Optimization of Partial-least-square Regression for Determination of Manganese in Low-alloy Steel by Single-shot Laser-induced Breakdown Spectroscopy

Abstract

Laser-induced breakdown spectroscopy (LIBS) is employed for ultra-high-speed determination of the manganese content in various steel specimens. LIBS is widely known as a method for elemental analysis with a rapid response. It has several advantages such that it can work under ambient pressure, and that specimens can be tested without any pre-treatment such as acid digestion, cleaning, or polishing of surface of the specimens. We applied a laboratory-build LIBS system for the determination of Mn in a series of low-alloy steel certified reference materials by a multivariate analysis using partial-least-square regression. Considering enough intensities of Mn emission lines and spectral interferences from emission lines of the iron matrix in these alloys, two wavelength ranges for the spectrograph could be employed. By minimizing the predicted residual sum of squares and the root mean square error of prediction, the analytical result of the Mn concentrations could be obtained with reasonable accuracy and precision.

1. Introduction

Steel products are widely utilized in modern society over various industrial fields. With progressing the demands of steel products to higher performance, advanced and strict control of the iron and steel making process is required. Especially, chemical compositions of C, Mn, Si, S, and P are quite significant demanding factors for the process control of iron and steel making. As a conventional way for obtaining the chemical compositions during the production process, spark-discharge optical emission spectroscopy (SD-OES) has been widely adopted. Related to SD-OES, a JIS standard has been enacted by G 1253.¹⁾ Though SD-OES has been recognized as a highly-reliable and robust method, it needs some time for preparing the specimens. In order to carry out ultra-fast and real-time monitoring of the chemical composition of steel products, laser-induced breakdown spectroscopy (LIBS) has been widely studied because of its advantages^2,3,4) for the on-site measurements. Various studies have been published for reporting the feasibility of LIBS in steel industry, such as the measurement of solid steel,^{5,6,7,8,9,10,11,12)} molten steel,^13,14,15) slags,^16,17,18) oxides,^19,20,21) scrap²²⁾ rapid in-situ analysis,^23,24,25,26) etc.

In determination of analytical values using inductive-statistical data analysis, partial-least-square regression (PLSR) was adopted in several LIBS researches^27,28,29) because of its fast computation for regression and multivariate analysis. PLSR was first published by H. Wold,³⁰⁾ and developed by his son, S. Wold.³¹⁾ The most important key concept of PLSR is projection of both explanatory and objective variables to subspace, so that we can eliminate not only multicollinearity in the explanatory variables, but also dimensions of both the variables for reduction of computational complexity. In general, spectra data contain many explanatory variables such as data points by wavelength, which are equivalent to measuring pixels of a multi-wavelength detector in a spectrometer. A basic model of the multivariate PLS model could be expressed as follows:   

$$\mathrm{X}=\mathrm{T}{\mathrm{P}}^{\mathrm{T}}+\mathrm{E}$$

  

$$\mathrm{Y}=\mathrm{U}{\mathrm{Q}}^{\mathrm{T}}+\mathrm{F}$$

All the uppercase letters in the equations above are expressed as matrices. X and Y represent n×m matrix of predictors and n×p matrix of responses, respectively. T and U, respectively, stand for projections of X and Y, and they are also called X scores and Y scores. P and Q are, respectively, m×l and p×l orthogonal loading matrices, and matrices of E and F show the errors, which are assumed to be mutually independent and identically distributed as random normal variables. In order to reflect the variances of original variables faithfully, decompositions of X and Y are carried out so as to maximize the covariance between T and U.

In this study, we conducted ultra-fast determination of Mn components including in low-alloy steel specimens by using laser-induced breakdown spectroscopy and the data treatment with PLSR. The main aim of this study is to establish a method of the determination of Mn in steel samples for a short measuring time.

2. Experimental

The LIBS system depicted in Fig. 1 comprised an Q-switched Nd: YAG pulse laser (LS2147/2, LOTIS TII, Minsk, Belarus), a three-dimensional sample stage, planoconvex lenses for focusing the laser (f = 150 mm) and collecting the emitted radiation from the generated laser-induced plasma (f = 70 mm), an optical fiber for ultraviolet region with a core diameter of 500 μm, an astigmatism-compensated Czerny-Turner-type imaging spectrometer (MS7504i, SOL Instruments, Minsk, Belarus), and an ICCD detector (DH334T-18F-03, Andor, Belfast, UK). The wavelength, pulse width, and pulse energy of the Nd:YAG pulse laser were 532 nm (SHG mode), 16–18 ns, and 470 mJ/pulse, respectively. The atomic/ionic emission from the laser-induced plasma was focused to the end-cap of the optical fiber along the radial direction, and then guided to the entrance slit of the spectrometer. The width of the entrance slit was set to 50 μm, which was adjusted to obtain emission spectra with the enough emission intensity as well as the spectral resolution to avoid spectral interferences. The spectrometer has 750 mm of focal length and 1/8.9 of F-number at the entrance slit. We installed a grating which had line density of 2400 grooves/mm, a blaze wavelength of 270 nm, and a reciprocal linear dispersion of 0.51 nm/mm to the spectrometer. There was no slit at the output port connected to the ICCD camera for spectrograph. The size and the number of pixels of the camera were 13 μm and 1024×1024, respectively, resulting in an effective active area of CCD of 13.3 mm × 13.3 mm. The effective wavelength ranges for emission spectra, according to the reciprocal linear dispersion of the installed grating, was calculated to be 6.8 nm (0.51 nm/mm × 13.3 mm). A full-vertical binning for pixels which were arrayed perpendicularly to the spectral direction of the ICCD camera was conducted in order to integrate the received emission intensity. Since gating control of the ICCD camera was needed in the measurement of atomic/ionic emissions from laser-induced plasma with a digital delay generator, we set a gate delay and a gate width to be 1 μs and 8 μs, respectively, as typical values for the following LIBS measurement in order to reduce the background emission by recombination emission and bremsstrahlung.³²⁾ Whereas the ICCD camera had an ability to amplify the emission signal more than 1000 times, the multiplying factors had to be adjusted approximately from 40 to 120, in order to prevent a ghosting of the camera by an excessive input of the emission. A center wavelength of the spectral region observed was set to be 291 nm or 401 nm, so that we could detect emission lines of Mn having enough emission intensities even in a co-existence of very strong atomic/ionic emissions from Fe. The exposure time was set to 10 μs as the minimum value of the camera, and the resulting emission spectra was replicated 6 times. The slit width, the analytical lines of Mn, and the number of measurement were selected as referred to JIS G 1253.

Fig. 1.

The block diagram of LIBS setup in this study.

For obtaining a training model by PLSR, we obtained emission spectra from a series of certified reference materials (CRMs) of low-alloy steels, NBS 1161–1168. After the PLS model for the determination of Mn was formed, the concentrations of Mn in the CRMs of JSS 150-1–155-1³³⁾ were predicted by the regression analysis with the model for checking the accuracy and precision of the model. The recovery rate, calculated as the equation below, and relative standard deviation were utilized as the following evaluation index:   

$$\text{R}\left(\text{\%}\right)= \frac{M{n}_{predicted}}{M{n}_{certified}}\times 100$$

The chemical compositions of all CRMs are summarized in Table 1. The values of “others” in the table are the sum of the certified contents other than C, Si, Mn, P, and S in each series of CRMs. The PLSR calculation was performed by scientific data analysis software (OriginPro 2017J SR2, OriginLab, Northampton, Massachusetts, USA). The PLSR calculation by the software was conducted on a windows-based laptop computer, which equipped 2.8 GHz CPU of Intel Core i7 7700HQ (Kaby Lake), and 16 GB RAM of DDR4-2400 SODIMM. The required time for each PLSR calculation could not be obtained accurately, but it could be estimated for less than 0.1 seconds.

Table 1. The certified chemical compositions of the NBS CRMs and the JSS CRMs.

Nos. of NBS	C/wt%	Si/wt%	Mn/wt%	P/wt%	S/wt%	Others/wt%	Fe/wt%
1161	0.15	0.047	0.36	0.053	0.02	2.9	Bal.
1162	0.4	0.28	0.94	0.045	0.02	2.3	Bal.
1163	0.19	0.41	1.15	0.031	0.02	2.2	Bal.
1164	0.54	0.48	1.32	0.017	0.02	0.9	Bal.
1165	0.037	0.029	0.032	0.008	0.01	0.5	Bal.
1166	0.065	0.025	0.113	0.012	0.01	0.2	Bal.
1167	0.11	0.26	0.275	0.033	0.01	1.8	Bal.
1168	0.26	0.075	0.47	0.023	0.02	2.5	Bal.

Nos. of JSS	C/wt%	Si/wt%	Mn/wt%	P/wt%	S/wt%	Others/wt%	Fe/wt%
150	0.25	0.36	0.20	0.017	0.023	4.9	Bal.
151	0.25	0.038	1.37	0.026	0.014	3.6	Bal.
152	0.28	0.15	0.44	0.032	0.017	4.0	Bal.
153	0.24	0.23	0.77	0.049	0.018	3.8	Bal.
154	0.29	0.57	1.04	0.015	0.016	3.7	Bal.
155	0.35	0.61	0.10	0.016	0.033	4.2	Bal.

3. Results and Discussion

Figures 2(a) and 2(b) shows a series of LIBS spectra of pure iron (MAIRON SHP, Toho Zinc co., Ltd.), NBS 1165 (Mn: 0.032 wt%), NBS 1161 (Mn: 0.36 wt%) and 1164 (Mn: 1.32 wt%). The center wavelengths of these spectra were 401 nm and 291 nm, respectively. Six Mn emission peaks identified in the spectra are marked in the figures, and their assignments are listed in Table 2(a) (401 nm) and Table 2(b) (291 nm). The intense Mn peaks in Fig. 2(a) were derived mainly from the atomic resonance lines, while those in Fig. 2(b) derived from the ionic lines. Since the assignment of the 403.573 nm line marked with an asterisk in Table 2 was not found in the Atomic Spectra Database by NIST,³⁴⁾ we estimated it by a calculation utilizing the excited energy levels and appropriate selection rules.³⁵⁾ Additionally, two V I lines and fifteen V II lines having higher intensities were found in NBS 1164 that contained 0.295 wt% of V. The emission peaks of V are also marked in Fig. 2(b).

Fig. 2.

The stacked spectra of NBS 1161, 1164, 1165 and MAIRON SHP. The center wavelength are 401 nm (a), and 291 nm (b).

Table 2. The assignments of atomic/ionic emissions of Mn found in Figs. 2(a) and 2(b).

401 nm	Wavelength/nm	Transition probability/×108	Assignment/eV
			Upper			Lower
Mn I	401.810	2.03	3d64p	6D7/2	5.1990	3d64s	6D9/2	2.1142
Mn I	403.076	1.4	3d54s4p	6P7/2	3.0751	3d54s2	6S5/2	0.0000
Mn I	403.307	0.99	3d54s4p	6P5/2	3.0733	3d54s2	6S5/2	0.0000
Mn I	403.449	0.632	3d54s4p	6P3/2	3.0722	3d54s2	6S5/2	0.0000
Mn I	403.573*	–	3d64p	6D5/2	5.2139	3d64s	6D7/2	2.1426
Mn I	404.136	7.87	3d64p	6D9/2	5.1812	3d64s	6D9/2	2.1142

291 nm	Wavelength/nm	Transition probability/×108	Assignment/eV
			Upper			Lower
Mn II	290.015	12	3d54p	3D3	16.2053	3d54s	3P2	11.9315
Mn II	293.305	6.12	3d54p	5P1	12.8744	3d54s	5S2	8.6085
Mn II	293.931	9.9	3 d54p	5P2	12.8254	3d54s	5S2	8.6085

Again, a key concept of the partial-least-square regression for multivariate analysis is that the number of explanatory variables are reduced as much as possible. In general, the number of the explanatory variables, called as a PLS factor, for formation of a PLS model should be optimized for avoiding overfitting. The most reasonable method for optimizing the number of the PLS factor is a cross-validation. The cross-validation is a method for selecting the appropriate number of the PLS factors for minimizing predicted errors among CRMs. When some of CRMs are picked up from the whole set of them, a PLS model is formed by the rest CRMs for the prediction of concentrations of Mn. Subsequently, the concentrations of Mn in the picked-up CRMs are predicted by using this PLS model, and the errors are calculated from a difference between the predicted concentration and the certified concentration of Mn. Finally, PRESS (Predicted REsidual Sum of Squares) is calculated by an equation below:   

$$PRESS={{\displaystyle \sum }}^{}{\left(y_{predicted}-y_{certified}\right)}^2$$

There are various ways for carrying out the cross-variation.³⁶⁾ In this study, a leave-one-out method was adopted since this method was available in the OriginPro software. After background reduction was performed by subtracting baselines which were obtained by asymmetric least square smoothing,³⁷⁾ the number of the PLS factor and the resulting PLS model were simultaneously determined by the singular value decomposition. A summary of the PLS model formation by using NBS CRMs is depicted in Figs. 3(a)–3(d). In Fig. 3(a), the root-mean PRESS is plotted with an increase in PLS factors as the result of cross-validation. Optimized numbers of the PLS factors are suggested as minimum points of the root-mean PRESS, and the values are decided to be 8 for the PLS model of 401 nm, and 6 for the PLS model of 291 nm, respectively. After the number of the PLS factor are optimized, a relationship between the certified and predicted concentrations of Mn were regressed as shown in Fig. 2(b) for 401 nm and Fig. 2(c) for 291 nm, respectively. It is clearly observed that the model of 401 nm is better than that of 291 nm with a slightly better fit of the linear regression between the certified and predicted concentrations of Mn, and less residuals for each leave-one-out prediction in Fig. 2(d). With these model, predictions of the Mn concentration are listed in Table 3. It could be obtained for a very fast detection time of 10 μs that the predicted concentrations of Mn by the model generally agreed to the certified ones. For evaluating the quality of these predictions, the Root-Mean-Square Error of Prediction (RMSEP) was calculated³³⁾ and the sum of them are also shown in Table 3. The values of ΣRMSEP were 2.046 wt% for 401 nm and 3.171 wt% for 291 nm, respectively; therefore, it could be judged that the PLS model of 401 nm provided a better result of the predicted Mn concentrations. Comparing the 291 nm model to the 401 nm model, explanatory variables for the 291-nm region seemed to be more complex in a co-existence of V atomic/ionic lines, which might give the lower prediction performance.

Fig. 3.

The changes in root mean squares towards the number of PLS factors (a), PLS regression models of 401 nm (b) and 291 nm (c), the residuals calculated from the models (d). The minimum values were underlined in (a).

Table 3. The predictions of Mn concentrations. The number of PLS factors were optimized by root mean PRESS.

401 nm	Mn in CRM/wt%	1st	2nd	3rd	4th	5th	6th	Average/wt%	Recovery rate/%	RSD/%
	0.20	0.21	0.21	0.18	0.19	0.20	0.22	0.200	100	6.8
PLS factors	1.37	1.43	1.35	1.38	1.37	1.36	1.43	1.386	101	2.2
8	0.44	0.40	0.37	0.33	0.36	0.37	0.37	0.367	83	6.2
	0.77	0.62	0.62	0.65	0.71	0.65	0.76	0.668	87	7.8
ΣRMSEP	1.04	1.29	1.07	1.18	1.04	1.16	1.15	1.147	110	7.1
2.046 wt%	0.10	0.11	0.11	0.13	0.10	0.14	0.13	0.121	121	9.7

291 nm	Mn in CRM/wt%	1st	2nd	3rd	4th	5th	6th	Average/wt%	Recovery rate/%	RSD/%
	0.20	0.24	0.22	0.20	0.20	0.21	0.23	0.215	108	6.6
PLS factors	1.37	1.50	1.53	1.59	1.58	1.57	1.56	1.554	113	2.0
6	0.44	0.33	0.44	0.42	0.47	0.42	0.41	0.414	94	10.1
	0.77	0.69	0.72	0.75	0.67	0.60	0.60	0.672	87	8.6
ΣRMSEP	1.04	0.93	0.87	0.87	0.87	0.85	0.80	0.865	83	4.3
3.171 wt%	0.10	0.10	0.08	0.08	0.10	0.06	0.07	0.077	77	20.1

It should be noted that the optimization method by cross-validation cannot be always applied directly for other series of specimens because PRESS is always calculated within the certified reference materials. On the other hand, ΣRMSEP is calculated by the sum of difference between the certified and predicted Mn concentrations; therefore, the relationship between PRESS/ΣRMSEP and the contribution percentage of explanatory variables X at 401 nm-PLS model are plotted in Fig. 4. The plots in Fig. 4 corresponds to the number of PLS factors of 1 to 10, from left to right. Contribution percentage of X represents the ratio between the partial and total variance of explanatory variables. It can be found that the values of ΣRMSEP was almost invariant when the contribution percentage of X was more than 80%. The number of PLS factor was optimized to be 8 by the cross-validation method described below. The value of ΣRMSEP was sufficiently low at the same time; therefore, the optimization of the number of PLS numbers by cross-validation method was available in the case of this study. For further improvement of the accuracy and precision for the Mn determination, emission signals from LIP may be integrated during multiple pulse laser shots, while the demands for rapid determination of Mn should be taken into consideration.

Fig. 4.

The changes in root mean PRESS and ΣRMSEP towards the contribution percentage of X.

4. Conclusion

In order to achieve an ultra-high-speed determination of Mn in low alloy steel CRMs, we suggested a prediction method of Mn concentrations by partial-least-square regression in laser-induced breakdown spectroscopy. Two center wavelengths for spectrograph were chosen to be 401 nm and 291 nm, so that enough intensities of Mn emission lines could be obtained and spectral inferences from iron lines could occur to a less extent. The optimum numbers for PLS factors, which were computed using the NBS CRMs at a center wavelength of 291 nm, could be directly adopted for the spectra analysis of the JSS CRMs, while such a relationship could not be utilized for the spectra at a center wavelength of 401 nm. The Mn concentration for the JSS CRMs could be well determined since the range of the recovery rate was from 83% to 121%, and the RSDs were within 10%.

Acknowledgement

This research is partially supported by a Grand-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan (No. 17906642). Several parts of the set-ups were installed under a support of I-Type collegium of real-time monitoring of molten steels by the Iron and Steel Institute of Japan.

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