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Analysis of influencing factors of the component analysis method for multivariate mixture
Optik - International Journal for Light and Electron Optics 183 (2019) 854–862
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Analysis of influencing factors of the component analysis method for multivariate mixture ⁎
T
⁎
Xinhao Hua,b, Yan Lva,b, Lijian Zhoua,b, , Qiushi Wanga,b, Müslüm Arıcıa,b, , Dong Lia a b
School of Architecture and Civil Engineering, Northeast Petroleum University, Daqing, China Engineering Faculty, Mechanical Engineering Department, Kocaeli University, Umuttepe Campus, 41380, Kocaeli, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: Quantitative optical analysis Refractive index spectra Mixed gradient Spectral pretreatment
The quantitative optical analysis method, as the development trend of mixture quantitative analysis, has the advantages of fast, non-contact and real time online detection. The refractive index spectra contain information of mixture, which are regarded as the basis for establishing quantitative optical analysis model. Compared with the traditional transmittance or reflectance spectra, the refractive index spectra based on mixing rule can reduce the scale of sample set, and achieves the purpose of simplifying the model. The refractive index spectra contain not only useful information, but also other useless information such as noise. Thus, the effect of spectral pretreatment on the prediction ability was compared due the existence of the interference information. The mixed gradient and the data type determine the scale of the sample set, which are all the determinant factors of the complexity of the model. Therefore, they were also regarded as factors that influence the prediction ability of the M-D-iPLS model. It is found that the M-D-iPLS0.05 model based on the spectral pretreatment has better prediction performance.
1. Introduction There is some relationship between the component content in the mixture and its corresponding optical characterization, such as the transmittance or the reflectance, which is the key to the quantitative analysis of the mixture. Quantitative analysis of multivariate mixture can be attributed to the solution of the linear relationship between two sets of data, that is, the independent variable set and the dependent variable set. The independent and dependent variable sets are represented by the X matrix and the Y matrix, respectively. Considering the role of samples in model, the dataset can be divided into calibration set and prediction set. The partial least squares (PLS) model is an efficient tool for establishing a linear model between the two matrices [1]. In general, the PLS model should be utilized on the entire independent variable set, that is, the X matrix. Guo et al. [2] built quantitative models by diffuse reflectance FTIR spectroscopy and used fifteen fusidic acid samples with different concentration ratio for calibration. They indicated that diffuse reflectance FTIR spectroscopy offered significant advantages in providing accurate measurement of polymorphic content in the fusidic acid binary mixtures. Chen et al. [3] combined the NIR spectroscopy and the interval-based PLS (iPLS) model for determining wool content in textile and a total of 108 cloth samples with wool content ranging from 0% to 100% (w/w) which were collected to compose data sets. In the above-mentioned studies, the measurement of large amounts of data is necessary for conventional quantitative analysis but there is no benefit for quantitative optical analysis [4]. The main reason for this problem is that the larger the number of samples collected, the larger the scale of the independent variable matrix, and the more complex the corresponding spectral information. ⁎
Corresponding authors at: School of Architecture and Civil Engineering, Northeast Petroleum University, Daqing, China. E-mail addresses: [email protected] (L. Zhou), [email protected] (M. Arıcı).
https://doi.org/10.1016/j.ijleo.2018.11.171 Received 11 October 2018; Accepted 19 November 2018 0030-4026/ © 2019 Published by Elsevier GmbH.
Optik - International Journal for Light and Electron Optics 183 (2019) 854–862
X. Hu, et al.
However, not all the spectral information is equally important and some information have nothing to do with the dependent variable set, that is, it is not related to the analyzed matter [5]. As a result, the interference of complex spectral information and other external factors such as noise lead to a decline in the prediction accuracy of the model [6]. To sum up, the pretreatment of spectral data, the selection of characteristic wavelength and the compression of the dataset scale are the key factors for improving the accuracy and stability of the quantitative optical analysis model. Moreover, the measurement of these parameters can elucidate the results more accurate and reduce the computational cost [7]. Pretreatment of spectra can enhance the useful information and reduce irrelevant information. The Savitzky-Golay differentiation is a data smoothing method based on partial least squares, which has been applied in the signal denoising [8,9] and data smoothing [10,11]. Guo et al. [12] presented an enhanced sensor fault detection and diagnosis method combining the Satizky-Golay (SG) method and the principal component analysis (PCA) method for the VRF system. Smoothing the original data using SG method is one of the main features of the method. The Savitzky-Golay differentiation which can eliminate baseline interference and improve spectral resolution is most commonly used method for spectral pretreatment [13]. Zheng et al. [14] utilized three NIR datasets including diesel, wheat and corn datasets to test the pretreatment method of fractional order Savitzky-Golay differentiation. The results showed that the fractional order SavitzkyGolay differentiation (FOSGD) improved the performance of the PLS model with smaller RMSECV and RMSEP than integral order ones. Sechenyh et al. [15] examined the performance of several mixing rules which were commonly used in modeling refractive index at two different wave lengths. It was reported that values predicted by the Newton mixing rule displayed a slightly better agreement with the experimental values. Shukla et al. [16] measured the refractive indices for the binary liquid mixtures under different temperatures and pressures and used Lorentz-Lorentz mixing rule to study the refractive indices. They showed the feasibility of mixing rules for solving the refractive indices of mixtures and reported that the studied models can elucidate molar refractivity, molecular interaction and association constant. The above researches prove that it is feasible to establish refractive index spectra according to a certain mixied gradient by using the refractive index mixing rule. The mixed gradient is one of the key factors affecting the structure and prediction ability of the quantitative analysis model. In this work, a quantitative optical analysis method was established by combining the PTT method, the Newton mixing rule and the iPLS model, and the influence of spectral pretreatment and mixed gradient on the model was analyzed taking the mixture of ethanol and distilled water as the research objects. 2. Experimental procedure 2.1. Material and transmittance spectra measurement Ethanol and distilled water purchased from Tianjin Da Mao chemical reagents factory, having a purity > 99%. The volume fraction of distilled water is defined asϕdistilled water , and the concentration gradients of the samples are shown in Table 1. Transmittance spectra under two optical paths (5 mm and 10 mm) were obtained by a TU-1900 two-beam ultraviolet-visible spectrophotometer, which are shown in Fig. 1. Transmittance spectra of samples under two optical paths of 5 mm (a) and 10 mm (b) 2.2. The solution of refractive index Optical measurement methods combined with iterative algorithms are available to obtain the refractive index of material. Ai et al. [17] proposed an improved double-thickness method combined with genetic algorithm was developed to determine the optical constants of liquid hydrocarbon fuels, and the optical constants of the distilled water were determined to verify the reliability of this method. Wang et al. [18] established a new method called the particle swarm double-thickness transmittance method (PTT), which combined the optical measurement method and the iterative algorithm. The method measures the transmittance of two optical paths and forms a closed set of equations, which are solved by the particle swarm optimization algorithm. The following equations constitute PTT method.
T1 =
T2 =
ρ=
(1 − ρ)2⋅e−α ⋅ L1 1 − ρ2 ⋅e−2 ⋅ α ⋅ L1
(1)
ρ)2⋅e−α ⋅ L2
(1 − 1 − ρ2 ⋅e−2 ⋅ α ⋅ L2 1)2
(2)
k2
(n − + (n + 1)2 + k 2
(3)
Table 1 The concentration gradients of the samples. Serial number ϕdistilled water
1
2
3
4
5
6
7
8
0
0.05
0.2
0.5
0.6
0.8
0.9
1.0
855
Optik - International Journal for Light and Electron Optics 183 (2019) 854–862
X. Hu, et al.
Fig. 1. The spectral band width is 1 nm, and the room temperature is 23.
α=
4⋅π⋅k λ
(4)
Here, n is the refractive index, k is the absorption index and λ is the wavelength. α and ρ are the absorption coefficient and reflectivity of material, respectively. T1 and T2 are the transmittance at different optical paths of 5 mm and 10 mm, respectively. Tm and Tc are the experimental measurement and calculated values of transmittance, respectively. Based on the above values, the closed-form of nonlinear equations can be constructed as follows:
Tc1 − Tm1 = 0
(5)
Tc 2 − Tm2 = 0
(6)
The evaluation function can be obtained as:
(Tm1 − Tc1)2 + (Tm2 − Tc 2)2 ≤ ε
(7)
The refractive indices are shown in Fig. 2. 2.3. The Newton mixture rule The expression of the Newton mixture rule is as follows: 2 2 (nm2 − 1) = (ndistilled water − 1) ϕdistilled water + (nethanol − 1) ϕethanol
ϕdistilled water + ϕethanol = 1
(8) (9)
ϕ1and
ϕ2 where nm , ndistilled water , n ethanol respectively represents the refractive index of mixture, pure distilled water and pure ethanol. are the volume fractions of distilled water and ethanol, respectively. The difference between the two adjacent samples of the volume fraction of distilled water is considered as a mixed gradient ( Δmix − gradient ). In the following study, 0.01 and 0.05 will be used to discuss the effect of the mixed gradient on the quantitative analysis model.
Fig. 2. Refractive index spectra of samples obtained by the PTT method. 856
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Table 2 The mixing data samples of the calibration set for the M-D-iPLS-0.01 model. No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 0.01 0.02 0.03 0.04 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39
41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 61
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.61
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.81
81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 —
0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 —
2.4. The M-D-iPLS model The M-D-iPLS model consists of the PTT method, the Newton mixing rule and the iPLS model. According to the difference of the mixed gradient, the M-D-iPLS model can be divided into the M-D-iPLS-0.01 model and the M-D-iPLS-0.05 model. The samples were divided into two types, the experimental data samples and the mixing data samples. The refractive indices of the former are obtained by the PTT method based on the transmittance spectra (see Section 2.2), and the refractive indices of the latter are obtained by the Newton mixing rule based on thr refractive indices of pure component (see Section 2.3). 3. The influencing factors 3.1. The mixed gradient The mixed gradient of 0.01 ( Δmix − gradient = 0.01) and mixed gradient of 0.05 ( Δmix − gradient = 0.05) were used to establish the quantitative analysis model, respectively. The refractive index of pure distilled water (ϕdistilled water = 1) and pure ethanol (ϕdistilled water = 0 ) were introduced into Eq. (8) to obtain the refractive index spectra. The mixing data samples and the experimental data samples of the calibration set are shown respectively in Tables 2 and 3, and the predicted set shown in Table 4 are all the experimental data samples. As a contrast, the new calibration set (Table 5) was established under the mixed gradient of 0.05. The refractive indices of samples No. 2, No. 5, No. 10, No. 12 and No. 17 were obtained by the PTT method while the refractive indices of the rest samples were obtained by the Newton mixing rule. The prediction set samples obtained by the PTT method are shown in Table 6. The refractive index spectra under the two mixed gradients are shown in Fig. 3. As can be seen in Fig. 3(a) and (b), the distributions of refractive indices of mixtures under the two gradients are similar, and the most significant difference of the two mixed gradients is the size of the data volume. It is obvious that when the mixed gradient is 0.05 ( Δmix − gradient = 0.05), the refractive index distribution is sparse and contains less data volume. 3.2. The spectral pretreatment The Savitzky-Golay differentiation is a commonly used spectral pretreatment method that can eliminate baseline interference and improve spectral resolution [19]. To reduce the noise and smoothing the spectra, the Savitzky-Golay differentiation was used to pretreat the spectra. The original and pretreated spectra of the correction sets are shown in Fig. 4(a) and (b), respectively. Fig. 4(a) and (b) show that the interference of noise significantly reduces and the spectral resolution improves after the spectral pretreatment, the refractive index curves become smooth. Table 3 The experimental data samples of the calibration set for the M-D-iPLS-0.01 model. No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
6
0.05
21
0.2
50
0.5
60
0.6
89
0.9
857
Optik - International Journal for Light and Electron Optics 183 (2019) 854–862
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Table 4 The experimental data samples of the prediction set for the M-D-iPLS0.01 model. Serial number Volume fraction
1
2
0.4
0.8
Table 5 Calibration set for the M-D-iPLS-0.05 model. No.
ϕvolume
No.
ϕvolume
1 2 3 4 5 6 7 8 9 10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5
11 12 13 14 15 16 17 18 19 —
0.55 0.6 0.65 0.7 0.75 0.85 0.9 0.95 1.0 —
Table 6 Prediction set for the M-D-iPLS-0.05 model. Serial number Volume fraction
1
2
0.4
0.8
Fig. 3. Refractive index spectra for Δmix − gradient = 0.01 (a) and Δmix − gradient = 0.05 (b).
4. Results and discussion 4.1. Effect of spectral pretreatment The effect of spectral pretreatment was examined by the M-D-iPLS-0.01 model using the calibration set and the prediction set shown in Tables 2–4 (see Section 3.1). The calibration set and the prediction set models based on the original spectra are plotted respectively in Figs. 5(a) and (b), and 6(a) and (b) show the calibration set and the prediction set models based on the pretreated spectra, respectively. The prediction results before and after the spectral pretreatment are presented in Tables 7 and 8, respectively. It can be seen from Figs. 5(a) and 6(a) that the calibration set model has almost the same evaluation parameters before and after the spectral pretreatment. The RMSEP of the predicted set model is reduced from 0.335 to 0.125 (Figs. 5(b) and 6(b)). After the spectral pretreatment, the relative error of the predicted value is greatly reduced. Especially for the samples No. 1 of the detection set, the relative error is reduced from 44.87% to 9.905%. The results show that the spectral pretreatment can abate the interference of 858
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Fig. 4. Pretreated refractive index spectra of the M-D-iPLS-0.01 model.
Fig. 5. Calibration set model (a) and prediction set model (b) the M-D-iPLS-0.01 model based on the original spectra.
Fig. 6. Calibration set model (a) and prediction set model (b) for the M-D-iPLS-0.01 model based on the pretreated spectra.
unrelated information and significantly improve the prediction accuracy of the model. In subsequent comparative studies, the spectra of all models were pretreated. 4.2. Effect of the mixed gradient On the basis of pretreatment spectra, the prediction ability of the models under the given two mixed gradients ( Δmix − gradient = 0.01 859
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Table 7 Prediction results of the M-D-iPLS-0.01 model based on the original spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.2205 0.3610
44.87 54.87
Table 8 Prediction results of the M-D-iPLS-0.01 model based on the pretreated spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.4396 0.6275
9.905 21.56
and Δmix − gradient = 0.05) were compared. The calibration set and the prediction set models under the mixed gradient of 0.05 are shown in Fig. 7(a) and (b), respectively. As can be seen in Figs. 6 and 7, the evaluation parameters of the model change marginally and the RMSEP of the prediction set model decreases slightly under the mixed gradient of 0.05, compared to the mixed gradient of 0.01. From the prediction results (Tables 8 and 9), it is observed that the prediction accuracy of the model is improved under the large mixed gradient ( Δmix − gradient = 0.05).
4.3. Analysis of principal component score The principal component score is one of the PLS outcome that indicates the relationship between samples. Samples in the score plot are placed with respect to their similarities (or differences) and the distance between them is directly proportional to the spectral difference [20,21]. The percentage of the variance explained by the first two principal components of each model can be seen in Fig. 8. Taking the M-D-iPLS-0.01 model as an example, the percentage of the variance explained by the principal component based on the pretreated spectra was analyzed. The first principal component score is PC1 = 94.02% and the second is PC2 = 4.34%, which can explain 98.36% of the total variability (see Fig. 8). The research object is two component mixture of distilled water and ethanol, and the volume fraction of distilled water is used as an index to classify the samples. Considering the percentage of variance explained by the first principal component (PC1) which is as high as 96.8%, it can be concluded that volume fraction of distilled water is the dominating determinant factor. Further analysis was performed about the similarity or difference between the spectra of the experimental data and the mixing rule data. We distributed the samples in the PC1/PC2 plane according to the scores of the first and second principal components. Fig. 9 shows the distribution of samples in the PC1/PC2 plane. For the M-D-iPLS-0.01 and M-D-iPLS-0.05 models, the samples consist of mixing data samples and experimental data samples. The experimental data samples and the mixing data samples can be distinguished clearly from the principal component score plot. The experimental and mixing data samples are circled by the blue and purple dotted lines in Fig. 9(a) and (b), respectively. The experimental data samples and mixing data samples have the same distribution rule along the PC1 axis. It can be seen that the addition of experimental data samples does not change the distribution law of the whole data on the PC1 axis. On the contrary, the overall
Fig. 7. Calibration set model (a) and prediction set model (b) for the M-D-iPLS-0.05 model. 860
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Table 9 Prediction results of the M-D-iPLS-0.05 model based on the pretreated spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.4253 0.6479
6.332 19.01
Fig. 8. Percentage of the variance explained by the first two principal components of each model.
Fig. 9. Principal component score plots (a) M-D-iPLS-0.01, (b) M-D-iPLS-0.05.
difference of sample data is increased while the effect of multiple collinearity is eliminated which implies that the number of experiments can be reduced. The above analysis proves that the joint modeling of the experimental data samples and the mixing data samples together reduces the multiple collinearity of the mixing data samples and avoids the model misalignment to some extent. This also reduces the number of experiments and simplifies the quantitative analysis of processes. In contrast, the M-D-iPLS-0.05 model has the best prediction ability and the complexity of the model is lower than that of the M-D-iPLS-0.01 model. In the models proposed in this paper, the M-D-iPLS-0.05 model has the highest accuracy and the complexity of the model is moderate. It is neither too simple nor too complicated and its stability is strong. 5. Conclusions Based on the refractive index spectra, a new kind of mixture quantitative analysis model was established, in which the refractive index was obtained by the PTT method and the Newton mixing rule, and the quantitative analysis was realized by the iPLS model. Samples of different volume fraction of distilled water were prepared with pure distilled water and pure ethanol, and the samples were divided into experimental data samples and mixing data samples according to the different acquisition methods of sample data. Then the samples were taken as research objects to investigate the predictive ability of the quantitative analysis model. 861
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We compared the prediction ability of the M-D-iPLS-0.01 model before and after the spectral pretreatment, and confirmed that the spectral pretreatment was helpful to improve the prediction accuracy of the model. Therefore, in subsequent studies, the influence of mixed gradient on the model was analyzed on the basis of pretreatment spectra. The stability and prediction ability of the model were compared and analyzed through the relative error of the prediction set samples. We further analyzed the distribution of samples by principal component score, and judged the feasibility of the model. The following conclusions can be drawn from the preceding results : 1) The spectral pretreatment can make the spectra more smooth, improve the resolution of the spectra and weaken the interference of irrelevant information. Taking the M-D-iPLS-0.01 model as an example, the prediction accuracy is obviously improved before and after spectral pretreatment. Especially for the sample 1 of the detection set, the relative error is reduced from 44.87% to 9.905%. The results show that the spectral pretreatment can abate the interference of unrelated information and significantly improve the prediction accuracy of the model. 2) Through the comprehensive analysis of the principal component score, the results show that the addition of the experimental data can effectively reduce the multiple collinearity of the mixed rule data, but does not destroy the distribution of the model on the PC1 axis. The addition of experimental data can effectively improve the generalization ability of the model and accept the data information other than the mixing rule, thus improving the stability and prediction ability of the model. 3) The M-D-iPLS model has the advantages of nondestructive and cost-efficient in optical analysis, and non-contact measurement does not pollute the samples. At the same time, the purpose of simplifying model can be achieved by improving the mixed gradient of the M-D-iPLS model and the prediction accuracy of the M-D-iPLS-0.05 model is improved. The relative error of the sample No. 1 and the sample No. 2 of the prediction set are 19.01% and 6.332%, respectively. In conclusion, the performance of the M-D-iPLS model is improved after the spectral pretreatment and the mixed gradient redistribution. The introduction of mixing rule can quickly construct the refractive index spectra without carrying out a large number of experimental measurements. The combination of optical analysis, mixing rule and quantitative analysis makes the optical analysis model of multicomponent mixture more practical. Funding The financial support provided by the Project Funded by China Postdoctoral Science Foundation through Grant No. 2018T110267, the PetroChina Innovation Foundation through Grant No. 2018D-5007-0608, and the Natural Science Foundation of Heilongjiang through Grant No. E2016010. References [1] L.M. Duarte, D. Paschoal, C.M.S. Izumi, M.D. Dolzan, V.R. Alves, G.A. Micke, H.F. Dos Santos, M.A.L. de Oliveira, Simultaneous determination of aspartame, cyclamate, saccharin and acesulfame-K in powder tabletop sweeteners by FT-Raman spectroscopy associated with the multivariate calibration: PLS, iPLS and siPLS models were compared, Food Res. Int. 99 (2017) 106–114. [2] C.Y. Guo, X.F. Luo, X.H. Zhou, B.J. Shi, J.J. Wang, J.Q. Zhao, X.X. Zhang, Quantitative analysis of binary polymorphs mixtures of fusidic acid by diffuse reflectance FTIR spectroscopy, diffuse reflectance FT-NIR spectroscopy, Raman spectroscopy and multivariate calibration, J. Pharm. Biomed. Anal. 140 (2017) 130–136. [3] H. 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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Analysis of influencing factors of the component analysis method for multivariate mixture ⁎
T
⁎
Xinhao Hua,b, Yan Lva,b, Lijian Zhoua,b, , Qiushi Wanga,b, Müslüm Arıcıa,b, , Dong Lia a b
School of Architecture and Civil Engineering, Northeast Petroleum University, Daqing, China Engineering Faculty, Mechanical Engineering Department, Kocaeli University, Umuttepe Campus, 41380, Kocaeli, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: Quantitative optical analysis Refractive index spectra Mixed gradient Spectral pretreatment
The quantitative optical analysis method, as the development trend of mixture quantitative analysis, has the advantages of fast, non-contact and real time online detection. The refractive index spectra contain information of mixture, which are regarded as the basis for establishing quantitative optical analysis model. Compared with the traditional transmittance or reflectance spectra, the refractive index spectra based on mixing rule can reduce the scale of sample set, and achieves the purpose of simplifying the model. The refractive index spectra contain not only useful information, but also other useless information such as noise. Thus, the effect of spectral pretreatment on the prediction ability was compared due the existence of the interference information. The mixed gradient and the data type determine the scale of the sample set, which are all the determinant factors of the complexity of the model. Therefore, they were also regarded as factors that influence the prediction ability of the M-D-iPLS model. It is found that the M-D-iPLS0.05 model based on the spectral pretreatment has better prediction performance.
1. Introduction There is some relationship between the component content in the mixture and its corresponding optical characterization, such as the transmittance or the reflectance, which is the key to the quantitative analysis of the mixture. Quantitative analysis of multivariate mixture can be attributed to the solution of the linear relationship between two sets of data, that is, the independent variable set and the dependent variable set. The independent and dependent variable sets are represented by the X matrix and the Y matrix, respectively. Considering the role of samples in model, the dataset can be divided into calibration set and prediction set. The partial least squares (PLS) model is an efficient tool for establishing a linear model between the two matrices [1]. In general, the PLS model should be utilized on the entire independent variable set, that is, the X matrix. Guo et al. [2] built quantitative models by diffuse reflectance FTIR spectroscopy and used fifteen fusidic acid samples with different concentration ratio for calibration. They indicated that diffuse reflectance FTIR spectroscopy offered significant advantages in providing accurate measurement of polymorphic content in the fusidic acid binary mixtures. Chen et al. [3] combined the NIR spectroscopy and the interval-based PLS (iPLS) model for determining wool content in textile and a total of 108 cloth samples with wool content ranging from 0% to 100% (w/w) which were collected to compose data sets. In the above-mentioned studies, the measurement of large amounts of data is necessary for conventional quantitative analysis but there is no benefit for quantitative optical analysis [4]. The main reason for this problem is that the larger the number of samples collected, the larger the scale of the independent variable matrix, and the more complex the corresponding spectral information. ⁎
Corresponding authors at: School of Architecture and Civil Engineering, Northeast Petroleum University, Daqing, China. E-mail addresses: [email protected] (L. Zhou), [email protected] (M. Arıcı).
https://doi.org/10.1016/j.ijleo.2018.11.171 Received 11 October 2018; Accepted 19 November 2018 0030-4026/ © 2019 Published by Elsevier GmbH.
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However, not all the spectral information is equally important and some information have nothing to do with the dependent variable set, that is, it is not related to the analyzed matter [5]. As a result, the interference of complex spectral information and other external factors such as noise lead to a decline in the prediction accuracy of the model [6]. To sum up, the pretreatment of spectral data, the selection of characteristic wavelength and the compression of the dataset scale are the key factors for improving the accuracy and stability of the quantitative optical analysis model. Moreover, the measurement of these parameters can elucidate the results more accurate and reduce the computational cost [7]. Pretreatment of spectra can enhance the useful information and reduce irrelevant information. The Savitzky-Golay differentiation is a data smoothing method based on partial least squares, which has been applied in the signal denoising [8,9] and data smoothing [10,11]. Guo et al. [12] presented an enhanced sensor fault detection and diagnosis method combining the Satizky-Golay (SG) method and the principal component analysis (PCA) method for the VRF system. Smoothing the original data using SG method is one of the main features of the method. The Savitzky-Golay differentiation which can eliminate baseline interference and improve spectral resolution is most commonly used method for spectral pretreatment [13]. Zheng et al. [14] utilized three NIR datasets including diesel, wheat and corn datasets to test the pretreatment method of fractional order Savitzky-Golay differentiation. The results showed that the fractional order SavitzkyGolay differentiation (FOSGD) improved the performance of the PLS model with smaller RMSECV and RMSEP than integral order ones. Sechenyh et al. [15] examined the performance of several mixing rules which were commonly used in modeling refractive index at two different wave lengths. It was reported that values predicted by the Newton mixing rule displayed a slightly better agreement with the experimental values. Shukla et al. [16] measured the refractive indices for the binary liquid mixtures under different temperatures and pressures and used Lorentz-Lorentz mixing rule to study the refractive indices. They showed the feasibility of mixing rules for solving the refractive indices of mixtures and reported that the studied models can elucidate molar refractivity, molecular interaction and association constant. The above researches prove that it is feasible to establish refractive index spectra according to a certain mixied gradient by using the refractive index mixing rule. The mixed gradient is one of the key factors affecting the structure and prediction ability of the quantitative analysis model. In this work, a quantitative optical analysis method was established by combining the PTT method, the Newton mixing rule and the iPLS model, and the influence of spectral pretreatment and mixed gradient on the model was analyzed taking the mixture of ethanol and distilled water as the research objects. 2. Experimental procedure 2.1. Material and transmittance spectra measurement Ethanol and distilled water purchased from Tianjin Da Mao chemical reagents factory, having a purity > 99%. The volume fraction of distilled water is defined asϕdistilled water , and the concentration gradients of the samples are shown in Table 1. Transmittance spectra under two optical paths (5 mm and 10 mm) were obtained by a TU-1900 two-beam ultraviolet-visible spectrophotometer, which are shown in Fig. 1. Transmittance spectra of samples under two optical paths of 5 mm (a) and 10 mm (b) 2.2. The solution of refractive index Optical measurement methods combined with iterative algorithms are available to obtain the refractive index of material. Ai et al. [17] proposed an improved double-thickness method combined with genetic algorithm was developed to determine the optical constants of liquid hydrocarbon fuels, and the optical constants of the distilled water were determined to verify the reliability of this method. Wang et al. [18] established a new method called the particle swarm double-thickness transmittance method (PTT), which combined the optical measurement method and the iterative algorithm. The method measures the transmittance of two optical paths and forms a closed set of equations, which are solved by the particle swarm optimization algorithm. The following equations constitute PTT method.
T1 =
T2 =
ρ=
(1 − ρ)2⋅e−α ⋅ L1 1 − ρ2 ⋅e−2 ⋅ α ⋅ L1
(1)
ρ)2⋅e−α ⋅ L2
(1 − 1 − ρ2 ⋅e−2 ⋅ α ⋅ L2 1)2
(2)
k2
(n − + (n + 1)2 + k 2
(3)
Table 1 The concentration gradients of the samples. Serial number ϕdistilled water
1
2
3
4
5
6
7
8
0
0.05
0.2
0.5
0.6
0.8
0.9
1.0
855
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Fig. 1. The spectral band width is 1 nm, and the room temperature is 23.
α=
4⋅π⋅k λ
(4)
Here, n is the refractive index, k is the absorption index and λ is the wavelength. α and ρ are the absorption coefficient and reflectivity of material, respectively. T1 and T2 are the transmittance at different optical paths of 5 mm and 10 mm, respectively. Tm and Tc are the experimental measurement and calculated values of transmittance, respectively. Based on the above values, the closed-form of nonlinear equations can be constructed as follows:
Tc1 − Tm1 = 0
(5)
Tc 2 − Tm2 = 0
(6)
The evaluation function can be obtained as:
(Tm1 − Tc1)2 + (Tm2 − Tc 2)2 ≤ ε
(7)
The refractive indices are shown in Fig. 2. 2.3. The Newton mixture rule The expression of the Newton mixture rule is as follows: 2 2 (nm2 − 1) = (ndistilled water − 1) ϕdistilled water + (nethanol − 1) ϕethanol
ϕdistilled water + ϕethanol = 1
(8) (9)
ϕ1and
ϕ2 where nm , ndistilled water , n ethanol respectively represents the refractive index of mixture, pure distilled water and pure ethanol. are the volume fractions of distilled water and ethanol, respectively. The difference between the two adjacent samples of the volume fraction of distilled water is considered as a mixed gradient ( Δmix − gradient ). In the following study, 0.01 and 0.05 will be used to discuss the effect of the mixed gradient on the quantitative analysis model.
Fig. 2. Refractive index spectra of samples obtained by the PTT method. 856
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Table 2 The mixing data samples of the calibration set for the M-D-iPLS-0.01 model. No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 0.01 0.02 0.03 0.04 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39
41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 61
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.61
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.81
81 82 83 84 85 86 87 88 90 91 92 93 94 95 96 97 98 99 —
0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 —
2.4. The M-D-iPLS model The M-D-iPLS model consists of the PTT method, the Newton mixing rule and the iPLS model. According to the difference of the mixed gradient, the M-D-iPLS model can be divided into the M-D-iPLS-0.01 model and the M-D-iPLS-0.05 model. The samples were divided into two types, the experimental data samples and the mixing data samples. The refractive indices of the former are obtained by the PTT method based on the transmittance spectra (see Section 2.2), and the refractive indices of the latter are obtained by the Newton mixing rule based on thr refractive indices of pure component (see Section 2.3). 3. The influencing factors 3.1. The mixed gradient The mixed gradient of 0.01 ( Δmix − gradient = 0.01) and mixed gradient of 0.05 ( Δmix − gradient = 0.05) were used to establish the quantitative analysis model, respectively. The refractive index of pure distilled water (ϕdistilled water = 1) and pure ethanol (ϕdistilled water = 0 ) were introduced into Eq. (8) to obtain the refractive index spectra. The mixing data samples and the experimental data samples of the calibration set are shown respectively in Tables 2 and 3, and the predicted set shown in Table 4 are all the experimental data samples. As a contrast, the new calibration set (Table 5) was established under the mixed gradient of 0.05. The refractive indices of samples No. 2, No. 5, No. 10, No. 12 and No. 17 were obtained by the PTT method while the refractive indices of the rest samples were obtained by the Newton mixing rule. The prediction set samples obtained by the PTT method are shown in Table 6. The refractive index spectra under the two mixed gradients are shown in Fig. 3. As can be seen in Fig. 3(a) and (b), the distributions of refractive indices of mixtures under the two gradients are similar, and the most significant difference of the two mixed gradients is the size of the data volume. It is obvious that when the mixed gradient is 0.05 ( Δmix − gradient = 0.05), the refractive index distribution is sparse and contains less data volume. 3.2. The spectral pretreatment The Savitzky-Golay differentiation is a commonly used spectral pretreatment method that can eliminate baseline interference and improve spectral resolution [19]. To reduce the noise and smoothing the spectra, the Savitzky-Golay differentiation was used to pretreat the spectra. The original and pretreated spectra of the correction sets are shown in Fig. 4(a) and (b), respectively. Fig. 4(a) and (b) show that the interference of noise significantly reduces and the spectral resolution improves after the spectral pretreatment, the refractive index curves become smooth. Table 3 The experimental data samples of the calibration set for the M-D-iPLS-0.01 model. No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
No.
ϕvolume
6
0.05
21
0.2
50
0.5
60
0.6
89
0.9
857
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Table 4 The experimental data samples of the prediction set for the M-D-iPLS0.01 model. Serial number Volume fraction
1
2
0.4
0.8
Table 5 Calibration set for the M-D-iPLS-0.05 model. No.
ϕvolume
No.
ϕvolume
1 2 3 4 5 6 7 8 9 10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45 0.5
11 12 13 14 15 16 17 18 19 —
0.55 0.6 0.65 0.7 0.75 0.85 0.9 0.95 1.0 —
Table 6 Prediction set for the M-D-iPLS-0.05 model. Serial number Volume fraction
1
2
0.4
0.8
Fig. 3. Refractive index spectra for Δmix − gradient = 0.01 (a) and Δmix − gradient = 0.05 (b).
4. Results and discussion 4.1. Effect of spectral pretreatment The effect of spectral pretreatment was examined by the M-D-iPLS-0.01 model using the calibration set and the prediction set shown in Tables 2–4 (see Section 3.1). The calibration set and the prediction set models based on the original spectra are plotted respectively in Figs. 5(a) and (b), and 6(a) and (b) show the calibration set and the prediction set models based on the pretreated spectra, respectively. The prediction results before and after the spectral pretreatment are presented in Tables 7 and 8, respectively. It can be seen from Figs. 5(a) and 6(a) that the calibration set model has almost the same evaluation parameters before and after the spectral pretreatment. The RMSEP of the predicted set model is reduced from 0.335 to 0.125 (Figs. 5(b) and 6(b)). After the spectral pretreatment, the relative error of the predicted value is greatly reduced. Especially for the samples No. 1 of the detection set, the relative error is reduced from 44.87% to 9.905%. The results show that the spectral pretreatment can abate the interference of 858
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Fig. 4. Pretreated refractive index spectra of the M-D-iPLS-0.01 model.
Fig. 5. Calibration set model (a) and prediction set model (b) the M-D-iPLS-0.01 model based on the original spectra.
Fig. 6. Calibration set model (a) and prediction set model (b) for the M-D-iPLS-0.01 model based on the pretreated spectra.
unrelated information and significantly improve the prediction accuracy of the model. In subsequent comparative studies, the spectra of all models were pretreated. 4.2. Effect of the mixed gradient On the basis of pretreatment spectra, the prediction ability of the models under the given two mixed gradients ( Δmix − gradient = 0.01 859
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Table 7 Prediction results of the M-D-iPLS-0.01 model based on the original spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.2205 0.3610
44.87 54.87
Table 8 Prediction results of the M-D-iPLS-0.01 model based on the pretreated spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.4396 0.6275
9.905 21.56
and Δmix − gradient = 0.05) were compared. The calibration set and the prediction set models under the mixed gradient of 0.05 are shown in Fig. 7(a) and (b), respectively. As can be seen in Figs. 6 and 7, the evaluation parameters of the model change marginally and the RMSEP of the prediction set model decreases slightly under the mixed gradient of 0.05, compared to the mixed gradient of 0.01. From the prediction results (Tables 8 and 9), it is observed that the prediction accuracy of the model is improved under the large mixed gradient ( Δmix − gradient = 0.05).
4.3. Analysis of principal component score The principal component score is one of the PLS outcome that indicates the relationship between samples. Samples in the score plot are placed with respect to their similarities (or differences) and the distance between them is directly proportional to the spectral difference [20,21]. The percentage of the variance explained by the first two principal components of each model can be seen in Fig. 8. Taking the M-D-iPLS-0.01 model as an example, the percentage of the variance explained by the principal component based on the pretreated spectra was analyzed. The first principal component score is PC1 = 94.02% and the second is PC2 = 4.34%, which can explain 98.36% of the total variability (see Fig. 8). The research object is two component mixture of distilled water and ethanol, and the volume fraction of distilled water is used as an index to classify the samples. Considering the percentage of variance explained by the first principal component (PC1) which is as high as 96.8%, it can be concluded that volume fraction of distilled water is the dominating determinant factor. Further analysis was performed about the similarity or difference between the spectra of the experimental data and the mixing rule data. We distributed the samples in the PC1/PC2 plane according to the scores of the first and second principal components. Fig. 9 shows the distribution of samples in the PC1/PC2 plane. For the M-D-iPLS-0.01 and M-D-iPLS-0.05 models, the samples consist of mixing data samples and experimental data samples. The experimental data samples and the mixing data samples can be distinguished clearly from the principal component score plot. The experimental and mixing data samples are circled by the blue and purple dotted lines in Fig. 9(a) and (b), respectively. The experimental data samples and mixing data samples have the same distribution rule along the PC1 axis. It can be seen that the addition of experimental data samples does not change the distribution law of the whole data on the PC1 axis. On the contrary, the overall
Fig. 7. Calibration set model (a) and prediction set model (b) for the M-D-iPLS-0.05 model. 860
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Table 9 Prediction results of the M-D-iPLS-0.05 model based on the pretreated spectra. Serial number
Real value
predicted value
relative error /%
1 2
0.4000 0.8000
0.4253 0.6479
6.332 19.01
Fig. 8. Percentage of the variance explained by the first two principal components of each model.
Fig. 9. Principal component score plots (a) M-D-iPLS-0.01, (b) M-D-iPLS-0.05.
difference of sample data is increased while the effect of multiple collinearity is eliminated which implies that the number of experiments can be reduced. The above analysis proves that the joint modeling of the experimental data samples and the mixing data samples together reduces the multiple collinearity of the mixing data samples and avoids the model misalignment to some extent. This also reduces the number of experiments and simplifies the quantitative analysis of processes. In contrast, the M-D-iPLS-0.05 model has the best prediction ability and the complexity of the model is lower than that of the M-D-iPLS-0.01 model. In the models proposed in this paper, the M-D-iPLS-0.05 model has the highest accuracy and the complexity of the model is moderate. It is neither too simple nor too complicated and its stability is strong. 5. Conclusions Based on the refractive index spectra, a new kind of mixture quantitative analysis model was established, in which the refractive index was obtained by the PTT method and the Newton mixing rule, and the quantitative analysis was realized by the iPLS model. Samples of different volume fraction of distilled water were prepared with pure distilled water and pure ethanol, and the samples were divided into experimental data samples and mixing data samples according to the different acquisition methods of sample data. Then the samples were taken as research objects to investigate the predictive ability of the quantitative analysis model. 861
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We compared the prediction ability of the M-D-iPLS-0.01 model before and after the spectral pretreatment, and confirmed that the spectral pretreatment was helpful to improve the prediction accuracy of the model. Therefore, in subsequent studies, the influence of mixed gradient on the model was analyzed on the basis of pretreatment spectra. The stability and prediction ability of the model were compared and analyzed through the relative error of the prediction set samples. We further analyzed the distribution of samples by principal component score, and judged the feasibility of the model. The following conclusions can be drawn from the preceding results : 1) The spectral pretreatment can make the spectra more smooth, improve the resolution of the spectra and weaken the interference of irrelevant information. Taking the M-D-iPLS-0.01 model as an example, the prediction accuracy is obviously improved before and after spectral pretreatment. Especially for the sample 1 of the detection set, the relative error is reduced from 44.87% to 9.905%. The results show that the spectral pretreatment can abate the interference of unrelated information and significantly improve the prediction accuracy of the model. 2) Through the comprehensive analysis of the principal component score, the results show that the addition of the experimental data can effectively reduce the multiple collinearity of the mixed rule data, but does not destroy the distribution of the model on the PC1 axis. The addition of experimental data can effectively improve the generalization ability of the model and accept the data information other than the mixing rule, thus improving the stability and prediction ability of the model. 3) The M-D-iPLS model has the advantages of nondestructive and cost-efficient in optical analysis, and non-contact measurement does not pollute the samples. At the same time, the purpose of simplifying model can be achieved by improving the mixed gradient of the M-D-iPLS model and the prediction accuracy of the M-D-iPLS-0.05 model is improved. The relative error of the sample No. 1 and the sample No. 2 of the prediction set are 19.01% and 6.332%, respectively. In conclusion, the performance of the M-D-iPLS model is improved after the spectral pretreatment and the mixed gradient redistribution. The introduction of mixing rule can quickly construct the refractive index spectra without carrying out a large number of experimental measurements. The combination of optical analysis, mixing rule and quantitative analysis makes the optical analysis model of multicomponent mixture more practical. 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