A computational intelligence scheme for estimating

Journal of Molecular Liquids 221 (2016) 624–632

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

A computational intelligence scheme for estimating electrical conductivity of ternary mixtures containing ionic liquids Mohsen Hosseinzadeh a, Abdolhossein Hemmati-Sarapardeh b,⁎, Forough Ameli c, Fereshteh Naderi d, Mohammadmahdi Dastgahi e a

Department of Chemical Engineering, Amirkabir University of Technology, Tehran, Iran Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran Chemical Engineering Department, Islamic Azad University, North Tehran Branch, Tehran, Iran d Department of Chemistry, Shahr-e Qods Branch, Islamic Azad University, Tehran, Iran e Department of Electrical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran b c

a r t i c l e

i n f o

Article history: Received 11 May 2016 Accepted 20 May 2016 Available online 1 June 2016 Keywords: Ionic liquids Ternary mixture Electrical conductivity Least square support vector machine

a b s t r a c t Due to unique physical and chemical properties of ionic liquids (ILs), they received lots of attention in many industrial fields and are widely under research. Ionic liquids, also are emerging as important components for applications in electrochemical devices. To develop their applications and achieving desire properties, they are usually mixed with organic solvents. Applying ionic liquids in many applications needs the accurate and reliable data of electrical conductivity of ILs and their mixtures. To this end, a total of 224 experimental data were collected from literature and divided randomly into two datasets: 179 data was selected as training set and the remained 45 data was used as a testing set. Afterwards, a reliable modeling technique is developed for modeling the electrical conductivity of the ILs ternary mixtures. This approach is called least square support vector machine (LSSVM). The model parameters were optimized using the method of couple simulated annealing (CSA). The input model parameters were, temperature of the system, melting point, molecular weight and mole percent of each component. A comprehensive error investigation was carried out, yielding the well accordance between the predictions of the model and experimental data. The presented model can predict the dependency of electrical conductivity variations with input variables. Moreover, the sensitivity analyses demonstrated that, among the selected input parameters, the average melting point of mixture has the largest effect on the electrical conductivity. Furthermore, suspected data were detected using the Leverage approach, residual, Williams plot and statistical hat matrix. Except seven data points, the all data appear to be reliable. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Ionic liquids (ILs) are known since 1914 [1], however the interest for them have been increased since 1992, when Wilkes and Zaworotko introduced new compounds with hydrolytically stable ions [2]. Due to distinctive physical and chemical properties of ionic liquids, they received lots of attention in the past decade. ILs are liquid salts with melting points normally below 373.15 K [3–5]. They are usually composed of nonsymmetrical organic cations and numerous different inorganic or organic anions. ILs have a series of outstanding properties, including large temperature range of liquid state, negligible vapor pressure, wide electrochemical windows, good solubility of organic and inorganic compounds, flame resistance, tunable nature, remarked catalytic property [6–12], strong solubilization power, recyclability, nonvolatility, ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (A. Hemmati-Sarapardeh).

http://dx.doi.org/10.1016/j.molliq.2016.05.059 0167-7322/© 2016 Elsevier B.V. All rights reserved.

high thermal stability (up to 673 K), low toxicity, and conduction properties [13–18]. ILs are used as nonflammable, nonvolatile ILs-based electrolytes in high temperature fuel cells and batteries [19–22]. Chemical and physical properties of ILs such as their polarity, hydrophobicity and viscosity can be selected by choosing the cationic or the anionic constituents [23]. Normally, ILs are composed of bulky, nonsymmetrical organic cations such as pyrrolidinium, pyridinium, quaternary ammonium or phosphonium, imidazolium, and numerous different inorganic or organic anions such as tetrafluoroborate, hexafluorophosphate, trifluoromethanesulfonate, bis(trifluoromethylsulfonyl)imide and bromide anions [4,24–30]. ILs can be used as recyclable “green solvents” [31,32] as well as conductors in electrochromic devices [33]. These liquids have been used for liquid electrolytes based on aliphatic tetraalkylammonium or aromatic pyrrolidinium and isoquinolinium cations [34,35]. Using ILs in the aforementioned applications requires knowledge of some physical and thermodynamic properties. The electrical conductivity of ILs is of paramount importance. Knowing the exact value of electrical conductivity of ILs in some applications such

M. Hosseinzadeh et al. / Journal of Molecular Liquids 221 (2016) 624–632

as electrolytes is crucial. Therefore, up to now, the properties of pure ILs have been studied by a number of scientists [33,36,37]. For example, in 2011, Kanakubo et al. [38] have measured the densities, viscosities and electrical conductivities of N-methoxymethyl-N-methylpyrrolidinium bis(trifluoromethanesulfonyl) amide over a wide temperature at atmospheric pressure. In 2013, Tomida et al. [39] have studied the densities and thermal conductivities of N-butylpyridinium tetrafluoroborate, N-hexylpyridinium tetrafluoroborate and N-octylpyridinium tetrafluoroborate. Then, in 2014, Makino et al. [40] have reported the densities, viscosities, and electrical conductivities of ethylimidazolium and 1ethyl-3-methylimidazolium ILs. In spite of numerous advantages of ILs, there are a strong electrostatic force and hydrogen-bonding interaction among ions, so there is a microscopic aggregation in ILs and because of that the viscosities of ILs are always very high [41–45]. Adding less viscous molecular solvents into ILs have a significant effect on the physical properties of ILs, and thus improves the large-scale applications of ILs [7,46, 47]. Therefore, a number of scientists have studied the physical properties of binary solutions of ILs [48–52]. Study of physicochemical properties of mixtures of ILs and molecular solvent is too necessary. It provides valuable data for the application of ILs as well as excellent information about the variation of microscopic structures of those mixtures [53,54]. Until now, most studies related to ILs are limited to pure ILs or binary mixtures. In spite of extensive studies on pure ILs properties such as density, viscosity, conductivity, etc., little information on thermodynamic properties of ternary mixtures have been published [55]. Zhang et al. determined the conductivities and viscosities of the room-temperature IL, 1-butyl-3methylimidazolium hexafluorophosphate ([bmim][PF6]) + water + ethanol and [bmim][PF6] + water + acetone ternary mixtures at different temperatures [55]. Chen et al. measured the densities, viscosities, and conductivities of the ternary solutions [N-EMP]Br (N-ethyl,methylpiperidinium bromide) + [NPMP]Br (N-propyl,methyl-piperidinium bromide) + H 2 O, [NEMP]Br + [NBMP] Br (N-butyl,methyl-piperidinium bromide) + H2 O, and [N-PMP]Br + [N-BMP]Br + H 2 O and their binary subsystems at different temperatures and atmospheric pressure. Recently, Hosseinzadeh and Hemmati-Sarapardeh proposed a model for prediction of viscosity of IL ternary mixtures. The variables of this equation were selected as molecular weight, composition, and normal boiling point [56]. The viscosity of ternary mixtures including ionic liquids was calculated based on a new, prompt, and precise technique namely least square support vector machine (LSSVM). In this study, we follow up our previously published work [56] and develop a computation method, namely LSSVM to predict electrical conductivities of ternary mixtures containing ILs at various temperatures and atmospheric pressure. Coupled simulated annealing (CSA) approach was applied for estimation of LSSVM model parameters. Afterward, statistical and graphical error analyses are utilized simultaneously to check the performance of the developed model. In addition, trend analysis is used to examine if the model can capture the physically expected trends. Besides, sensitivity analysis is employed through the relevancy factor to deepen our understanding about the relative effect of input parameters on electrical conductivity. Finally, the Leverage

625

approach is used to find the applicability domain of the developed model as well as to assess the quality of experimental data points. 2. Data gathering The model accuracy and robustness depend highly on precaution of the dataset used for forecasting the determined properties. To this end, our attempt is to assemble all available experimental data on the electrical conductivities of ternary mixtures containing IL data sets, over temperatures ranging from 288.15 to 308.15 K which are available in open literature sources [55,57]. A total of 224 experimental published data points were gathered from the reliable literatures. The component of each ternary mixtures are tabulated in Table 1 and the properties of the selected compounds are gathered in Table 2, which were provided form these references [58–63]. The input variables of this model are selected temperature, melting point (Tf), molecular weight (Mw) of the compounds, and the composition of the two first substances. It should be noticed that the third component's mole fraction is a dependent variable, since the summation of all the mole fractions is equal to one.
 μ ¼ f T; Mwi ; T f i ; X 1 ; X 2

ð1Þ

The developed model based on these data could be reliable and efficient for predicting electrical conductivity of other ternary mixtures containing IL. 3. Model development The support vector machine (SVM) is a proper mathematical tool to develop nonlinear relationships among the available experimental data considered as inputs of the model and the desired output. This theory which has been applied in many fields, mainly has been discussed in computational science for a set of related supervised learning methods that analyze data and recognize patterns and are used for regression analysis [64–66]. Among the large number of machine learning methods which have already been used to solve a wide range of difficulties in science and engineering, artificial neural networks (ANNs) have yielded highly accurate results [67]. However, ANN has lost its favor after appearance of SVM [68,69]. Moreover, ANN usually encounters over-fitting or under-fitting issues due to their empirical risk minimization characteristics; however, such possibilities have been minimized in SVM paradigm by incorporating a structural risk minimization strategy. As a result, SVM can overcome several shortage difficulties in ANN models mentioned earlier. These reasons attract the attention of several researchers to SVM models [70–73]. Recently, a modified version of SVM has been presented [69] called least squares support vector machine (LSSVM) trying to minimize its complexity and improve its convergence speed. LSSVM employs equality constraints rather than inequality ones. This reformulation introduces a system of linear equations which can be iteratively solved in several consecutive steps [67,69]. Moreover, only a portion of support vectors are applied to construct an approximation model because of spars feature of SVM, whereas LSSVM utilizes all data points in order

Table 1 Statistical parameters of the ternary mixtures at different conditions. Component 1

Component 2

Component 3

Temperature range, K

Number of data points

Ref

[bmim][PF6] [bmim][PF6] H2O H2O H2O

H2O H2O NaNO3 [C6mim][Cl] [C6mim][Cl]

Ethanol Acetone [C6mim][BF4] NaCl [C6mim][BF4]

288.15–308.15 288.15–308.16 273.15–274.32 273.16–275.27 272.85–273.15

56 49 33 30 56

[55] [55] [57] [57] [57]

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Table 2 Components' properties of each ternary mixtures. Component

Molar weight (g/gmol)

Melting point (K)

Ref.

Water NaCl Ethanol NaNO3 Acetone [c4mim][PF6] [C4mim][BF4] [C6mim][Cl] [C6mim][BF4]

18.01 58.44 46.07 84.9947 58.08 284.18 226.03 202.73 254.08

273.15 1074 159.15 581 178.15 283.1 192.15 198.15 191.15

[58] [59] [60] [61] [58] [62] [63] [63] [63]

to find a satisfactory approximation [69]. The function f(x) based on SVM principle is defined as below: f ðxÞ ¼ wT ϕðxÞ þ b

N
X 1 T  w þc ξk −ξk 2 k¼1

ð3Þ

Subjected to the following constraints: 8 < yk −wT ϕðxk Þ−b≤ε þ ξk ; k ¼ 1; 2; …; N  wT ϕðxk Þ þ b−yk ≤ε þ ξk ; k ¼ 1; 2; …; N :  ξk ; ξk ≥0; k ¼ 1; 2; …; N

ð4Þ

where xk and yk stand for kth data input and kth data output, respectively. Moreover, ε expresses the fixed precision of the function approximation, and ξk and ξk⁎ are slack variables. Using a small value for ε would lead to creating an exact model that would lead to placing some data point out of the ε precision. This would lead to impractical solution. As a result, the stack variables would apply for designation of the permissible limit for the error. The deviation quantity from the preferable ε is evaluated using the tuning parameter of SVM (the c N 0 in Eq. (3)). The lagrangian as follows is applied for minimization of the cost function which is presented in Eq. (3) with constraints in Eq. (4): Lða; a Þ ¼ − þ

N
N
X 
  1X a −ak al −al K ðxk ; xl Þ−ε ak −ak 2 k;l¼1 k k¼1 N X




 yk ak −ak

ð5Þ

 ak −ak ¼ 0; ak ; ak ∈½0; c

ð5bÞ

where αk and α∗k show Lagrangian multipliers. Eventually, the final version of SVM can be expressed as follows: f ðxÞ ¼

N
X

 ak −ak K ðx; xk Þ þ b

ð7Þ

yk ¼ wT φðxk Þ þ b þ ek

ð8Þ

where γ is tuning parameter in LSSVM method, and ek is the error variable. The Lagrangian for this problem is: Lðw; b; e; aÞ ¼

N N X
 1 T 1 X w wþ γ e2 − a wT φðxk Þ þ b þ ek −yk 2 2 k¼1 k k¼1 k

ð9Þ

where ak represents the multipliers of Lagrangian. To solve the problem the derivatives of Eq. (9) are equated to zero. This leads to the following equations: 8 N X > ∂L > > ak ϕðxk Þ ¼ 0⇒w ¼ > > > ∂w > k¼1 > > > N > X > ∂L > < ak ¼ 0 ¼ 0⇒w ¼ ∂b k¼1 > > ∂L > > > ¼ 0⇒ak ¼ γek ; k ¼ 1; 2; …; N > > ∂e > k > > > ∂L > > : ¼ 0⇒wT ϕðxk Þ þ b þ ek −y ¼ 0; k ¼ 1; 2; …; N ∂ak

ð10Þ

As it is obvious, the number of equations and the unknowns are 2 N + 2 (ak, ek, w, and b). Solving the system of equations (Eq. (10)), leads to obtaining the LSSVM parameters. As mentioned before, there is a tuning parameter γ in LSSVM. The parameters of the kernel functions are introduced as the second tuning parameter, as both LSSVM and SVM are kernel-based techniques. For the present study, the RBF kernel function is introduced as below:   K ðx; xk Þ ¼ exp −kxk −xk2 =σ 2

ð11Þ

Another tuning parameter is σ2. As a result, there are two tuning parameters in LSSVM method using the RBF kernel functions. These are obtained by minimization of the deviations of LSSVM technique using the experimental values [75]. The root mean square error (RMSE) from the outputs of LSSVM algorithm is determined as below: B B RMSE ¼ B B @

11=2 n X ðOrep:=predi −O expi Þ2 C C i¼1 C C n A

ð12Þ

ð5aÞ

k¼1

K ðxk ; xl Þ ¼ ϕðxk ÞT ϕðxl Þ; k ¼ 1; 2; …; N

N 1 T 1 X w wþ γ e2 2 2 k¼1 k

Subjected to the following constraint:

0

k¼1 N
X

Cost Function ¼

ð2Þ

where ϕ(x) and wT are the kernel function and the transposed output layer, respectively, and b is the bias, the intercept of the linear regression in the modified SVM method (LSSVM). x shows the input vector of the model parameters. It has a dimension of N × n, where N and n represent the number of data points and input parameters, respectively. Vapnik proposed minimization of the below cost function in order to calculate w and b: Cost Function ¼

the modification of SVM technique least square to assist the main SVM technique. The SVM method is reformulated in LSSVM method which is presented as below [75]:

ð6Þ

where O is the output, subscripts rep./pred. and exp. stand for the represented/predicted, and experimental values, respectively, and n shows the number of samples from the initial population. In this study, the LSSVM algorithm developed by Suykens and Vandewalle [69] has been used. Coupled simulated annealing is applied for optimizing LSSVM parameters. 4. Model evaluation

k¼1

The above stated parameters namely, ak , ak⁎ and b are determined by solving the quadratic programming problem. This is a difficult task. To overcome this problem, Suykens and Vandewalle [69,74] introduced

4.1. Statistical and graphical error analyses To evaluate the accuracy and robustness of the developed model, several statistical parameters have been used consisting of root mean

M. Hosseinzadeh et al. / Journal of Molecular Liquids 221 (2016) 624–632

square error (RMSE), average percent relative error (APRE), average absolute percent relative error (AAPRE), and coefficient of determination (R2). Definitions and equations of these parameters are given below: 1. Root mean square error vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X
2 O −Oirep:=pred RMSE ¼ t N i¼1 i exp

ð13Þ

627

constant value of 3(k + 1)/n, where n denotes the number of data points and k stands for the number of model parameters. The cut-off value of three is regarded to R, which would select the data points in the range of ±3 of standard deviation to cover 99% of the distributed data. To develop a reliable model, the values of data points should be located in the criteria of 0 ≤ H ≤ H* and −3 ≤ R ≤ 3. This approach would lead to a valid model from statistical point of view. The “Good High Leverage” points are located in H* ≤ H and −3 ≤ R ≤ 3 domain. The data points which are located out of the applicability domain are called “Good High Leverage”. If the data points are placed in the range of R b −3 or 3 b R, they are called “Bad High Leverage” points which are outliers.

2. Average percent relative error 5. Results and discussion

Er ¼

n 1X Ei N i¼1

ð14Þ

where Ei (percent relative error) is determined as follows: Ei ¼

  ðOÞi exp −ðOÞirep:=pred Oi exp

 100⇒i ¼ 1; 2; 3; …; n

ð15Þ

3. Average absolute percent relative error

Ea ¼

n 1X jEi j N i¼1

ð16Þ

4. Coefficient of determination (R2)

n
X

Oi exp −Oi rep:=pred

2

R2 ¼ 1− i¼1n  2 X Oi rep:=pred −O

ð17Þ

i¼1

where O is the average of the experimental data points. For model evaluation and visualization of the results, two graphical analyses namely crossplot and error distribution curve were used. The definition of the graphical analyses can be found elsewhere [76,77]. 4.2. Identifying outliers in experimental data Outliers detection are of paramount importance for model development [78]. For detection of outliers, numerical and graphical methods are introduced [78–82]. Leverage approach for detection of outliers, determines the residual values and Hat matrix (H) [78,79,81,82]. In this study, the algorithm is represented to determine the data of interest. The Hat or leverage indices are determined based on Hat matrix (H), using the following equation [78–81]:
−1 t H ¼ X Xt X X

In the present work, CSA-LSSVM algorithm was implemented to construct an accurate, reliable and robust model for predicting electrical conductivity of IL ternary mixtures. As mentioned earlier, the input model parameters were temperature, molecular weight (Mw), melting point (Tf), and composition of the components. In fact, there are nine inputs consist of temperature, molecular weight and melting point of three components and mole fraction of the two first components. It is worth noting that as the summation of the mole fractions of components is equal to one, the third compound's mole fraction would not be an independent variable. Using various sources, a large data bank of the ternary mixtures was selected. The collected data was initially divided into two subsets including training and testing sets. The former set is employed to perform and generate the model structure. The latter set is used to investigate the final performance and validity of the proposed model for unseen data. To increase the model applicability and robustness, the whole database was divided randomly into two subsets: 80% (179 data points) and 20% (45 data points) as training and testing sets, respectively. To optimize the LSSVM parameters, coupled simulated annealing technique was applied. The optimization process has been performed repeatedly to obtain the most adequate probable optimum of the objective function, resulting in 0.0446 for σ2 and 269,724.1282 for γ, where σ2 and γ are the two main parameters of this model. The proposed model was evaluated using the statistical parameters, including RMSE, APRE, R2, and AAPRE, which are reported in Table 3. This table shows the statistical parameters for both of the “Training” and “Test” sets. These parameters show the accuracy of the model in all data sets including the “Training” and “Test” sets. The results reveal that the developed model, reports the electrical conductivity of IL mixtures with APRE and AAPRE = −2.54% and 5.42%, respectively. Moreover, the values of other statistical parameters are reported as R2 = 0.9998 and RMSE = 0.51. In addition to statistical error analysis, graphical error analysis was performed to visualize the accuracy and performance of the proposed CSA-LSSVM model. Fig. 1 shows the crossplot of experimentally measured electrical conductivity values versus predicted values by the proposed model. As it is obvious, a light cloud of points around the unit slope line reveals the precision and validity of the developed model without underestimation or overestimation in the training and testing data sets, demonstrating the superior proficiency of the proposed model. In general, the 45° straight line between the experimental values

ð18Þ

where X is an (n × k) matrix consisting of n data and k parameters of the model, and t denotes the transpose matrix. The Hat values of the data are the diagonal elements of the H value. Graphical identification of the outliers is usually carried out through sketching the William plot based on the H values calculated from Eq. (18). Correlation of Hat indices and standardized cross-validated residuals (R) is represented by this plot. A warning Leverage (H⁎) has a

Table 3 statistical error analysis of the proposed model in both training and testing sets. Data sets

APRE (%)

AAPRE (%)

RMSE

R2

This study, training set This study, test set This study, total

−3.06 −0.46 −2.54

5.85 3.75 5.42

0.3999 0.8276 0.5152

0.9999 0.9999 0.9999

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M. Hosseinzadeh et al. / Journal of Molecular Liquids 221 (2016) 624–632

Fig. 1. Crossplot for electrical conductivity of ternary mixtures in this study.

and represented/predicted data shows the perfect model line. The closer the plotted data to the 45° perfect model line, the higher is the consistency of the model. Deviations of the electrical conductivity values predicted by the model from experimentally measured data are also represented in Figs. 2 and 3, for training and test sets, respectively. As indicated in Fig. 2, a well accordance exists between electrical conductivity values predicted by CSA-LSSVM model and electrical conductivity from experimental data set in training set. As it is obvious, the developed model predicts the values of electrical conductivity data accurately in the training set. Besides, Fig. 3 illustrates the excellent results of the test set using the developed model. Using the graphical and statistical error results, it is clear that this model can accurately determine the value of electrical conductivity of the ternary mixtures including ILs; therefore, the developed model is an excellent candidate for determining the electrical conductivity of IL mixtures, instead of performing the expensive and time consuming experimental techniques. The percent relative error from experimental values of the training and test datasets are shown in Fig. 4. As can be seen in Fig. 4, at very low electrical conductivities, or high concentration of ionic liquids the deviations from experimental values are generally larger than for higher values of

Fig. 2. Comparison between CSA-LSSVM model predictions with the experimental data in training phase.

electrical conductivities or lower concentration of ionic liquids. The estimated electrical conductivities by the proposed LSSVM model with the experimental ones in different ranges of temperature and composition for the [bmim][PF6] + H2O (2) + ethanol (3) and [bmim][PF6] + H2O (2) + acetone (3) systems were compared in Figs. 5 and 6, respectively. As can be seen in both of the systems, by variation of the composition and temperature, the developed model follows the trends that are physically expected and the predicted data match well with the corresponding experimental ones. In order to detect and identify the suspected data, the Leverage statistical approach was accomplished in this study, in which the residuals of the model, Williams Plot, and statistical Hat matrix lead to recognition of possible outliers. The h values were calculated through Eq. (18). Moreover, the Williams plot has been sketched in Fig. 7 for the output obtained from CSA-LSSVM. Being the majority of data points within the ranges 0 b H b 0.134 and −3 b R b 3 reveals that the proposed model is statistically valid and accurate. As shown in Fig. 7, the entire data point except seven points are located within the applicable domain for the presented CSA-LSSVM model. Accordingly, these seven points can be stated as probable doubtful datum. These doubtful data are presented in Table 4. It is interesting to demonstrate that the qualities of the electrical conductivity data are not the same. In other words, lower value of R, and H would lead to more accurate experimental data. To evaluate the influence of input parameters (i.e., melting point, molecular weight and mole percent of each compound) on the output, a sensitivity analyses was performed. For this purpose, to evaluate the impact of each parameter on the electrical conductivity of ternary mixture, the relevancy factor (r) [83–85] was introduced. Higher values for r between input and output parameters demonstrate the higher impact of that input variable on the output result. The negative or positive impact of input parameters on ternary mixture electrical conductivity, nevertheless, is not demonstrated by the absolute value of r. The relevancy factor values with directionality causes a clearer and sensational comprehension about the entire impact, which was calculated in the present study. The values of r are determined as follows: Xn
 
Inpk;i −Inpk λi −λ
 i¼1 r Inpk ; λg ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn
2 2 Xn
Inpk;i −Inpk λi −λ i¼1 i¼1

ð19Þ

where Inpk, i and Inpkare the ith value and the average value of the of the kth input variable, respectively (k = average melting point and average molecular weight), λi represents ith value of the reported electrical

Fig. 3. Comparison between CSA-LSSVM model predictions with the experimental data in testing phase.

M. Hosseinzadeh et al. / Journal of Molecular Liquids 221 (2016) 624–632

629

Fig. 4. Relative error versus experimental electrical conductivity in both traring and testing phases.

Fig. 5. The estimated electrical conductivities by the proposed LSSVM model with the experimental ones in different ranges of temperature and composition for the [bmim][PF6] + H2O (2) + ethanol (3).

Fig. 6. The estimated electrical conductivities by the proposed LSSVM model with the experimental ones in different ranges of temperature and composition for [bmim][PF6] + H2O (2) + acetone (3) systems.

630

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Fig. 7. Detecting the probable outlier data and applicability domain of the LSSVM model.

Table 4 The suspected experimental data based on the Leverage approach. Ref

Component 1

Component 2

Component 3

Tf 1

Tf 2

Tf 3

MW 1

MW 2

MW 3

T (K)

X1

X2

Hat

R

[55] [55] [55] [55] [57] [57] [57]

[bmim][PF6] [bmim][PF6] [bmim][PF6] [bmim][PF6] H2O H2O H2O

H2O H2O H2O H2O [C6mim][Cl] [C6mim][Cl] [C6mim][Cl]

Ethanol Ethanol Ethanol Ethanol [C6mim][BF4] [C6mim][BF4] [C6mim][BF4]

283.1 283.1 283.1 283.1 273.15 273.15 273.15

273.15 273.15 273.15 273.15 192.15 192.15 192.15

159.15 178.15 178.15 178.15 191.15 191.15 191.15

284.18 284.18 284.18 284.18 18.01 18.01 18.01

18.01 18.01 18.01 18.01 226.0 226.0 226.0

46.07 58.08 58.08 58.08 254.0 254.0 254.0

308.15 288.15 308.15 308.15 298.15 298.15 298.15

0.8 0.42 0.52 0.52 0.99729 0.99640 0.99639

0.10 0.17 0.34 0.14 0.002032 0.001803 0.00271

0.047 0.061 0.070 0.048 0.018 0.018 0.018

3.679 −3.524 4.682 3.197 3.122 3.036 3.438

conductivity, and λ denotes the average value of the presented electrical conductivity. First of all, it was aimed to figure out the relevancy factor between each of the nine input parameters and the electrical conductivity; nevertheless, not meaningful outcome was resulted, because the combination of these parameters affects electrical conductivity of ternary mixtures. To overcome this problem, the relevancy factor between temperature, average melting point (Tfa) of the ternary mixtures and average molecular weight (MWa) of the ternary mixtures with electrical conductivity was calculated. These average parameters were determined using the following formula:

Tfa ¼

i¼3 X

T f i  xi

ð20Þ

MWa ¼

i¼3 X

MWi  xi

ð21Þ

i¼1

The result of the sensitivity analysis conducted in this study is shown in Fig. 8. As it is demonstrated obviously in the figure, either of temperature and average molecular weight has positive impacts on electrical conductivity of ternary mixture, while average melting point has a negative impact on ternary mixtures electrical conductivity. As can be seen in this figure, the relevancy factors are 0.56, −0.79 and 0.05 for the average molecular weight, average melting point and temperature, respectively.

i¼1

6. Conclusion

Fig. 8. Relevancy factor of temperature, average molecular weight and average melting point of ternary mixtures with electrical conductivity.

In this article, a large data set of ternary mixtures containing ILs was prepared from literature sources and a supervised learning algorithm which is least square support vector machine has been presented to estimate the electrical conductivity of IL ternary mixtures as a function of temperature, composition, molecular weight and melting points of components. About 80% of the whole dataset (179 data points) were used for training and 20% (45 data points) of the whole dataset were applied to test the model performance. The developed CSA-LSSVM model predicts electrical conductivity of ternary mixtures containing ILs with a good accuracy. Graphical and statistical error analyses shows that the developed model is efficient and reliable. Predictions of the model were compared with experimental measurements and well accordance was observed, and the overall R2 of 0.9999 and AAPRE of 5.42% were obtained. The relevancy factor revealed that the average melting point has the greatest impact on the ternary mixture electrical conductivities. At the end, the Leverage approach revealed that the developed model is valid and reliable form statistical point of view.

M. Hosseinzadeh et al. / Journal of Molecular Liquids 221 (2016) 624–632

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