water volume, temperature, speed, and relative pressure. In each batch, 180 groups of data points can be collected by automatic instruments. The one batch is used as training set, and the other one batch is used as test set. Before constructing soft sensors, experimental data have to be normalized. Moreover, their dimensions should be reduced by variable selection methods for simplification. In this research, PCA method was used to select variables (King et al., 1999). Finally, five highly related variables selected as input variables are tabulated in Table 1. Biomass concentration, difficult to measure online, was chosen as the primary variable in soft sensing modeling. Table 1: Input variables in the erythromycin fermentation process No. Input variable Unit 1 Temperature K 2 PH - 3 Agitator power W 4 Dissolved oxygen concentration g/L 5 relative pressure N 1840 Table 2: Results of normality test Number of clusters Meet the normal distribution of variables (p>0.05) Number of variables 1 1, 4 2 2 1, 2, 3 3 3 1, 2, 3, 4, 5 5 4 1, 2, 4, 5 4 5 1, 3, 4,5 4 By the proposed normality test method, the optimal number of Gaussian components can be achieved on training set. From Table 2, the optimal number of mixture components is obtained as optK = 3. Meanwhile, the GMR and MPA-GMR are also constructed for comparisons of the quality prediction performance. The prediction results of the two different soft sensor approaches are shown in Fig. 1 and Fig. 2. It gives predicted values of the different model and original measurements, where the red line denotes the fitted biomass concentration measurements, the blue line represents the prediction results of the dynamic GMR soft sensor and the black dotted line depicts the 95% confidence interval of estimates. 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 Time(h) B io m as s co nc en tr at io n( g/ L ) measurements predicted value confidence interval Figure 1: Simulation results of GMR model 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 Time(h) B io m as s co nc en tr at io n( g/ L ) fitted analysing values predicted values confidence intervals Figure 2: Simulation results of MPA-GMR model 4. Conclusions In order to reduce the model calibration frequency of GMR model during modeling, the problems are solved about the confidence limits (or threshold) setting of the model and the difficulty of correction. This paper develops the modeling method based on performance evaluation model of GMR (MPA-GMR) and discusses the effect on improving the performance of the model. The simulation was carried out to study, where erythromycin fermentation reaction process as the research object. This modeling method generates 1841 confidence limits of the model automatically according to initial characteristics process; and results of an evaluation of model performance, selectively activating correction model calibration and confidence limits. Simulation results show that the MPA-GMR modeling method can improve the accuracy of the model and the computational efficiency. Acknowledgments This work is supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD [2011]6), Open Research Foundation of Key Laboratory of Modern Agricultural Equipment and Technology in Jiangsu University(NZ201301), and Natural Science Foundation of Jiangsu Province of China (BK20130531, BK20151345). References Birol G., Undey C., Cinar A., 2002, A modular simulation package for fed-batch fermentation: penicillin production, Computers & Chemical Engineering, 26, 1553-1565. Cui L., Xie P., Sun J., Yu T., Yuan J., 2012, Data-driven prediction of the product formation in industrial 2-keto- l-gulonic acid fermentation, Computers & Chemical Engineering, 36, 386-391. Jin H., Chen X., Yang J., Wang L., Wu L., 2015, Online local learning based adaptive soft sensor and its application to an industrial fed-batch chlortetracycline fermentation process, Chemometrics and Intelligent Laboratory Systems, 143, 58-78. 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