Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm

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JTICE-1032; No. of Pages 9 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx

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Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm Hossein Ziaee a,*, Seyyed Mohsen Hosseini b, Abdolmajid Sharafpoor c, Mohammad Fazavi b, Mohammad Mahdi Ghiasi d, Alireza Bahadori e,** a

Young Researchers and Elites Club, North Tehran Branch, Islamic Azad University, Tehran, Iran Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Chemical Engineering, Iran University of Science & Technology, Tehran, Iran d National Iranian Gas Company (NIGC), South Pars Gas Complex (SPGC), Asaluyeh, Iran e School of Environment, Science & Engineering, Southern Cross University, Lismore, NSW, Australia b c

A R T I C L E I N F O

A B S T R A C T

Article history: Received 13 May 2014 Received in revised form 9 September 2014 Accepted 14 September 2014 Available online xxx

This paper concerns with implementation of support vector machine algorithm for developing improved models capable of predicting the solubility of CO2 in five different polymers namely polystyrene (PS), poly vinyl acetate (PVAC), polypropylene (PP), poly butylene succinate-co-adipate (PBSA) and poly butylene succinate (PBS). Validity of the presented models has been evaluated by utilizing several statistical parameters. The predictions of the developed models for polymers of PS, PVAC, PP, PBSA, PBS are in excellent agreement with corresponding experimental data with the average absolute relative deviation percent (%AARD) equal to %0.151, %0.500, %1.381, %0.158, %0.239 and R2 values of greater than 0.999. Furthermore, the estimation capability of the proposed models has been compared to a wellknown equation of state (EOS) as well as artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) models. According to the results of comparative studies, it was found that the developed models are more robust, reliable and efficient than other existing techniques for improved analysis and design of polymer processing technology. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Solubility SVM model Polystyrene, poly vinyl acetate, polypropylene, poly butylene succinate-coadipate Poly butylene succinate

1. Introduction Solubility of gases in polymers have been of continuous experimental and theoretical interest because gas and vapor separation membranes are based on the so-called solution– diffusion mechanism, according to which thermodynamic parameters of gas sorption determine the mass transfer driving force [1]. This subject is one of the most important interests of many chemical engineers due to its wide application in industries such as polymer foaming processes. During the past decades, several attempts have been made to describe the solubility of gases in polymers. The main experimental methods for determination of gas solubility in polymers are volumetric method [2,3], gravimetric method [4], chromatographic

* Corresponding author at: Young Researchers and Elites Club, North Tehran Branch, Islamic Azad University, Tehran, Iran. Tel.: Tel.: +98 916 3647678. ** Corresponding author at: Southern Cross University, School of Environment, Science and Engineering, Lismore, NSW, Australia. Tel.: +61 2 6626 9412. E-mail addresses: [email protected] (H. Ziaee), [email protected] (A. Bahadori).

method [5], separation method [5] etc. Due to the fact that development of suitable conditions for implementation these experiments is very difficult, time consuming and costly process, several studies have been focused on development of new ways for simple and accurate determination of solubility of gases in polymers [6,7]. Carbon dioxide also has a great use in polymer processing. Micro-molecular foams can be formed by using CO2 as a blowing agent [8]. Some polymers (polypropylene) can be used in heat insulators and support materials after foaming with gases like CO2 [8]. For these reasons the prediction of CO2 solubility in different polymers has been a research topic for many years [9–11]. In recent years new intelligent methods have been developed to predict the answer of a problem without difficulty of experimental works [7,11–14]. For example in artificial neural networks (ANNs) the experimental data are used for learning the network and due to high computational rate of this method it can improve the accuracy of prediction in compared to thermodynamic models [11]. Another intelligent model is adapted neuro-fuzzy inference system (ANFIS) which has a potential for solving nonlinear systems and reduces the computational time [7]. For more examples of intelligent

http://dx.doi.org/10.1016/j.jtice.2014.09.015 1876-1070/ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: Ziaee H, et al. Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.09.015

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JTICE-1032; No. of Pages 9 2

H. Ziaee et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx

Nomenclature AARD b C K(x,xk) LSSVM MSE R2 RMSE SA T w y

ai g s2 PS PP PVAC PBS PBSA

= average absolute relative deviations (%) = bias term = a positive constant = Kernel function = least-squares supported vector machine = mean square error = R-squared = root mean square error = simulated annealing = transpose = a nonlinear function = outputs = Lagrange multipliers = relative weight of the summation of the regression errors = squared bandwidth = polystyrene = polypropylene = poly vinyl acetate = poly butylene succinate = poly butylene succinate-co-adipate

including polystyrene (PS), poly vinyl acetate (PVAC), polypropylene (PP), poly butylene succinate (PBS) and poly butylene succinate-co-adipate (PBSA). To follow our objective in this study, statistical error analysis (i.e., APRE, AAPRE, RMSE and R2) and graphical error analysis (i.e., cross plots) have been performed. In addition accuracy of the developed models is compared with other intelligent models (i.e., ANN and ANFIS) as well as EOS. Moreover, the applied data base is evaluated by the leverage value statistical approach. 2. LSSVM modeling The SVM is a supervised learning model which was proposed by Vapnik [30]. SVMs use the spirit of the structural risk minimization principle [30,31]. However the major drawback of the SVM is its higher computational burden because of required constrained optimization programing [29]. Suykens and Vandewalle [29] presented a modification to the traditional SVM called leastsquares SVM (LSSVM) so as to facilitate the solution of the original SVM framework. LSSVM appears to offer advantages similar to those of SVM, but its great advantage is that LSSVM applies a set of linear equations (linear programming), instead of quadratic programming problems in order to reduce the complexity of optimization process [29]. Considering the problem of approximating a given dataset fðx1 ; y1 Þ; ðx2 ; y2 Þ; . . . ; ðxN ; yN Þg with a nonlinear function: f ðxÞ ¼ hw; FðxÞi þ b

systems, fuzzy logic, wavelet analysis, colony optimization algorithm, particle swarm optimization algorithm, radial basis neural networks and support vector machine can also be mentioned [12,15–23]. Khajeh et al. [7] applied adaptive neuro-fuzzy inference systems for prediction of CO2 solubility in seven different polymers and compared the prediction results with artificial neural network and EOSs. They concluded that the prediction of ANFIS is more accurate than other predictive tools. More recently, Li et al. [6] developed an intelligent model for solubility prediction of different gases in polymers using radial basis function neural network based on chaotic self-adaptive particle swarm optimization and a clustering method. The statistical quality measures showed that their model is accurate for gas-polymer solubility prediction. In another study Li et al. [24] attempted to improve back propagation artificial neural network model by chaotic self-adaptive particle swarm optimization techniques (CSPSO) for the same purpose. Their results showed that developed model has high capability for prediction of solubility of CO2 and N2 in polystyrene and polypropylene. Khajeh et al. [25] also used an adaptive neurofuzzy inference system and radial basis function neural network for solubility prediction of some gases in polystyrene such as butane, isobutene, carbon dioxide, 1,1,1,2-tetrafluoroethane, 1-chloro-1,1difluoroethane,1,1-difluoroethane. In the past decades, the developments in various statistical and intelligent methods make them attractive for modeling of complex systems [12,26,27]. Nowadays, support vector machine (SVM) is gaining more popularity among researchers [12,14,28]. The SVM is a new and supervised machine learning technique which works based on the statistical learning theory [29]. The least square version of the SVM (LSSVM) which widely used in complex system studies for modeling, regression or parameter prediction, was described in Suykens and Vandewalle [29]. However its application to the gas-polymer systems is very limited. Moreover, to the best of authors’ knowledge, no work has been published on the subject of modeling of CO2 gas solubility prediction in polymer with this approach. Hence, In this work the LSSVM algorithm was applied for accurate determination of CO2 solubility in five different polymers

(1)

where, h:; :i represent dot product; FðxÞ represents the nonlinear function that performs linear regression; w and b are weight vector and bias terms, respectively. In LSSVM for function prediction, the optimization problem is expressed as [32]: min jðw; eÞ ¼ w;b;e

N 1 1 X kwk2 þ g e2 2 2 k¼1 k

s:t: yk ¼ hw; Fðxk Þi þ b þ ek

(2)

k ¼ 1; . . . ; N

(3)

where, ek 2 R are error variables; g  0 is a regularization constant. The Lagrangian of the problem is defined by [32]: LLSSVM ¼

N N X 1 1 X kwk2 þ g e2k  ak fhw; Fðxk Þi þ b þ ek  yk g 2 2 k¼1 k¼1

(4)

with Lagrange multipliers ak 2 R. The condition for optimally is determined by [32]: 8 N X > @LLSSVM > > > ¼ 0!w ¼ ak Fðxk Þ > > @w > > k¼1 > > N > X > @LLSSVM > < ¼ 0! ak ¼ 0 @b (5) k¼1 > > > @ LLSSVM > > ¼ 0 ! ak ¼ g ek > > @ek > > > > @LLSSVM > > ¼ 0 ! hw; Fðxk Þi þ b þ ek  yk ¼ 0 :

@ak

By specifying Y ¼ ½y1 ; . . . ; yN ; 1v ¼ ½1; . . . ; 1; a ¼ ½a1 ; . . . ; aN  and eliminating ek and w, following equations are achieved [32]:      0 b 0 1TN ¼ (6) 1 Y a 1N V þ g IN where, 1N is an N  N identity matrix, and V 2 RNN is the kernel matrix presented by:

Vkl ¼ Fðxk ÞFðxl Þ ¼ K ðxk ; xl Þ;

k; l ¼ 1; . . . N

(7)

As mentioned earlier, has a tuning parameter g . From the other point, as the LSSVM is a kernel based method, parameters of kernel

Please cite this article in press as: Ziaee H, et al. Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.09.015

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Table 1 Applied experimental data for modeling Co2 solubility in polymers. Polymer

Pressure (MPa)

Temperature (K)

Solubility (g/kg)

N*

Reference

PS PVAC PP PBSA PBS

2.159–20.151 0.199–17.449 4.96–24.91 1.098–20.127 1.025–20.144

373.15–473.15 313.15–373.15 313.2–483.7 323.15–453.15 323.15–452.15

7.14–121.84 14.65–551 29.9–242.9 11.84–174.1 8.76–176.1

35 31 25 44 53

Sato et al. [4] Sato et al. [4] Lei et al. [8] Sato et al. [41] Sato et al. [41]

*

Number of data points.

Table 2 Optimized parameters of the developed LSSVM models for various polymer systems. Polymer

g

s2

PS PVAC PP PBSA PBS

3.537E + 04 3.615E + 10 1.235E + 03 6.071E + 11 1.115E + 08

3.734E + 00 4.503E + 02 9.254E  01 1.172E + 02 1.124E + 01

function should be considered. For this purpose, there are many kernel function including linear, polynomial, radial basis function (RBF), etc [33,34]. However, most widely used kernel functions are RBF (Eq. (8)) and polynomial (Eq. (9)). ! kxk  xl k2 K ðxk ; xl Þ ¼ exp (8) 2

as the number of datasets are shown. For modeling purpose, two independent variables including temperature and pressure have been selected as input parameters and value of solubility was assigned as the target (output) variable. 3.2. Pre-processing In the next step the data sets of each category was divided into two sub datasets including ‘‘training set’’ and ‘‘testing set’’. Generally, the training data are employed for construction and training the model and validation of the model accuracy for prediction of results is also checked by testing data [14,27]. This division commonly performed randomly. For this purpose about 80% of each category of data was assigned for the training set and the remaining 20% applied for testing set.

s

3.3. Designing LSSVM models for solubility prediction  d K ðxk ; xl Þ ¼ xTk xl þ t

(9)

where, s 2 is the squared variance of the Gaussian function and t and d is the intercept and polynomial degree, respectively. In the case of RBF kernel, we have another tuning parameter (i.e., s2) While for the polynomial kernel function there are two additional parameters (i.e., t and d). As the result, the LSSVM model with the RBF kernel function has two tuning parameters and that with the polynomial kernel has three tuning parameters which should be obtained by minimization of the deviation of the LSSVM model from experimental values.

To develop LSSVM models for precise prediction of CO2 solubility in various polymers including PS, PVAC, PP, PBSA and PBS pressure and temperature are assumed as the correlating variables as follow: CO2 solubility ¼ f ðT; PÞ

(10)

The mean square error (MSE) between the developed model results and corresponding experimental values reported in the literature, as defined by Eq. (11), was considered as objective function during model computation [12,14,35,36].

3. Data acquisition and processing MSE ¼

3.1. Data gathering Experimental data which was applied in this study were collected from literature and reported in Table 1. In this table for each modeling system the ranges of input/output variables as well

n  2 1X t  oj n j¼1 j

(11)

in which, t and o are target value and estimated value, respectively; and n is number of the data points.

Table 3 Statistical parameters of the developed LSSVM models to predict the CO2 solubility in polymers. Polymer

Training

Testing

ARDa PS PVAC PP PBSA PBS a

0.010 0.010 0.052 0.003 0.004  N  exp pred X Y Y 100 i i ARD% ¼ N . exp Y

i¼1

b

RMSEc

R2

ARD

AARD

RMSE

R2

ARD

AARD

RMSE

R2

0.136 0.334 1.140 0.130 0.140

0.048 0.583 0.992 0.084 0.129

0.9999 0.9999 0.9999 0.9999 0.9999

0.038 0.772 0.648 0.106 0.439

0.211 1.193 2.347 0.268 0.614

0.065 0.682 0.904 0.496 0.231

0.9999 0.9999 0.9997 0.9999 0.9999

0.001 0.157 0.088 0.020 0.088

0.151 0.500 1.381 0.158 0.239

0.052 0.603 0.975 0.237 0.155

0.9999 0.9999 0.9998 0.9999 0.9999

i

 N  exp pred  X i  Yi Y AARD% ¼ 100  Y exp  . N i

i¼1

c

Total

AARDb

RMSE ¼

 !1=2

PN 

exp pred 2 Y Y i¼1 i i

N

.

Please cite this article in press as: Ziaee H, et al. Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.09.015

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250

Predicted solubility (g/Kg)

200

150

100

PS polymer PBS polymer PBSA polymer PVAC polymer PP polymer Y=X

50

0

0

50

100 150 Experimental solubility (g/Kg)

200

250

Fig. 1. LSSVM predicted CO2 solubility for all the studied systems vs. corresponding experimental values for the test data.

PS

PVAC

PP

PBSA

PBS

250 Experimental value Predicted value

150

2

CO solubility (g/kg)

200

100

50

0

0

5

10

15

20 Index

25

30

35

40

Fig. 2. Prediction of presented LSSVM model in comparison with the target data for the test data.

Please cite this article in press as: Ziaee H, et al. Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.09.015

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JTICE-1032; No. of Pages 9 H. Ziaee et al. / Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx Table 4 Optimized parameters of the developed LSSVM models with polynomial kernel function for various polymer systems. polymer

g

t

d

PS PVAC PP PBSA PBS

2.6461E  04 5.2310E + 03 0.8551E + 00 0.0033E + 00 1.6820E + 00

1.4095E + 05 7.9678E + 01 2.3261E + 00 1.3861E + 01 0.9822E + 00

3 5 3 3 3

4. Results and discussion 4.1. Parameters of the developed LSSVM models The optimum values of the parameters of LSSVM algorithm namely s2 and g have been evaluated by a proper optimization method. For this purpose, coupled simulated annealing (CSA) optimization technique [37] was applied. The obtained optimized values of s2 and g by applying RBF kernel function are reported in Table 2. The step-by-step procedure to use the LSSVM model is presented in Appendix A.

5

4.2. Graphical and statistical evaluation of the developed LSSVM models Table 3 indicates the statistical parameters of the developed intelligent models for predicting the CO2 solubility in various polymers including R-squared (R2), average absolute relative deviation (AARD), average relative deviation (ARD) and root mean square error (RMSE). Tabulated results in Table 3 clearly demonstrate the capability of the utilized algorithm, LSSVM, in predicting the values of CO2 solubility in polymers accurately. Fig. 1 shows the cross plot diagram, demonstrating the relation between the results of the developed intelligent models and corresponding experimental data for the test data. A tight cloud of points about 458 line for training and testing data sets indicate the robustness of the proposed models. In Fig. 2, the output predictions of developed networks have been compared with the target test data. As it is clear from Fig. 2, the predicted solubility values of CO2 in all the studied polymers are in excellent agreement with experimental data.

Table 5 Average relative deviation (ARD) for ANN, ANFIS, equation of state, and LSSVM models. Polymer

ANN

ANFIS

EOS

LSSVM (RBF kernel function)

LSSVM (polynomial kernel function)

PS PVAC PP PBSA PBS

0.599a 0.583a 1.535a 0.503a 0.635a

0.254b 0.67b 1.881b 0.351b 0.341b

1.6c 3.6c 5d 1.9e 2.2e

0.1511 0.5003 1.3813 0.1581 0.2385

0.7013 1.2427 6.7081 0.7697 1.571

a b c d e

Khajeh et al. [11]. Khajeh et al. [7]. Sato et al. [4]. Lei et al. [8]. Sato et al. [41].

4

3

Standardized residuals

2

1

0

−1

−2

Valid data Upper suspected data limit Lower suspected data limit Leverage limit

−3

−4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Hat diagonal Fig. 3. Detection of probable outliers in the dataset of PS +CO2 system.

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4

3

Standardized residuals

2

1

0

−1

−2

Valid data Upper suspected data limit Lower suspected data limit Leverage limit

−3

−4

0

0.05

0.1

0.15

0.2 Hat diagonal

0.25

0.3

0.35

0.4

Fig. 4. Detection of probable outliers in the dataset of PVAC +CO2 system.

4

3

Standardized residuals

2

1

0

−1

−2

Valid data Upper suspected data limit Lower suspected data limit Leverage limit

−3

−4

0

0.05

0.1

0.15

0.2

0.25 Hat diagonal

0.3

0.35

0.4

0.45

0.5

Fig. 5. Detection of probable outliers in the dataset of PP +CO2 system.

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Fig. 6. Detection of probable outliers in the dataset of PBSA +CO2 system.

4.3. Comparison of the developed models against other models In order to show the predictivity of the present LSSVM model, the results in this modeling approach were compared with the

accuracy of other predictive models. For this purpose, in Table 4 the results of LSSVM algorithm developed based on RBF kernel function were compared with the results of ANN, ANFIS models as well as Sanchez & Lacombe EOS (S-L EOS). As it was previously

Fig. 7. Detection of probable outliers in the dataset of PBS +CO2 system.

Please cite this article in press as: Ziaee H, et al. Prediction of solubility of carbon dioxide in different polymers using support vector machine algorithm. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.09.015

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mentioned, For LSSVM, there are other types of kernel function in addition to widely used RBF kernel function. Here, in order to see the performance of another type of kernel functions, i.e., polynomial kernel function, LSSVM modeling for all the systems with this type of kernel function were done and model’s parameters including g ; t; and d are reported in Table 4. Errors of the predictions of these networks were also reported in Table 5. It is seen in Table 5 that the ANN model has better accuracy than the S-L EOS for all mentioned polymers but the ANFIS model only improve the prediction about PBS and PBSA polymers. To make a judgment based on reported AARD% in Table 5 it can be concluded at this point that for all CO2 gas-polymer systems studied in this study, the prediction of LSSVM models developed based on RBF kernel function have better performance with higher accuracy and have better correlation and robustness. 4.4. Outlier detection It is interesting to evaluate the data base applied in this study, since the any uncertainty affects the prediction capability of the model. To this end, leverage value statistical approach [38] was implemented in this study. The graphical detection of the suspended data is undertaken through sketching the Williams plot. A detailed description of equations and computational procedure of this method can be found elsewhere [23,39,40]. The Williams plot has been sketched in Figs. 3–7 for the results using the developed models. It shows that the whole data are located within the applicability domains of the applied model. Therefore, the used dataset in the case study is statistically correct and valid. 5. Conclusions Solubility of gases in polymers is one of the most important interests of many chemical engineers due to its wide application in industries. In this study an attempt was made to develop new intelligent models based on LSSVM algorithm for improved prediction of CO2 solubility in five various polymers. To this end, wide ranges of experimental dataset were utilized to develop accurate predictive models. Comparing result of LSSVM with the results obtained by ANN, ANFIS, and S-L EOS indicate that LSSVM models are more accurate than other predictive models with correlation coefficients (R2) greater than 0.999. The results indicate that the LSSVM modeling algorithm is a reliable and accurate model for the solubility of CO2 in polymers, and is a practicable method for the analysis and design of polymer processing technology. Appendix A. Appendix A. Instructions for using the LSSVM model A computer program is organized to use the developed LSSVM model. At first, the LSSVM toolbox for MATLAB should be installed, and then, the directory of the toolbox should be inserted as the main directory in the MATLAB environment. After that, the model file is dragged and dropped in the MATLAB workspace. Following example will clearly illustrate the commands/function call for using the LSSVM model Example Predict the solubility of carbon dioxide in Polystyrene polymer at T = 373.15 K and P = 12.631 MPa. Solution: The solubility of carbon dioxide in Polystyrene polymer at the specified conditions is calculated simply using the below command line in the MATLAB command window: Data = [373.15 12.631]; %Input vector Kernel = ‘RBF_kernel’; % Kernel type type = ‘function estimation’; gam = 3.53668110E + 04; %Optimized parameter (Table 2)

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