CN106483077B - Method for measuring element content of combustion body based on principal component and multiple linear regression - Google Patents

Abstract

The invention proposes a method for determining element content of a combustion body based on principal components and multiple linear regression. First, data screening is performed on the light intensity value of the emission spectrum of the combustion body, and the filtered light intensity value is denoised; then, according to The main component score of the light intensity value is obtained from the denoised light intensity value; finally, the content of the element to be tested in the combustion body is calculated according to the main component score of the light intensity value. In the invention, the element content is determined by analyzing the light intensity value of the emission spectrum of the combustion body, the determination method is simple and efficient, and the measurement accuracy is high.

Description

Determination method of element content in combustion body based on principal components and multiple linear regression

technical field

The invention belongs to the technical field of optics, and in particular relates to a method for measuring element content of a combustion body based on principal component analysis and multiple linear regression method.

Background technique

For the combustion body, in order to determine which elements and the content of each element contained in it, complex chemical measurement and analysis methods are required. Elements and their contents are difficult to measure by this method. At the same time, for combustion bodies containing inorganic elements, it is difficult to perform elemental analysis on the residues after combustion. In addition, the cost of measuring elements and their contents by methods in the chemical field is relatively high. high, and the measurement cost is high.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for measuring the element content of a combustion body based on principal component analysis and multiple linear regression. The element content is measured by analyzing the light intensity value of the emission spectrum of the combustion body.

In order to solve the above-mentioned technical problems, the present invention provides a method for measuring element content of a combustion body based on principal components and multiple linear regression. First, data screening is performed on the light intensity value of the emission spectrum of the combustion body, and the filtered light intensity value is removed. Noise processing; then, the principal component score of the light intensity value is obtained according to the denoised light intensity value; finally, the content of the element to be tested in the combustion body is calculated according to the principal component score of the light intensity value, and the calculation method is shown in the formula ,

Y=b ₀ +b ₁ S ₁ +...+b _p S _p

Among them, Y represents the content of the element Y to be measured in the combustion body, S ₁ , S ₂ , … _.Sp represents the principal component score, b ₀ is a constant term, b ₁ … b _p is the regression coefficient, b ₀ and b ₁ … _bp are all known quantities.

Compared with the prior art, the present invention has significant advantages in that (1) it does not need to adopt the analysis and measurement method in the field of chemistry, and it is easy to realize in practice; (2) the principal component analysis method is adopted, and the measurement accuracy is relatively high; (3) ) using the mathematical modeling method, the method is more innovative; (4) theoretically can be used to measure most of the elements in nature, the method is widely used.

Detailed ways

It is easy to understand that, according to the technical solution of the present invention, without changing the essential spirit of the present invention, those skilled in the art can imagine the multiplication of the method for determining the element content of the combustion body based on principal component analysis and multiple linear regression of the present invention. an implementation. Therefore, the following specific embodiments are only exemplary descriptions of the technical solutions of the present invention, and should not be regarded as the whole of the present invention or as limitations or restrictions on the technical solutions of the present invention.

The present invention performs multiple linear regression analysis on the principal component _score variables S ₁ , S ₂ , . As shown in formula (1):

Y=b ₀ +b ₁ S ₁ +...+b _p S _p (1)

In formula (1), b ₀ is a constant term; b ₁ ... b _p is the regression coefficient, and b ₁ is S ₁ , S ₂ ... When fixed, the effect of each additional unit of S ₁ on Y is the effect of S ₁ on Y. Partial regression coefficient; Similarly, when b ₂ is S ₁ , S ₂ . . . is fixed, the effect of each additional unit of S ₂ on Y is the partial regression coefficient of S ₂ on Y and so on. The regression coefficient can be obtained through experiments. When the content of a certain element to be measured in the combustion body is carried out each time, it is only necessary to _obtain the principal component scores S ₁ , S ₂ . . . It can realize the determination of the content of a certain element Y in the combustion body.

1. The regression coefficient is obtained through multiple experiments in advance. Each time it is used, it can be used only by looking up the table. The process of obtaining the regression coefficient is as follows:

In many experiments, the light intensity value of the emission spectrum of the combustion body containing element Y was measured, and the content value of the element Y to be measured in the combustion body was measured very difficultly by other means before the experiment (that is, the prior information). ), suppose that in each test, the light intensity values corresponding to _L light wavelengths in the emission spectrum of the combustion body are X ₁ , X ₂ , . . . XL , and the wavelength difference of the light wavelengths from X ₁ to _XL is a constant.

The first step is to perform a screening operation on the light intensity value of the emission spectrum according to the relevant spectral theoretical knowledge, and obtain the light intensity value of the filtered emission spectrum.

According to the research on relevant spectral theoretical knowledge, it can be known that the correctness of spectral data screening has a certain influence on the subsequent processing and analysis of the data. Assuming that the element Y needs to be measured, according to the existing knowledge, the wavelength of the combustion emission spectrum of element Y should be lamda1, lamda2..., lamdan, a total of n wavelength values. Since the wavelength value of the emission spectrum may shift to a certain extent during the spectral line measurement, it is necessary to firstly compare the _L light wavelength values corresponding to the light intensity values X ₁ , X ₂ , ... XL with the known standard n wavelengths respectively. Compare the values of lamda1, lamda2… The spectral line corresponding to this wavelength is not the emission spectral line of the element Y, so it is rejected; it is assumed that after the aforementioned _screening , the light intensity values X ₁ , X ₂ , . . .

In the second step, according to the relevant denoising theory, denoising preprocessing is performed on the light intensity values X ₁ , X ₂ , . . . X _m of the filtered emission spectrum. The emission spectrum data is often affected by many factors during the collection, so that the spectrum data contains not only useful information, but also irrelevant information such as background and instrument noise. When analyzing and processing spectral data, these noises will adversely affect the analysis results and reduce the accuracy of spectral analysis, so it is necessary to perform denoising preprocessing first. As a preferred solution, the denoising preprocessing of the present invention adopts a signal smoothing method to remove high-frequency noise and improve the signal-to-noise ratio of the spectral signal. The moving average smoothing (MAS) method is specifically used for signal smoothing. Set the light intensity corresponding to m light wavelengths after denoising preprocessing as

进行主成分分析，获得最终的主成分得分S_i(i＝1,2,...p)。In the third step, according to the principal component analysis method, the light intensity value after denoising preprocessing is

Principal component analysis is performed to obtain the final principal component score S _i (i=1,2,...p).

The light intensity value of the emission spectrum has a large amount of data, which contains rich material composition and structural information. The spectral data is analyzed by the principal component analysis method, and a few principal component variables are used to replace the original variables. The principal component variables are the linear combination of the original variables. , which can best represent the information of the original variable. The principal component analysis method makes the principal components uncorrelated with each other through orthogonal transformation, so as to overcome the problem of data collinearity. The specific analysis process of principal component analysis is as follows:

共m个光强度值，每一个波长对应的光强度值有n个数据(即n次试验获得的结果)，将n次试验获得光强度值组成自变量矩阵X如式(2)所示，其中x_nm代表第n次试验中第m个波长对应的光强度值。1.1 Assuming that for

There are m light intensity values in total, and the light intensity value corresponding to each wavelength has n data (that is, the results obtained from n tests), and the light intensity values obtained from n tests are composed of an independent variable matrix X, as shown in formula (2), where x _nm represents the light intensity value corresponding to the mth wavelength in the nth experiment.

n represents the number of rows of matrix X, m represents the number of columns of matrix X

1.2 Calculate the correlation coefficient matrix R of the independent variable matrix X, and the correlation coefficient matrix R is shown in formula (3):

和

的相关系数，其计算公式如式(4)所示：In formula (3), r _ij (i,j=1,2,...,m) is the light intensity value

and

The correlation coefficient of , and its calculation formula is shown in formula (4):

表示变量x_i的平均值，x_i表示矩阵X中第i列数据；k表示矩阵X的行数，k＝1,2,...,n。in,

Represents the average value of variable x _i , x _i represents the data in the i-th column of matrix X; k represents the number of rows in matrix X, k=1,2,...,n.

1.3 According to the correlation coefficient matrix R, the eigenvalues λ ₁ , λ ₂ ... λ _m required to be used for the characteristic equation |λI-R|=0 are solved, where I is a unit matrix, and the eigenvalues λ ₁ , λ ₂ are solved. ..λ _m The commonly used method is the Jacobi method, and the solved eigenvalues λ ₁ , λ ₂ ... λ _m are arranged in descending order λ ₁ ≥λ ₂ ≥...≥ λ _m ≥ 0

1.4 Substitute the eigenvalues λ ₁ , λ ₂ ... λ _m into the characteristic equation |λI-R|=0 to obtain the eigenvectors e _i corresponding to the eigenvalues λ _i respectively, and the eigenvectors e _i are a matrix with m rows and 1 column, where i is an integer in the range of 1 to m.

1.5 Calculate the contribution rate and cumulative contribution rate of the principal component according to the eigenvalues λ ₁ , λ ₂ ... λ _m , the contribution rate calculation formula (5) is shown, and the cumulative contribution rate is shown in the formula (6):

The cumulative contribution rate indicates the degree of original data information contained in the new variable (ie principal component) generated. In order to represent the original variable to a greater extent, the eigenvalues λ ₁ , λ ₂ with the cumulative contribution rate of the principal component ≥95% are generally selected. ,...λ _p corresponds to the principal component, p≤m.

1.6 Calculate each principal component load I _i according to the eigenvalues λ ₁ , λ ₂ ... λ _m and the eigenvectors e _i , the calculation formula is shown in formula (7):

The principal component load I _i is a matrix with m rows and 1 column;

Then, according to the obtained load I _i of each principal component, the score S _i of each principal component is calculated, as shown in formula (8),

Among them, Si is a matrix with _n rows and 1 column, which represents the score value of the principal component variable Si under _n trials.

It can be seen from the above that principal component analysis is a lossy dimensionality reduction method. When too few principal components are selected, the amount of information loss will be relatively large. In the calculation process, the number of principal component variables should be reasonably selected according to the needs.

The fourth step is based on the principal component score Si ( _i =1,2,...p) obtained by the above principal component analysis and the known content value of element Y in the combustion body Yi ( _i =1,2, ...n), the least squares method is used for fitting, and the regression coefficients b ₁ ...b _p can be obtained.

2. Determination of principal component scores S ₁ , S ₂ ... _Sp

将

组成一个1行m列的矩阵Z，根据公式9可获得主成分得分的值S₁、S₂…S_p。Assuming that the light intensity values of the emission spectrum of the combustion body whose content of element Y needs to be determined at present are _{z 1} , z ₂ , .

Will

A matrix Z with 1 row and m columns is formed, and the values S ₁ , S ₂ . . . S _p of the principal component scores can be obtained according to formula 9.

where i=1,2,...p.

The present invention can be further illustrated by the following experiments.

The wavelength range of the emission spectrum of the combustion body in this experiment is 245nm to 1044nm, with a total of 3648 light wavelength values. The variables w1, w2...w3648 respectively represent the light intensity value corresponding to a certain wavelength value, and the light wavelength difference between w1 and w3648 is a constant. , the light intensity value is a multiple of a certain fixed reference intensity, and each variable has the results of 19 trials. Based on this, the determination of the carbon content in the combustion body is completed.

The first step is to screen the light intensity value data of the emission spectrum for the original light intensity values of the 19 tests (not the combustion body to be tested this time), and obtain the filtered light intensity value, which is emitted by the carbon element inquired. Spectral data can be obtained: the wavelength range of carbon emission spectrum is 175.136nm ~ 453.18nm (see appendix for the specific wavelength value). According to the relevant spectral theory knowledge, the wavelength value of the emission spectrum may shift to a certain extent during the measurement of the spectral line. Therefore, the spectral lines at intervals of less than or equal to 0.2 nm near the wavelength range of the emission spectrum of carbon element are selected as the emission spectral lines of carbon element. The filtered spectral data contains 954 wavelengths, and w1, w2...w954 respectively represent the light intensity of the corresponding wavelength. value.

The second step is to perform data preprocessing on the light intensity values of the emission spectrum screened in the first step to obtain the preprocessed light intensity values, and use the moving average smoothing method to preprocess the spectral data. The light intensity value corresponding to the light wavelength is

为预处理后的光强度值变量，共有19次试验的结果，将19次试验的结果组成矩阵W(19行954列的数据)如下，其中n为19，m为954，w_nm代表第n次试验中第m个波长对应的光强度值。The third step is to perform principal component analysis (PCA) on the light intensity value preprocessed in the second step to obtain the principal component score value, and use the principal component analysis method in Matlab (a mathematical analysis and calculation software) to solve,

For the variable of light intensity value after preprocessing, there are a total of 19 test results, the results of 19 tests are composed of matrix W (data in 19 rows and 954 columns) as follows, where n is 19, m is 954, and w _nm represents the nth The light intensity value corresponding to the mth wavelength in this experiment.

(1) Normalize the data matrix W by column (the data in each column is divided by the sum of the data in each column) to obtain the normalized data matrix.

(2) For the standardized data matrix W in (1), obtain the correlation coefficient matrix R of the matrix W according to the following formula,

和

的相关系数，其计算公式如下所示，

代表变量w_i的平均值。In the formula, r _ij (i,j=1,2,...,m) is the original variable

and

The correlation coefficient of , and its calculation formula is as follows:

represents the mean of the variable _wi .

(3) Solve the eigenvalue λ _i and the corresponding eigenvector e _i (i=1, 2....954) for the correlation coefficient matrix R in (2), calculate the contribution rate and the cumulative contribution rate at the same time, and take the cumulative contribution When the rate reaches 96%, the eigenvalues λ ₁ , λ ₂ ,…,λ _p correspond to the first, second,…, pth (p≤954) principal components, and find the corresponding The contribution rate is shown in Table 1, and a total of 16 principal components were obtained.

Table 1 Contribution rate of each principal component

The contribution rate of each principal component principal component variable 1 0.698016 Principal Component Variable 2 0.215754 Principal Component Variable 3 0.007537 Principal Component Variable 4 0.006690 Principal Component Variables 5 0.006384 Principal Component Variables 6 0.006227 Principal Component Variables 7 0.006124 Principal Component Variables 8 0.005849 Principal Component Variables 9 0.005556 Principal Component Variables 10 0.005476 Principal Component Variables 11 0.005339 Principal Component Variables 12 0.004911 Principal Component Variables 13 0.004832 Principal Component Variables 14 0.004634 Principal Component Variables 15 0.004504 Principal Component Variables 16 0.004327

(4) Calculate the principal component variable load according to the eigenvalues and eigenvectors obtained in (3), and the calculation formula is:

(954行1列的矩阵)

(Matrix with 954 rows and 1 column)

The score of each principal component is calculated according to the load of each principal component:

Among them, _Si is a matrix with 19 rows and 1 column, which represents the score of the principal component variable _Si under 19 sample trials.

Use the above formula to obtain the score of each principal component variable (16 in total), and each principal component variable has the results of 19 trials.

The fourth step is to perform multiple linear regression fitting on the 16 principal component score variables S ₁ , S ₂ , ... S ₁₆ obtained in the third step and the prior information in the experiment (the carbon content measured before the experiment) Y , 16 principal component score variables are used as independent variables, and carbon element content is used as a dependent variable, and the relationship between carbon content Y and 16 principal component variables is obtained:

Y=b ₀ +b ₁ S ₁ +...+b ₁₆ S ₁₆

Call the regress function (least square fitting method) in Matlab to obtain the coefficients in the above formula as shown in Table 2:

Table 2 Multiple linear regression equation fitting coefficients

Multiple Linear Regression Fit Coefficients b₀ 7.000564 b₁ 0.346845 b₂ -0.015874 b₃ -46.693979 b₄ -33.287644 b₅ 55.128680 b₆ 7.827610 b₇ 2.922307 b₈ 4.568449 b₉ -20.499940 b₁₀ -40.594263 b₁₁ 96.927876 b₁₂ 5.312227 b₁₃ 14.558073 b₁₄ 123.090712 b₁₅ -58.641999 b₁₆ 9.248874

The relationship model equation obtained after fitting is used to solve the carbon content. The real value of the carbon content of the combustion body measured before the experiment is shown in Table 3:

Table 3 Carbon content true value

number of trials n True value of carbon content 1 21.000000 2 20.000000 3 18.000000 4 20.000000 5 18.480000 6 25.040000 7 29.350000 8 19.000000 9 24.000000 10 19.600000 11 25.000000 12 25.310000 13 30.000000 14 27.750000 15 15.170000 16 20.000000 17 17.920000 18 20.440000 19 14.280000

The carbon content calculated using the fitted equation is shown in Table 4:

Table 4 Carbon Content Measurements

number of trials n Carbon content measurement 1 22.214923 2 20.845078 3 18.373913 4 20.355424 5 18.524505 6 23.565235 7 28.691861 8 18.742529 9 22.509877 10 19.190907 11 23.602100 12 26.447184 13 31.459607 14 27.598998 15 15.613033 16 21.484846 17 18.472477 18 21.342110 19 15.508944

It can be seen from the analysis that the present invention determines the element content by establishing a principal component analysis model and a multiple linear regression model, and analyzing the light intensity value of the emission spectrum of the combustion body. The precision is high, and the method is simple and efficient.

Claims (1)

1. A method for measuring the element content of a combustion body based on principal components and multiple linear regression is characterized in that,

carrying out data screening on the light intensity value of the emission spectrum of the combustion body, and carrying out denoising treatment on the screened light intensity value;

obtaining a principal component score of the light intensity value according to the de-noised light intensity value;

the content of the element to be measured in the combustion body is obtained by calculation according to the principal component score of the light intensity value, the calculation mode is shown as formula (1),

Y＝b₀+b₁S₁+...+b_pS_p(1)

wherein Y represents the content of the element to be measured in the combustion body, and S₁，S₂，….S_pRepresents the score of the principal component, b₀Is a constant term, b₁…b_pAs a regression coefficient, b₀And b₁…b_pAre all known amounts;

the regression coefficient acquisition process is as follows:

firstly, screening the light intensity value of the emission spectrum of the combustion body, and denoising the screened light intensity value to obtain the light intensity values corresponding to the m light wavelengths subjected to screening and denoising treatment

L is the number of light wavelengths in the emission spectrum of the combustion body;

secondly, according to the principal component analysis method, the light intensity value after denoising pretreatment is carried out

Performing principal component analysis to obtain principal component score S_iP, p is the number of principal components;

1.1 is provided for

M light intensity values are totally obtained, the light intensity values obtained by n times of tests form an independent variable matrix X shown as a formula (2), wherein X_nmRepresents the light intensity value corresponding to the mth wavelength in the nth test,

where n denotes the number of rows of the matrix X and m denotes the number of columns of the matrix X

1.2, calculating a correlation coefficient matrix R of the independent variable matrix X, wherein the correlation coefficient matrix R is shown as a formula (3):

in the formula (3), r_ij，i,j＝1,2,...,m，r_ijAs a light intensity value

And

the calculation formula of the correlation coefficient (c) is shown in formula (4):

wherein,

represents the variable x_iAverage value of (1), x_iRepresents the ith column of data in the matrix X; k represents the number of rows of matrix X, k being 1,2.., n;

1.3 solving the eigenvalue lambda according to the correlation coefficient matrix R₁,λ₂...λ_m；

1.4 dividing the characteristic value lambda₁,λ₂...λ_mSubstituting the characteristic equation | λ I-R | ═ 0 to respectively obtain characteristic values λ_iCorresponding feature vector e_iFeature vector e_iIs a matrix with m rows and 1 column, wherein i is an integer ranging from 1 to m;

1.5 according to the characteristic value lambda₁,λ₂...λ_mCalculating the contribution rate and the cumulative contribution rate of the principal component, wherein the contribution rate is calculated by formula (5), and the cumulative contribution rate is expressed by formula (6):

selecting p characteristic values lambda of which the accumulated contribution rate of the main components is more than or equal to 95 percent₁,λ₂,...λ_pTaking the corresponding principal component as the final principal component score, wherein p is less than or equal to m;

1.6 according to the characteristic value lambda₁,λ₂...λ_mAnd a feature vector e_iCalculating the loads I of the main components_iThe calculation formula is shown in formula (7):

principal component load I_iA matrix of m rows and 1 column;

then, based on the obtained loads I of the respective principal components_iCalculating to obtain score S of each principal component_iAs shown in the formula (8),

wherein S is_iIs a matrix of n rows and 1 column, and represents a principal component variable S_iScore values under n trials;

thirdly, according to the score S of the principal component_i(i ═ 1,2.. p) and the known content value Y of the element Y in the combustion body_i(i ═ 1,2.. n), fitting was performed by the least square method, and the regression coefficient b was determined₁…b_p；

Screening and denoising light intensity values of emission spectra of elements to be detected in a combustion body to form a matrix Z with 1 row and m columns, and obtaining principal component scores according to the following formula (9):