Quality prediction for polypropylene production process based on. CLGPR model

Transcription

1 uality predictio for polypropylee productio process based o CLGPR model Zhiiag Ge 1, ao Che, Zhihua Sog 1 1 State Key Laboratory of Idustrial Cotrol echology, Istitute of Idustrial Process Cotrol, Zhejiag Uiversity, Hagzhou 317, Zhejiag, Chia Divisio of Civil, Chemical ad Evirometal Egieerig, Uiversity of Surrey, Guildford GU 7XH, UK Abstract Olie measuremet of the melt idex is typically uavailable i idustrial polypropylee productio processes, soft sesig models are therefore reuired for estimatio ad predictio of this importat uality variable. Polymerizatio is a highly oliear process, which usually produces products with multiple uality grades. I the preset paper, a effective soft sesor, amed Combied Local Gaussia Process Regressio (CLGPR), is developed for predictio of the melt idex. While the itroduced Gaussia process regressio model ca well address the high oliearity of the process data i each operatio mode, the local modelig structure ca be effectively exteded to processes with multiple operatio modes. Feasibility ad efficiecy of the proposed soft sesor are demostrated through the applicatio to a idustrial polypropylee productio process. Keywords: Melt idex; uality predictio; Gaussia process regressio; Pricipal compoet aalysis; Multiple operatio modes. 1. Itroductio As a importat material, polypropylee has bee widely used i may differet fields icludig chemical, optical ad medical sectors. he maufacture of polypropylee is a billio-dollar busiess, which has see about 5% aual growth rate i cosumptio i recet years (Shi et al., 6). he uality of polypropylee is covetioally assessed by the melt idex i practical idustrial processes (Kiparissides et al., 1993). However, due to the challeged egieerig activity ad the complexity of the process, the melt idex is usually obtaied through a offlie aalytical procedure, which may take up to two hours. herefore, this will cause a time delay to the uality cotrol system, sice the process is without ay uality idictor durig this period of time. A alterative is to istall a olie aalyzer, such as those based o ear ifrared spectroscopy or ultrasoud, for measurig the melt idex (Coates et al., 3). However, curret olie aalyzers are very expesive ad reuire cosiderable maiteace effort, resultig i limited adoptio i practical plats. Recetly, with the wide utilizatio of the distributed cotrol system (DCS) i idustrial processes, a large amout of process data ca be routiely recorded. Amog these recorded process data, some process variables that are highly correlated with the process uality ca be used to estimate ad predict the uality variable. he method of iferrig difficult-to-measure o whom all correspodece should be addressed. zge@iipc.zju.edu.c; gezhiiag@gmail.com - 1 -

2 uatities by easy-to-measure variables is kow as soft sesor, iferetial sesor or virtual sesor (Kadlec et al., 9). Particularly, i the polypropylee productio process, it was show that the melt idex ca be predicted by some related process variables that sigificatly reflect the process coditio ad the product uality, such as hydroge cocetratio of the reactor, propylee feed rate, reactio temperature, amog others (Shi et al., 6; Kiparissides et al., 1993; Coates et al., 3; Zhag et al., 1998; Ohshima & aigaki, ; Liu, 7). o date, differet data-based soft sesors have bee developed for uality predictio purpose, icludig pricipal compoet regressio (PCR) ad partial least suares (PLS) for liear processes, artificial eural etwork (ANN) ad support vector machie (SVM) for oliear processes ad etc (Kadlec et al., 9; Gozaga et al., 9; Kao et al., 8; Yag & Gao, 6; Gao & Re, 1). he polypropylee productio process is a well-kow highly oliear process as evideced by mechaistic aalysis of the reactios ad plats (Liu, 7). herefore, oliear soft sesors should be cosidered. Besides, practical idustrial processes are always cotamiated by oises, ad thus those measured process variables are iheretly radom variables. I this case, it is more appropriate to make statistical iferece ad predictio decisios based o probabilistic models. Ufortuately, most traditioal soft sesor modelig methods were costructed i a determiistic maer. Recetly, a probabilistic modelig method amely Gaussia process regressio (GPR) has gaied much attetio i both statistical ad egieerig areas, which is iitially proposed by (O Haga, 1978). It is demostrated that a large class of ANN based Bayesia regressio models will fially coverge to a approximate Gaussia process. herefore, GPR model has bee cosidered as a alterative method for oliear system modelig. Alog last several decades, lots of comparative studies have show that the GPR model performs better tha other oliear modelig approaches (Csato & Opper, ; Chu & Ghahramai, 5; Rasmusse & Williams, 6; Likar & Kocija, 7; Shi et al., 7; Che & Re, 9; ag et al., 1). Aother advatage of the GPR model is due to its probabilistic model structure, which ca successfully icorporate the oise iformatio ad provide a ucertaity predictio result for the process. o our best kowledge, GPR has rarely bee reported for soft sesor modelig i the process system egieerig area. Due to its efficiecy for oliear system modelig, the GPR model is employed for soft sesor costructio i the preset paper. For dimesio compressio of process data, ad also to address high correlatios betwee differet variables, the traditioal PCA method ca be iitially performed, which meas the GPR model will be costructed upo score variables of the PCA model. I this sese, the PCA-GPR method ca be cosidered as a probabilistic form of oliear PCA based regressio model. Beside of the oliear behavior, the polypropylee productio process also exhibits multiple productio grades, which is probably drive by differet market reuiremets (Liu, 7). herefore, this process always has multiple operatio modes. A straightforward idea is to build multiple GPR models uder differet operatio coditios. However, a importat issue is how to select the appropriate GPR model for the curret data sample. If the plat egieer kows the operatio mode of the process, he/she ca simply select the correspodig local GPR model for predictio. However, automatic process operatio with miimal huma itervetio is usually desired i moder processig plats. Besides, if the curret data belog - -

3 to the trasitio betwee differet operatio modes, automatic weightig of multiple local models should also be cosidered. o address this issue, differet forms of mixture Gaussia process models have bee developed, such as ifiite mixtures of Gaussia process experts ad hierarchical Gaussia process mixture model (Rasmusse & Ghahramai, ; Shi et al., 5; Ou & Marti, 8). However, most of these mixture Gaussia process models ivolve computatioally expesive Mote Carlo methods, such as Gibbs samplig ad hybrid Markov chai Mote Carlo (MCMC) samplig. he algorithmic complexity of the mixture Gaussia process models is the major reaso to limit their applicatio to large datasets ad/or high dimesioal processes. Besides, whe some operatio mode is idetified, the traditioal mixture Gaussia process models eed to be re-traied, demadig cosiderable computatioal resource ad maiteace effort. Other similar techiues for multimode modelig iclude the Gaussia mixture model (GMM) approach ad the fuzzy modelig method (Choi et al., 4; Yu & i, 8; Yu & i, 9; Rog et al., 6; Huag & Hah, 9). he GMM method ca also gives a probabilistic model structure for differet operatio modes. However, most of GMM approaches are limited i the liear case. Although the fuzzy modelig method ca address the oliear behavior of the process data, its performace depeds o the modelig structure of oliear models. Besides, the fuzzy modelig method may have some user-defied parameters, which are difficult to determie. I this paper, we ited to build multiple local PCA-GPR model based soft sesors for differet operatio modes i the first step. he, a soft assigmet ad combiatio strategy is proposed for result fusio i differet operatio modes for the data sample. Compared to traditioal mixture Gaussia process models, the implemetatio of our method is much easier, thus, it is more useful for practical applicatio. It is oted that this assigmet ad combiatio strategy ca perform automatically without reuiremet of additioal iformatio of the process. Besides, whe some operatio mode is available for modelig, we ca simply build a local PCA-GPR model for this operatio mode, ad put it ito the model pool. herefore, compared to the traditioal mixture Gaussia process model, model updatig accordig to the chage of process coditios is much easier i our proposed method. he rest of this paper is orgaized as follows. I sectio, some prelimiaries of the traditioal PCA ad GPR models are itroduced, which is followed by the detailed descriptio of the proposed soft sesor for uality predictio i the ext sectio. I sectio 4, a idustrial applicatio case study of the polypropylee productio process is provided performace evaluatio of the proposed method. Fially, some coclusios are made.. Some prelimiaries.1. Pricipal compoet aalysis (PCA) Give a dataset m X R, where m is the umber of process variables, ad is the sample umber for each variable, PCA is carried out upo the covariace matrix of X. raditioally, the sigular value decompositio (SVD) method ca be employed for costructio of the PCA model. Suppose k pricipal compoets have bee selected i the PCA model, X ca be decomposed as (i, 3) - 3 -

4 X P P P E (1) ( m k) R k where R ad are score matrices i the pricipal compoet subspace m k (PCS) ad the residual subspace (RS), P R m ( m k) ad P R correspod to loadig m matrices i PCS ad RS. E R is the residual matrix... Gaussia process regressio (GPR) Cosider a traiig dataset m X R ad y R, where X{ x m i R } i 1,,, is the iput data samples with m dimesios, ad y { yi R} i 1,,, is the output data sample, the aim of the regressio model is to build a fuctioal relatioship betwee x ad y. Particularly, a Gaussia process regressio model is defied such that the regressio fuctio y f( x ) has a Gaussia prior distributio with zero mea, which is give as y [ f ( x ), f ( x ),, f ( x )] ~ GP(, C ) () 1 where C is a covariace matrix, with its ij-th elemet defied as C C( x, x ). o ij i j calculate the GPR model, differet covariace fuctios ca be selected. A commoly used covariace fuctio is the suared-expoetial covariace fuctio, which is give as (Rasmusse & Williams, 6) 1 C( xi, x j ) f exp{ ( xi x j ) M( xi x j )} ij (3) where ij 1 if i j, otherwise ij, M I, I is a idetify matrix with appropriate dimesio. he expoetial term is similar to the form of a radial basis fuctio, with its legth-scale, ad the terms f ad correspod to sigal variace ad oise variace. o idetify the GPR model, or precisely, to determie the value of the hyperparameter set (,, ), the followig log-likelihood fuctio ca be maximized f 1 1 L log( ) log(det( )) 1 C y C y (4) As a result, the hyperparameter set (,, ) ca be obtaied. A alterative way to f determie the value of the hyperparameter is to use the samplig methods such as Markov chai Mote Carlo (MCMC) samplig ad Gibbs samplig, which geerate samples for approximatio of the posterior distributio of the hyperparameter. For a iput data sample x, the predictive distributio of its correspodig output y is also Gaussia, the mea ad variace values of which are calculated as follows 1 y k ( x ) C y (5) - 4 -

5 C( x, x ) k ( x ) C k( x ) (6) 1 y where k( x ) [ (, 1), (, ),, (, )] C x x C x x C x x. 3. uality predictio based o local GPR model I this sectio, the detailed descriptio of the proposed method is provided. First, multiple local GPR model based soft sesors are costructio i differet operatio modes, depedig o which the olie uality predictio strategy is formulated. Fially, some discussios are made Costructio of local GPR model based soft sesor Suppose the while process cosist of operatio modes, we the represet the dataset as [ 1,,, ] m R X X X X ad [ 1,,, ] m y y y y, where X { x i R } i 1,,, is the iput dataset i the -th operatio modes, with its data sample umber as, y is the correspodig output dataset of the -th operatio modes, ad { yi R} i 1,,, 1. Before the implemetatio of the GPR modelig procedure, a iitial PCA pre-processig step ca be used for reduce the dimesioality of the iput variables. herefore, a total of PCA models ca be built, which are give as X P E X P E (7) X P E k R where ad P are score ad loadig matrices of the -th operatio mode, R mk m R E is the residual matrix, k is the selected umber of pricipal compoets i the -th operatio mode, which ca be easily determied by the cumulative percetage variace method. After the PCA iformatio extractio step, the dimesioality of the iput variables ca be greatly reduced. I the followig GPR modelig step, we ca oly focus o the score k matrices R, 1,,, i differet operatio modes. Followig the modelig procedure of the GPR algorithm, the regressio model betwee k the score matrix { t i, R } i1,,, ad the output data vector y { y, R} 1,,, i i - 5 -

6 ca be formulated as follows y [ y, y,, y ] [ f ( t ), f ( t ),, f ( t )] ~ GP(, C ) (8) 1,,, 1,,, where 1,,,, C is the covariace matrix of the -th GPR model. he geeral form of the covariace matrices for differet operatio modes ca be selected as e. (3). However, the hyperparameter values for differet operatio modes (, f,,, ), 1,,, are differetiated from each other, depedig o the GPR model optimizatio. 3.. Olie uality predictio through soft combiatio strategy After the local GPR model has bee costructed i each operatio mode, they ca be used for olie uality predictio of the iput data sample x. However, a importat issue is how to select appropriate PCA ad GPR models for dimesioality reductio ad uality predictio. Without additioal process iformatio, we do ot kow which operatio mode the data sample x belogs to. I the preset paper, we ited to propose a method to select the GPR model automatically, which ca softly assig the data sample x to differet operatio modes with correspodig probabilities. Based o the PCA model, a ormal operatio regio ca be built for each operatio mode. Particularly, the statistic which is covetioally used for process moitorig purpose ca be costructed as (i, 3) 1 i, i, i, t Λ t (9) where 1,,,, i 1,,,, Λ diag{ 1,,, } is a diagoal matrix with its k elemets as the eigevalues of the correspodig PCA model. o determie the operatio regio of the -th grade, the cotrol limit of the statistic ca be calculated as k ( 1) F lim, k,( k ), k where Fk,( k ), represets F-distributio, with its two parameters k ad k, is (1) sigificace level. herefore, by checkig if the holds, we ca easily determie i, lim, the operatio mode of the data sample. For the iput data sample x, we first calculate the score vector by each PCA model, ad the the value of the statistic ca be determied i each operatio mode, thus t P x,, 1,,, (11) 1,,, t Λ t (1) - 6 -

7 o determie the probability of the data sample Bayesia iferece ca be icorporated, which is give as follows P(, x) P( x ) P( ) P ( x) P( x) { P( x j) P( j)} j1 x i each operatio mode, a where 1,,,. o calculate the posterior probability value, two terms i the right side of e. (13) should be defied, which are kow as prior probability ad coditioal probability. Without ay process or expert kowledge, the prior probability for each operatio mode ca be simply defied as he coditioal probability ca be defied based o the (13) P( ) / (14), lim, statistic, which is give as P( x ) exp{ } (15) his is a expoetial fuctio, based o which the value of the coditioal probability is restricted betwee ad 1. Although the distributio of the coditioal probability does ot exactly expect to be a expoetial distributio, the expoetial form of the fuctio seems to be effective for modelig of such distributio. herefore, e. (13) becomes, exp{ } lim,, j j j1 lim, j P ( x ) (16) { exp{ }} After the probability of the data sample x i each operatio mode has bee determied, the predictive distributio of its correspodig output Py ( ) i each operatio mode ca the be calculated, which is also Gaussia. he mea ad variace values of the predictive distributio for x i differet operatio modes are give as follows y k ( t ) C y 1,, k ( t ) [ C ( t, t ), C ( t, t ),, C ( t, t )] y,, 1,,,,, [ y, y,, y ] 1,,, (17) C ( t, t ) k ( t ) C k ( t ) (18) 1 y,,,,, where 1,,,. Fially, the overall predictive distributio of the data sample ca be formulated - 7 -

8 through the weighted combiatio strategy, give as P( y ) P( y ) P( x ) (19) 1 he mea ad variace values of the fial predictive distributio ca be calculated as 1, ( ) (, ) ( ) 1 1 y y P x k t C y P x () 1 y y P, C,,,, P 1 1 ( x ) {[ ( t, t ) k ( t ) C k ( t )] ( x )} (1) It is oted that i e. (1), we have assumed that differet operatio modes are idepedet from each other, thus the variace of the fial predictio result is a simple weighted summatio of the variaces i differet operatio modes Discussios So far, the local GPR model based soft sesor has bee developed. Compared to existig soft sesors, such as PCR, PLS ad SVM, the soft sesor ca ot oly address the oliear behavior of the process, but also it ca provide a probabilistic predictio result for the uality variable. Besides, the soft sesor is specially desiged for uality predictio of processes which may have several differet operatio coditios. From a egieerig stadpoit, this proposed method is easy for practical implemetatio, ad the iterpretatio of the predictio result is also straightforward. While the sigle GPR model is oly efficiet for uality predictio i its specific regio, the combied local GPR model ca hadle overlappig operatio modes problem, which may exist i may idustrial processes. By softly assigig the data sample to differet operatio modes with correspodig weights, a probabilistic predictio result ca be geerated. Differet from the GMM approach which always limited i the liear case, the local GPR model ca model the oliear relatioship i each operatio mode of the process. herefore, the proposed method is more appropriate for multimode modelig i oliear processes. Before the modelig procedure of the soft sesor, we have assumed that the whole process dataset has bee partitioed ito sub-datasets accordig to differet operatio modes. However, this may be ot available i practice, sice the operatio mode iformatio is ot always feasible. o address this problem, the clusterig method ca be employed, traditioally, such as the K-mea method, fiite Gaussia mixture model based method, ad etc. Compared to the K-mea method, the fiite Gaussia mixture model is more appropriate for mode clusterig, sice it does ot reuire the prior process kowledge o the total umber of operatio modes. herefore, the fiite Gaussia mixture model is used for mode clusterig i the preset work. Aother importat issue of the PCA-GPR modelig strategy is that the performace of the PCA iformatio extractio step may be iflueced by some outliers or disturbaces. his ca be solved by employig the robust PCA method, data screeig ad filterig, or data recociliatio methods. However, i the preset work, it is assumed that ay outlier or data disturbace has already bee removed. Based o the proposed soft assigmet ad combiatio strategy, the fial uality predictio result ca be geerated automatically, which meas we do ot eed to kow the - 8 -

9 exact mode iformatio of the iput data sample. However, the mode iformatio (posterior probability) of the data sample ca be obtaied simultaeously withi the approach, which ca be calculated through e. (16). I our opiio, although the mode iformatio is ot reuired for uality predictio, it is importat for mode localizatio ad idetificatio of the process data, which may play a sigificat role i process uderstadig, moitorig, desig ad improvemet. Fially, it ca be oted i e. (1) that the predictive variace of the uality variable ca be compressed after the soft combiatio step. Sice the posterior probability of the process data obeys P( x ) 1, 1,,,, ad P ( x ) 1, the followig 1 result ca be easily obtaied y ( ) [ ] y P, y, 1,,, 1 x () It ca be see that the variace of the predictio is less tha the variace of the predictios based o ay sigle predictio model. I other words, compared to a sigle local soft sesor, the predictio ucertaity of the combied soft sesor has bee improved. 4. A idustrial case study A typical polypropylee productio device always cotais a catalytic body system, which cosists of icl 4, triethylalumium (EAL), ad dipheyldimethoxysilae (DONOR), a series of three reactors are coected. he flowchart of the polypropylee productio process is show i Figure 1. o record the data characteristic of this process, over 4 variables are measured olie. I this study, the all data samples are collected from the process daily records ad the correspodig laboratory aalysis of oe polypropylee productio compay i Chia. For predictio of the melt idex i this process, a total of 14 process variables have bee selected, which are highly correlated with the uality variable. hese 14 selected iput process variables are listed i able 1. Catalyst Catalytic body system hydroge Propylee Reactor #1 Reactor # Reactor #3 Figure 1: Flowchart of the polypropylee productio process Polypropylee - 9 -

10 hird Variable Melt Idex able 1. Selected variables i polypropylee productio process for uality predictio No. Measured variables No. Measured variables 1 Hydroge cocetratio of the first reactor 8 Propylee feed of the first reactor Hydroge cocetratio of the secod reactor 9 Propylee feed of the secod reactor 3 Desity of the first reactor 1 Power for the first reactor 4 Desity of the secod reactor 11 Power for the secod reactor 5 EAL flow 1 Lever of the secod reactor 6 DONOR flow 13 emperature of the first reactor 7 Atmer-163 flow 14 emperature of the secod reactor I this process, three operatio modes have bee carried out. 1 data samples of each operatio mode have bee selected for modelig traiig ad of each are used for performace evaluatio. herefore, a total of 3 data samples are used for costructio of the soft sesor, ad 6 data samples are used for testig purpose. Sice a lot of comparative studies have show that the GPR model performs better tha other oliear modelig approaches, this case study is maily focused o the multiple operatio mode behavior of the process data, ad compares the predictio performace of the combied strategy based soft sesor with that of differet local GPR model based soft sesors ad the global GPR model based soft sesor. Besides, detailed illustratios ad iterpretatios of the process data behavior, operatio mode iformatio, ad the predictio ucertaity are also provided. o examie the data behavior of the traiig dataset, the scatter plot of two iput variables is show i Figure (a). It ca be see that three clusters have clearly exhibited, which are highlighted i ellipses. Correspodigly, the characteristic of the uality variable is give i Figure (b), i which three differet data behaviors ca also be idetified. Before the GPR model costructio uder each operatio mode of the process, a iitial PCA pre-processig step is carried out. o determie the umber of retaied pricipal compoets i each PCA model, the CPV rule has bee used, which esures that these retaied pricipal compoets ca explai over 85% iformatio of the process data. Detailed explaatio percetages of retaied pricipal compoet i each local PCA model are show i Figure 3. As ca bee see, 7 pricipal compoets have bee retaied i the first ad secod local PCA models, while 8 pricipal compoets should be selected to explai over 85% of the data iformatio i the third operatio mode. However, it should be oted that with the icrease of the model complexity, the overfittig problem may happeed. herefore, attetios should be paid o this issue if too may pricipal compoets are selected First Variable

11 Explaatio Rate Explaatio Rate Explaatio Rate (a) Figure : Data characteristic of the traiig dataset, (a) Iput data; (b) uality data. (b) Pricipal Compoets Pricipal Compoets (a) (b) Pricipal Compoets (c) Figure 3: Explaatio rates of pricipal compoets i each local PCA model, (a) First model; (b) Secod model; (c) hird model. Next, depedig o the score matrices obtaied by the three PCA models, local GPR models ca be costructed i each operatio mode. hree parameters i each local GPR model are optimized through the traditioal gradiet based optimizatio method. After about 3 steps, the optimal parameters values ca be obtaied. For compariso, a global GPR model has also bee developed, which icorporates the iformatio of all 3 data samples. o evaluate the predictio performace of the developed soft sesors, the root mea suared error (RMSE) criterio is used, which is defied as follows RMSE _ te y j1 j yˆ _ te j (3) where j 1,,, _ te, y j ad y ˆ j are real ad predicted values, respectively, _ te is the total umber of test data samples. Detailed predictio results of the soft sesors based o combied local GPR model (CLGPR), sigle local GPR model (SLGPR), ad global GPR model (GGPR) are tabulated together i able. Compared to GGPR ad SLGPR based soft

12 Melt Idex Melt Idex Melt Idex Melt Idex sesors, the CLGPR based soft sesor performs much better, sice the RMSE value is much smaller. It ca be see that the GGPR model based soft sesor has better performace tha that of the three SLGPR model based soft sesors. his is because the GGPR model has used all iformatio of the traiig dataset, while the SLGPR model has oly icorporated a portio of the traiig data iformatio. Detailed predictio results of these three differet types of soft sesor are give i Figure 4. Differet from the sigle local model based approach, the global ad combied local model based soft sesors ca both track the grade chage of the process, ad thus perform much better. able. uality predictio results (RMSE) of differet soft sesors Soft sesors RMSE First SLGPR.949 SLGPR Secod SLGPR hird SLGPR.4866 GGPR.35 CLGPR Real Predicted Real Predicted (a) (b) Real Predicted Real Predicted (c) (d) - 1 -

13 Melt Idex Real Predicted (e) Figure 4: Detailed predictio results of differet soft sesors, (a) First SLGPR; (b) Secod SLGPR; (c) hird SLGPR; (d) GGPR; (e) CLGPR. Although the sigle local model based soft sesor has much worse performace, i specific operatio mode, it ca perform very well. For example, whe testig data samples are geerated from the first operatio mode, the predictio performace of the first SLGPR model based soft sesor will be very high. However, whe it is used for predictio of other data samples that belogs to the secod or third operatio grades, the performace will be sigificatly deteriorated. Amog the testig dataset used i this study, the first data samples are from the operatio mode oe, while data samples 1-4 ad 41-6 belog to the secod ad third operatio modes, respectively. RMSE values of the three local model based soft sesors i these specific operatio modes are tabulated i able 3. hrough this table, it ca be see that the local GPR model ca perform well i its correspodig operatio mode based o which the model has bee costructed. If we combied the predictio results i differet operatio modes which are obtaied by their correspodig local GPR model, the uality predictio result is the same as the CLGPR model. his is because there is o overlappig data sample i the testig dataset. However, compared to the multiple SLGPR models based soft sesor, the CLGPR method does ot eed to switch the predictio model if the operatio coditio has bee chaged, which meas its automatio level is higher tha that of the SLGPR model. I order to evaluate the advatage of the GPR model, the compared results of the CLGPR model with other methods, such as the GMM model, fuzzy-learig based model, multiple local PLS, ANN, ad SVR model based soft sesors are give i able 4. It ca be see that the best predictio result has bee obtaied by the CLGPR model based soft sesor. his result is i accordace with previous research studies o the GPR model. able 3. uality predictio results (RMSE) of SLGPR for differet operatio modes Sample umber First SLGPR Secod SLGPR hird SLGPR able 4. uality predictio results (RMSE) comparisos of differet soft sesors

14 Soft sesor models CLGPR GMM Fuzzy-PLS Multiple PLS Multiple ANN Multiple SVR RMSE o examie the mode iformatio of testig data samples, moitorig results of the statistic by the three local PCA models are give i Figure 5. Compared to data samples 1-6, the first data samples have much smaller statistic values i Figure 5 (a), which meas that these data samples have high probabilities i the first operatio mode. Similarly, it ca also be iferred that data samples 1-4 ad 41-6 have high probabilities i the secod ad third operatio modes. Precisely, the posterior probability value of each testig data sample uder the three differet operatio modes ca be examied, which are show i Figure 6. As ca be see, the results are cosistet with the moitorig results preseted i Figure (a) (b) (c) Figure 5: Moitorig results of the statistic, (a) First PCA model; (b) Secod PCA model; (c) hird PCA model

15 Predictive Variace Predictive Variace Posterior Probability Posterior Probability Posterior Probability (a) (b) (c) Figure 6: Posterior probability value of each testig data sample i differet operatio modes, (a) First grade; (b) Secod grade; (c) hird grade. Fially, the ucertaity iformatio of the predictio result is examied. Predictive variaces of both local ad combied soft sesors are demostrated i Figure 7 (a-c) ad Figure 7 (d). Compared to the result of sigle local model based soft sesor, the predictive variace of the combied local model based soft sesor has bee greatly reduced. Actually, it ca be iferred from e. (1) that the combied predictive variace ca be sigificatly reduced if differet weights have bee take by local models. he most ideal case is that a eual weight is take by each of the local model, thus the predictive variace ca be reduced to the smallest value

16 Predictive Variace Predictive Variace (a) (b) (c) (d) Figure 7: Predictive variace of local ad combied soft sesors, (a) First SLGPR; (b) Secod SLGPR; (c) hird SLGPR; (d) CLGPR. 5. Coclusios I the preset paper, a combiatio form of the local Gaussia process regressio model based soft sesor has bee developed for uality predictio of the polypropylee productio process. Differet from existig soft sesors, the soft sesor ca simultaeously address the oliear ad multiple operatio mode characteristics i the process. Besides, based o the structure of the Gaussia process model, the soft sesor ca also provide a predictive distributio for the uality variable, which ca exhibit the ucertaity iformatio of the soft sesor. hrough a real idustrial applicatio case study, the feasibility ad efficiecy of the proposed soft sesor have both bee cofirmed. I our opiio, further researches for soft sesor modelig o this topic may be focused o the dyamic ad time-varyig extesios of the Gaussia process regressio model. Besides, icorporatio of the developed soft sesor ito the feedback cotrol system is also a iterestig research topic i the future. Ackowledgemet his work was supported by the Natioal Natural Sciece Foudatio of Chia (614134), the Natioal 863 High echology Research ad Developmet Program of Chia (9AA4Z154), ad Chia Postdoctoral Sciece Foudatio (946137). Refereces Che,., & Re, J. (9). Baggig for Gaussia process regressio. Neurocomputig, 7, Choi, S. W., Park, J. H., & Lee, I. B. (4). Process moitorig usig a Gaussia mixture model via pricipal compoet aalysis ad discrimiat aalysis. Computers ad Chemical Egieerig, 8, Chu, W., & Ghahramai, Z. (5). Gaussia processes for ordial regressio. Joural of Machie Learig Research, 6,

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