The partial least squares regression is multivariate statistical analysis method which is widely used. It focuses on multivariate regression modeling of multiple variables. The technology for synthesis and screening of information is used in PLS modeling process, combined with the functions of multivariate linear regression analysis, typical correlation analysis and principal component analysis [4] [5]. Then the modeling method of partial least squares regression using in the spectral analysis is introduced.

The spectral response matrix Y and its corresponding gas concentration matrix X are simultaneously decomposed into principal components, and new synthetic variables are obtained as follows:

where T & U are load matrixes of X & Y respectively, and P & Q are scoring matrixes of X & Y respectively. E & F are the errors introduced by using PLS method to fit X & Y respectively.

The regression model is built with PLS, which uses the characteristic spectral response matrix T and the characteristic concentration matrix U of which the vectors are orthogonal to each other.

The regression coefficient matrix B is as follows, which is also called the correlation matrix.

Therefore, the main steps of PLS include the principal component decomposition for the variable matrix Y and the corresponding independent variable matrix X, and the calculation of the correlation matrix B.

Although the PLS method has more advantages than the traditional multivariate regression method, there is still low efficiency when analyzing the absorption spectra of multi-component gas, and the accuracy of the regression results is affected by the noise and sample distribution [6]. In order to improve the prediction accuracy, it is necessary to assign different weights to the samples in calibration sets. By analyzing the error and recovery rate of the calibration sets, the partial least squares (PLS) method can be further improved to into the error weighted partial least squares (EWPLS) and the variance weighted partial least squares (VWPLS) [7].

Assume that Y_c Є R^M*K is the concentration matrix of the calibration set calculated by the PLS method, then the error of the recovery rate can be obtained as the following

where“./”represents the division between the corresponding elements of matrices. E_c Є R^M*K represents the recovery rate errors of K organic matters in M calibration samples. If {r₁, r₂, …, r_M} is composed of the maximum error of the recovery rate of every row in E_c, the Gauss weight corresponding to the maximum error of the recovery rate is set as follows.

where α is the step adjustment parameter and is the weight vector of the M calibration samples. The predictive results can be improved to some extent by adjusting the maximum error of the recovery rate. In addition, the weight vector of the PLS method can also be constructed by the variance of the recovery rate. If represents the variance of the recovery rate in E_c, then the Gauss weight is as follows.

where β is the step adjustment parameter and is the variance weight vector of the recovery rate. Then the fluorescence intensity matrix and the concentration matrix can be updated by the weight vector.

where is a diagonal matrix and the diagonal elements are . And the new calibration sets X_cr and Y_cr are produced. Similarly, if the calibration sets are modified by the weight vector the new calibration sets X_cv and Y_cv can also be obtained. Next the EWPLS method is carried out as follows as an example.

1) Set initial vector u, calculate the weight w of X_cr, , calculate the score matrix t of X_cr,.

2) calculate the weight c of Y_cr, , calculate the score matrix u of Y_cr,.

3) If the convergence, has not reached, return to Steps 1 and 2, and otherwise continue with Step 4;

4) Remove the calculated components from X_cr, Y_cr, then .

5) Return to Step 1, until all the components are extracted.

6) According to Equation (4), calculate the regression factor, then calculate.

7) If the convergence has not reached, calculate W_r, X_cr, Y_cr, return to Step 1, Otherwise continue with Step 8.

8) calculate.

As shown in Figure 2, the experimental platform for absorption spectra acquisition of NO_X from the vehicle exhaust has been built. It’s composed of a gas distribution unit and a measuring unit. The gas distribution unit can mix two gases of different concentrations with a precision of 1%. And the measuring unit consists of a flashing xenon lamp, an ultraviolet spectrometer, a sample gas chamber and a data processing terminal and so on, which can measure the ultraviolet spectrum data of the sample gas effectively at a certain temperature and pressure.

In order to avoid multicollinearity, the orthogonal principle is followed in the sample concentration design for NO & NO₂. Samples of different concentration are designed as shown in Table 1.

Then a series of designed NO or NO₂ UV absorption spectra (200 nm - 440 nm) of different concentrations have been obtained with the platform, also with their mixture absorption spectra. The NO absorption spectra of different concentrations are shown in Figure 3(a), meanwhile the NO₂ absorption spectra of different concentrations are shown in Figure 3(b) and Figure 3(c) also with a series of their mixture absorption spectra in Figure 3(d).

Here, the concentration inversion of NO and NO₂ components from the vehicle exhaust with ultraviolet absorption spectrum based on WPLS is actually a partial least squares regression problem with three independent variables and two dependent variables, as shown in Table 2. The regression equation of Y₁ - Y₂ for X₁ - X₃ should be established to determine the relationship between the concentration of the two components gas, Y and their absorbance, X.

According to Steps 1-8 in 2.2, a data processing program has been compiled with MATLAB, and then the regression models have been established with the obtained spectral sample data from which one spectral data has been selected and taken out, then the concentration have been inversed by put the selected spectral data into the model. That’s the same with every concentration. Finally the experimental results are shown in Figures 4-9.

As seen from the above experimental results, using either the spectra of the individual components of NO or NO2 or mixed spectra of the two components or even all the individual and mixed spectra for the regression modeling and concentration inversion based on WPLS, the experimental results are all excellent.

The approximation between the inversion results and real sample concentration are all above 99.4%, of which the highest can reach 99.97%. So it can be concluded that with WPLS algorithm the components’ characteristics of the mixed spectral in which there’re overlapped absorption with NO & NO₂ can be separated, and then each concentration of the samples can be inversed accurately.

In this experiment, It’s not considered that the modeling optimization [6], the inversion band and spectral data denoising [8]. The spectral data used here in WPLS are the direct output of the spectrometer, of which the band is the whole detecting range of the spectrometer (its wavelength range: 198 - 438 nm, a total of 3648 sampling points). Meanwhile in the whole mixed spectra, there’re not overlapped absorption at all sampling points, and there are also many places with zero absorbance, which will affect the accuracy of WPLS modeling.