CN106483077B - Method for measuring element content of combustion body based on principal component and multiple linear regression - Google Patents
Method for measuring element content of combustion body based on principal component and multiple linear regression Download PDFInfo
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Abstract
The invention provides a method for measuring the element content of a combustion body based on principal components and multiple linear regression, which comprises the following steps of firstly, carrying out data screening on the luminous intensity value of the emission spectrum of the combustion body, and carrying out denoising treatment on the screened luminous intensity value; then, obtaining a principal component score of the light intensity value according to the light intensity value after the denoising processing; and finally, calculating the content of the element to be detected in the combustion body according to the principal component score of the light intensity value. The method for measuring the content of the elements by analyzing the luminous intensity value of the emission spectrum of the combustion body is simple and efficient, and the measurement precision is high.
Description
Technical Field
The invention belongs to the technical field of optics, and particularly relates to a method for measuring the element content of a combustion body based on principal component analysis and a multiple linear regression method.
Background
For a combustion body, in order to determine which elements and the content of each element contained in the combustion body, a complex chemical measurement analysis method is needed, and a common method is to perform element analysis on residues after substances are combusted, so that the elements and the content of the elements in the original combustion body can be analyzed.
Disclosure of Invention
The invention aims to provide a method for measuring the element content of a combustion body based on principal component analysis and multiple linear regression, which measures the element content by analyzing the luminous intensity value of the emission spectrum of the combustion body, and has the advantages of simplicity, high efficiency and high measurement precision.
In order to solve the technical problems, the invention provides a method for measuring the element content of a combustion body based on principal components and multiple linear regression, which comprises the steps of firstly, screening the light intensity value of the emission spectrum of the combustion body, and denoising the screened light intensity value; then, obtaining a principal component score of the light intensity value according to the light intensity value after the denoising processing; finally, 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 a formula,
Y＝b0+b1S1+...+bpSp
wherein Y represents the content of the element Y to be measured in the combustion body, and S1，S2，….SpRepresents the score of the principal component, b0Is a constant term, b1…bpAs a regression coefficient, b0And b1…bpAre all known quantities.
Compared with the prior art, the method has the obvious advantages that (1) an analysis and measurement method in the chemical field is not needed, and the method is easy to realize in practice; (2) the measurement precision is higher by adopting a principal component analysis method; (3) the method is more innovative by adopting a mathematical modeling method; (4) theoretically, the method can be used for measuring most elements in nature, and has wide application.
Detailed Description
It is easily understood that, according to the technical solution of the present invention, those skilled in the art can imagine various embodiments of the method for measuring the content of elements in a combustion body based on principal component analysis and multiple linear regression of the present invention without changing the essential spirit of the present invention. Therefore, the following detailed description is merely illustrative of the technical solutions of the present invention, and should not be construed as being all of the present invention or limiting or restricting the technical solutions of the present invention.
The invention scores a variable S for a principal component1，S2，….SpAnd performing multiple linear regression analysis on the dependent variable Y (namely the content of the element Y to be detected in the combustion body), and establishing a multiple linear regression model equation between the dependent variable Y and the dependent variable Y, wherein the multiple linear regression model equation is shown as a formula (1):
Y＝b0+b1S1+...+bpSp(1)
in the formula (1), b0Is a constant term; b1…bpAs a regression coefficient, b1Is S1,S2… at fixed time, S1The effect on Y per added unit, i.e. S1Partial regression coefficients for Y; same principle b2Is S1,S2… at fixed time, S2The effect on Y per added unit, i.e. S2Partial regression coefficients for Y, etc. The regression coefficient can be obtained through experiments, and only the principal component score S needs to be obtained according to the light intensity value when the content of a certain element to be measured in the combustion body is carried out each time1、S2…SpAnd (3) substituting the principal component score into the formula (1) to realize the measurement of the content of a certain element Y in the combustion body.
The regression coefficient is obtained in advance through a plurality of tests, and can be used only by looking up a table when used every time, wherein the regression coefficient obtaining process is as follows:
in multiple tests, the measured light intensity value of the emission spectrum of the combustion body containing the element Y is obtained, the content value (namely prior information) of the element Y to be measured in the combustion body is very difficultly measured in other modes before the test, and the light intensity values corresponding to L light wavelengths in the emission spectrum of the combustion body are set as X light intensity values in each test1，X2，…XLAnd X1～XLThe wavelength difference of the light wavelengths of (a) and (b) is a constant.
Firstly, according to the related spectrum theory knowledge, screening operation is carried out on the light intensity value of the emission spectrum, and the light intensity value of the screened emission spectrum is obtained.
According to the research on the related spectrum theory knowledge, whether the screening of the spectrum data is correct or not can be known to have certain influence on the subsequent processing and analysis of the data. Assuming that element Y is to be measured, it is known from the prior knowledge that the wavelength of the combustion emission spectrum of element Y should be n wavelength values of lamda1, lamda2 … and lamdan. Since the wavelength value of the emission spectrum may shift a certain amount during the spectral line measurement, the light intensity value X needs to be measured first1，X2，…XLThe corresponding L light wavelength values are respectively compared with n wavelength values lamda1 and lamda2 … lamdan of the known standard, if the difference value of the two is within a preset error delta L range, the spectral line corresponding to the wavelength is considered to be the emission spectral line of the element Y and is reserved, otherwise, the spectral line corresponding to the wavelength is considered not to be the emission spectral line of the element Y and is removed; setting the light intensity value X corresponding to m light wavelengths reserved after the screening1，X2，…Xm，m<L。
Secondly, according to the relevant denoising theory, the light intensity value X of the screened emission spectrum is subjected to1，X2，…XmAnd (5) carrying out denoising pretreatment. Emission spectrum data is often influenced by a plurality of factors when being collected, so that the spectrum data not only contains useful information, but also contains irrelevant information such as background and instrument noise. When spectral data is analyzed and processed, these noises adversely affect the analysis result and reduce the spectral analysis accuracy, so that it is necessary to perform denoising preprocessing in advance. As a preferable scheme, the denoising preprocessing of the invention adopts a signal Smoothing method to remove high-frequency noise and improve the signal-to-noise ratio of the spectral signal, and a Moving Average Smoothing (MAS) method is specifically adopted in the signal Smoothing. Setting the light intensity values corresponding to the m optical wavelengths after the denoising pretreatment as
Thirdly, according to the principal component analysis method, the light intensity value after the denoising pretreatment is the value
Performing principal component analysis to obtain final principal component score Si(i＝1,2,...p)。
The light intensity value data volume of the emission spectrum is large, the emission spectrum contains abundant material components and structural information, the spectral data are analyzed through a principal component analysis method, a few principal component variables are used for replacing original variables, the principal component variables are linear combinations of the original variables, and the information of the original variables can be represented to the maximum extent. The principal component analysis method makes the principal components mutually independent through orthogonal transformation, thereby overcoming the problem of data collinearity. The specific analysis process of the principal component analysis is as follows:
1.1 is provided for
The total number of the light intensity values is m, the light intensity value corresponding to each wavelength has n data (namely the result obtained by n times of tests), and n times of tests are carried outThe obtained light intensity values are tested to form an independent variable matrix X shown as a formula (2), wherein XnmRepresenting the light intensity value corresponding to the mth wavelength in the nth test.
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), rij(i, j ═ 1,2.. times, m) are light intensity values
And
the calculation formula of the correlation coefficient (c) is shown in formula (4):
wherein,
represents the variable xiAverage value of (1), xiRepresents the ith column of data in the matrix X; k denotes the number of rows of the matrix X, k being 1,2.
1.3 solving the eigenvalue λ needed to use the eigenvalue | λ I-R | ═ 0 according to the correlation coefficient matrix R1,λ2...λmWhere I is an identity matrix, solving for the eigenvalues λ1,λ2...λmThe commonly used method is the Jacobi (Jacobi) method, and the solved characteristic value lambda is used1,λ2...λmArranging λ in order from big to small1≥λ2≥...≥λm≥0
1.4 dividing the characteristic value lambda1,λ2...λmSubstituting the characteristic equation | λ I-R | ═ 0 to respectively obtain characteristic values λiCorresponding feature vector eiFeature vector eiIs a matrix of m rows and 1 column, where i takes an integer in the range of 1 to m.
1.5 according to the characteristic value lambda1,λ2...λmCalculating 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):
the cumulative contribution rate represents the degree of original data information contained in the generated new variable (i.e. principal component), and in order to represent the original variable to a greater extent, a characteristic value λ with a principal component cumulative contribution rate of 95% or more is generally selected1,λ2,...λpCorresponding main component, p is less than or equal to m.
1.6 according to the characteristic value lambda1,λ2...λmAnd a feature vector eiCalculating the loads I of the main componentsiThe calculation formula is shown in formula (7):
principal component load IiA matrix of m rows and 1 column;
then, based on the obtained loads I of the respective principal componentsiCalculating to obtain score S of each principal componentiAs shown in the formula (8),
wherein S isiIs a matrix of n rows and 1 column, and represents a principal component variable SiScore number under n testsThe value is obtained.
From the above, principal component analysis is a method with loss of dimension reduction, when the principal component is selected too little, the information loss will be large, and when the number of the principal component is selected too much, the computation load of the computer will be greatly increased, so that the number of the principal component variables should be reasonably selected according to the needs in the actual calculation process.
The fourth step, the principal component score S obtained from the above principal component analysisi(i ═ 1,2.. p) and the known content value Y of the element Y in the combustion bodyi(i ═ 1,2.. n), fitting was performed by the least square method, and the regression coefficient b was determined1…bp。
Second, principal component score S1、S2…SpIs determined
Let z be the light intensity value of the emission spectrum for the combustion body for which the elemental Y content currently needs to be determined1，z2，…zLPerforming corresponding processing according to the screening and preprocessing method to obtain the processed light intensity value
Will be provided with
Forming a matrix Z of 1 row and m columns, and obtaining the score S of the principal component according to equation 91、S2…Sp。
Wherein i is 1,2.
The invention can be further illustrated by the following experiments.
The emission spectrum wavelength range of the combustion body in the experiment is 245 nm-1044 nm, 3648 light wavelength values are totally obtained, variables w1 and w2 … w3648 respectively represent light intensity values corresponding to certain wavelength values, the light wavelength difference between w 1-w 3648 is a constant, the light intensity value is a multiple of certain fixed reference intensity, each variable has 19 test results, and the determination of the content of carbon elements in the combustion body is completed based on the results.
Firstly, screening the light intensity value data of the emission spectrum on the original light intensity value of 19 tests (not the combustion body to be tested at this time) to obtain the screened light intensity value, wherein the light intensity value can be obtained from the inquired carbon element emission spectrum data: the wavelength range of the carbon element emission spectrum is 175.136 nm-453.18 nm (the specific wavelength value is shown in the appendix), the wavelength value of the emission spectrum may shift to some extent when the spectrum line is measured according to related spectrum theory knowledge, so the spectrum line with the interval less than or equal to 0.2nm near the wavelength range of the carbon element emission spectrum is selected as the emission spectrum line of the carbon element, the screened spectrum data comprises 954 wavelengths, and w1 and w2 … w954 represent the light intensity values of the corresponding wavelengths respectively.
Secondly, preprocessing the light intensity values of the emission spectra screened in the first step to obtain preprocessed light intensity values, preprocessing the spectrum data by adopting a moving average smoothing method, wherein the preprocessed 954 light wavelength light intensity values correspond to each other
Thirdly, performing Principal Component Analysis (PCA) on the light intensity value preprocessed in the second step to obtain a principal component score value, solving the principal component score value in Matlab (a mathematical analysis and calculation software) by using a principal component analysis method,
the pretreated variation of the light intensity value has 19 test results, and the 19 test results are combined into a matrix W (data of 19 rows 954 columns), wherein n is 19, m is 954, and W isnmRepresenting the light intensity value corresponding to the mth wavelength in the nth test.
(1) The data matrix W is normalized by column (data in each column divided by the sum of data in each column) to obtain a normalized data matrix.
(2) For the normalized data matrix W in (1), the correlation coefficient matrix R of the matrix W is obtained according to the following formula,
in the formula, rij(i, j ═ 1,2.. times, m) as the original variable
And
the calculation formula of the correlation coefficient of (2) is as follows,
represents the variable wiAverage value of (a).
(3) Solving the characteristic value lambda of the correlation coefficient matrix R in the step (2)iAnd corresponding feature vector ei(i 1,2.. 954), calculating the contribution rate and the accumulated contribution rate at the same time, and taking a characteristic value lambda when the accumulated contribution rate reaches 96%1,λ2,…,λpThe first, second, …, and p (p.ltoreq.954) th principal components are corresponded to, and the contribution ratios corresponding to the principal component variables are determined to obtain 16 principal components in total as shown in table 1.
TABLE 1 contribution rate of each principal component
| Contribution ratio 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 variable 5 | 0.006384 |
| Principal component variable 6 | 0.006227 |
| Principal component variable 7 | 0.006124 |
| Principal component variable 8 | 0.005849 |
| Principal component variable 9 | 0.005556 |
| Principal component variable 10 | 0.005476 |
| Change of principal componentQuantity 11 | 0.005339 |
| Principal component variable 12 | 0.004911 |
| Principal component variable 13 | 0.004832 |
| Principal component variable 14 | 0.004634 |
| Principal component variable 15 | 0.004504 |
| Principal component variable 16 | 0.004327 |
(4) Calculating principal component variable load according to the characteristic value and the characteristic vector obtained in the step (3), wherein the calculation formula is as follows:
And calculating the score of each principal component according to the load of each principal component:
wherein S isiIs a matrix of 19 rows and 1 columns, and shows a principal component variable SiScores under 19 sample trials.
The above formula was used to obtain scores for each principal component variable (16 in total), each of which had the results of 19 trials.
The fourth step is to calculate the 16 principal component score variables S obtained in the third step1，S2，…S16And prior information (carbon element content measured before the experiment) Y in the experiment is subjected to multiple linear regression fitting, 16 principal component score variables are used as independent variables, the carbon element content is used as dependent variables, and the relation between the carbon content Y and the 16 principal component variables is obtained:
Y＝b0+b1S1+...+b16S16
the following table 2 shows the coefficients obtained by calling the regression function (least square fitting method) in Matlab:
TABLE 2 fitting coefficients of multiple linear regression equations
| Multiple linear regression fitting coefficient | |
| 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 carbon content was solved using the fitted relational model equation, and the true value of the carbon content of the combustion body measured before the experiment is shown in table 3:
TABLE 3 true carbon content values
| Number of tests 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 found using the fitted equation is shown in table 4:
TABLE 4 carbon content measurements
| Number of tests 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 |
According to analysis, the method disclosed by the invention can be used for determining the element content by establishing the principal component analysis model and the multiple linear regression model and analyzing the light intensity value of the emission spectrum of the combustion body, so that the error between the real value and the predicted value of the carbon content is less, the measurement precision is higher, 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+...+bpSp(1)
wherein Y represents the content of the element to be measured in the combustion body, and S1，S2，….SpRepresents the score of the principal component, b0Is a constant term, b1…bpAs a regression coefficient, b0And b1…bpAre 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 SiP, 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 XnmRepresents 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), rij，i,j＝1,2,...,m，rijAs a light intensity value
And
the calculation formula of the correlation coefficient (c) is shown in formula (4):
wherein,
represents the variable xiAverage value of (1), xiRepresents 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...λm；
1.4 dividing the characteristic value lambda1,λ2...λmSubstituting the characteristic equation | λ I-R | ═ 0 to respectively obtain characteristic values λiCorresponding feature vector eiFeature vector eiIs 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...λmCalculating 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,...λpTaking 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...λmAnd a feature vector eiCalculating the loads I of the main componentsiThe calculation formula is shown in formula (7):
principal component load IiA matrix of m rows and 1 column;
then, based on the obtained loads I of the respective principal componentsiCalculating to obtain score S of each principal componentiAs shown in the formula (8),
wherein S isiIs a matrix of n rows and 1 column, and represents a principal component variable SiScore values under n trials;
thirdly, according to the score S of the principal componenti(i ═ 1,2.. p) and the known content value Y of the element Y in the combustion bodyi(i ═ 1,2.. n), fitting was performed by the least square method, and the regression coefficient b was determined1…bp；
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):
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