CN112834448A - Spectral decomposition method for water pollutant analysis - Google Patents

Spectral decomposition method for water pollutant analysis Download PDF

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CN112834448A
CN112834448A CN202011632128.1A CN202011632128A CN112834448A CN 112834448 A CN112834448 A CN 112834448A CN 202011632128 A CN202011632128 A CN 202011632128A CN 112834448 A CN112834448 A CN 112834448A
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张慧
沈杰
黄付岭
陈思云
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Hangzhou Luopanxing Technology Co ltd
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Abstract

A spectrum decomposition method for water pollutant analysis, which takes a spectrum to be decomposed as a total probability density function to replace specific observation data; taking a double-Gaussian model bi-Gaussian as a subcomponent model to adapt to the asymmetry of a spectrum peak; and iteratively updating the sub-model parameters to finally obtain the mixed probability distribution consisting of a plurality of sub-components, thereby realizing the accurate decomposition and description of spectral lines. The invention normalizes the spectrum data and directly uses the normalized spectrum data as the probability density distribution of the observation data, avoids the error caused by introducing random numbers in the common method and has good function approximation capability. The double-Gaussian function bi-Gaussian with adjustable left-right width is used as a subcomponent model, so that the method is more flexible, higher in precision and capable of being widely applied to spectrums with serious overlapping.

Description

Spectral decomposition method for water pollutant analysis
Technical Field
The invention relates to a spectral decomposition method for water pollutant analysis.
Background
In recent years, with the continuous development of industry, the pollution of water resources is increased. In the water pollution prevention and control work, the method is vital to timely and quickly locate the pollutant types and take corresponding treatment measures. The ultraviolet-visible absorption spectrum technology has the advantages of simplicity, in-situ, rapidness, no secondary pollution, reagent-free, accuracy, stability and the like, and is widely researched and applied in the field of water environment monitoring.
The principle of the ultraviolet-visible absorption spectroscopy technology is to perform qualitative and quantitative analysis on the selective absorption of photons with different wavelengths by substance molecules. Since different species of substances have different functional groups and molecular spatial structures and have different selective absorption of photons, information of different species is recorded in the absorption spectrum and can be used for reflecting the species and related characteristics of the substances. However, in the absorption spectrum of most substances, there is often more than one absorption peak, and there is a difference in overlap between the multiple peaks, and in fact, these peaks represent information of different groups in the molecule, so the situation is more complicated in the water quality analysis process. Therefore, it is necessary to decompose each peak component in the spectrum and study the corresponding functional group, so as to obtain more detailed, real and accurate spectral characteristics in water quality analysis, and to make the subsequent model establishment more specific.
Most current analysis methods ignore the situation of multimodal overlap and directly extract some sensitive band data with strong correlation for analysis. Although some useful information can be obtained in the spectrum, the methods are poor in pertinence, and the model is easy to interfere when the material composition in the water body changes. In part of methods, a Gaussian least square fitting, a full spectrum filtering technology, a principal component analysis and the like are adopted to decompose spectral peaks or extract characteristic peaks, but the decomposition effect is poor when the number of absorption peaks is large and the overlapping is serious.
Therefore, a high-precision spectral decomposition method needs to be found for water pollutant analysis. As a pretreatment process, so as to realize more detailed qualitative analysis and more targeted quantitative analysis.
Disclosure of Invention
In order to overcome the defects of the prior art and realize more accurate spectral decomposition, the invention provides a spectral decomposition method for water pollutant analysis, which adopts a probability statistics idea to take a spectrum to be decomposed as a total probability density function to replace specific observation data; taking a double-Gaussian model bi-Gaussian as a subcomponent model to adapt to the asymmetry of a spectrum peak; and iteratively updating the sub-model parameters to finally obtain the mixed probability distribution consisting of a plurality of sub-components, thereby realizing the accurate decomposition and description of spectral lines. The spectral decomposition method has good function approximation capability and is beneficial to further analysis and detection of water environment pollutants.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a method of spectral decomposition for analysis of water contaminants, comprising the steps of:
1) carrying out probabilistic observation data conversion on the spectrum to be decomposed, wherein the process is as follows:
the spectral data are normalized according to a formula (1) so as to be converted into a probability density function of the observed data lambda, which is equivalent to replacing a specific value of the observed data with probability distribution, so that the error problem caused by introducing random numbers in a conventional probability method is avoided;
Figure BDA0002875141450000021
wherein SP (λ) is spectral data, the denominator is the integral of the spectral data with respect to wavelength, the value is equal to the area of the region enclosed by the spectrum and the wavelength axis, since the spectral data is discrete, a summation sign is used, so that f (λ) is taken as a probability density function of the observed data λ, the integral thereof is equal to 1, indicating that the sum of the probabilities of occurrence of all wavelengths λ is 1;
2) the observation data was considered as a mixture of K bi-Gaussian subcomponents as follows:
selecting a bigaussian function bi-Gaussian as the subcomponent model, with parameters including θ ═ μ, Σ1,Σ2A) respectively representing the mean value, the variance of the left Gaussian function and the variance of the right Gaussian function of the left and right Gaussian functions, and the proportion of the subcomponents in the total; suppose the observed data comes from K subcomponents (K2, 3, …, K)max) The initial parameters are respectively (theta)1，θ2，θ3，...，θk，...，θK) Probability density function phi (lambda | theta) of the kth sub-componentk) Comprises the following steps:
Figure BDA0002875141450000031
wherein beta iskIs the normalized coefficient of the sub-component function model, whose value is:
Figure BDA0002875141450000032
when sigma1kSum-sigma2kWhen the values are equal, the values are the same as the common Gaussian function, and the subcomponent models are axisymmetric; on the contrary, the left and right widths are different;
3) iteratively updating each subcomponent parameter, wherein the process is as follows;
31) calculating the probability that the observed data comes from each subcomponent, comprising:
the probability that the observed data λ is from the kth sub-component is expressed as:
Figure BDA0002875141450000033
here, phi (lambda | theta) in step 2)k) Substituting into the formula (4) for calculation;
32) updating each subcomponent parameter, including:
updated k-th sub-component akAnd mukRespectively as follows:
ak new＝∑p(k|λ,θk)f(λ)δλ (5)
Figure BDA0002875141450000034
here f (λ) in step 1) and p (k | λ, θ) in step 31) are comparedk) Calculated by substituting into equations (5) and (6). Introduction of f (lambda) shows that the actual distribution pairs of the spectral data to be decomposed are utilizedAdjusting parameters;
reuse of f (λ), p (k | λ, θ)k) And updated μk newCalculating updated sigma in sequence1kSum sigma2k：
Figure BDA0002875141450000035
Figure BDA0002875141450000036
Then, the updated kth sub-component parameter is obtained as θk new＝(μk new,Σ1k new,Σ2k new,ak new)
33) Judging a convergence condition, comprising:
determining whether the deviation between the updated subcomponent parameter and the original parameter is less than a small positive number epsilon, i.e.
Figure BDA0002875141450000041
If the deviation is larger than or equal to epsilon, the sub-component parameter theta updated in the step 32) is addedk newRepeating the calculation of p (k | λ, θ) in the sequential steps 31) and 32)k) And thetak newUntil the updating deviation is less than epsilon; when the updating deviation is smaller than epsilon, the parameter change is very small after one iteration, the iteration is ended, and whether the current sub-component number K reaches the preset upper limit K or not is continuously judgedmaxIf K is reachedmaxStep 4) is entered, otherwise, K is added by 1, and the step 2) is returned;
4) and comparing the decomposition errors, and selecting the optimal subcomponent number K according to the following process:
the decomposition error for each sub-component number K is calculated according to equation (10), and K with the smallest error is selected as the best component number.
Figure BDA0002875141450000042
5) The parameter theta of iterative convergence under the condition of the optimal sub-component number K obtained in the step 4)k newConverting into spectral data to obtain a spectral decomposition result, wherein the process is as follows:
the iterative convergence parameter theta when the best sub-component number K is givenk newSubstituting the formula (2) to obtain the probability density function phi (lambda | theta) of each componentk) Then, the data is converted back to a spectrum data form by using a formula (11), and each decomposed sub-component is as follows:
spk(λ)＝akφ(λ|θk)·∑SP(λ)δλ (11)。
further, in the step 2), the upper limit K of the number of subcomponents ismaxIs 3. Larger integers may be chosen when the spectrum is more complex.
Still further, in the step 2), the initial parameter of the subcomponent is θ1＝(220,49,100,1/K),θ2＝(272,144,81,1/K),θ3＝(300,169,169,1/K)。
Preferably, in the step 33), the update deviation threshold e is 0.01.
In the invention, the normalized spectrum data is directly used as the probability distribution of the observation data for subsequent iterative operation, so that a specific observation data value is replaced, and the operation of introducing random numbers is avoided. The subcomponent model selects a double-Gaussian function to adapt to asymmetry in the actual spectral shape. In the updating of the sub-component parameters, the probability distribution of the observation data is substituted to participate in the operation, so that the parameters can be adjusted according to the actual characteristics of the data.
The invention has the following beneficial effects:
1. the invention normalizes the spectrum data and directly uses the spectrum data as the probability density distribution of the observation data, and assumes that the spectrum data is composed of a plurality of sub-components, thereby avoiding the error caused by introducing random numbers in the general probability statistical method.
2. The double Gaussian function bi-Gaussian is used as a subcomponent model, and the widths of two sides of the function are variable, so that the method is more flexible and more suitable for the asymmetry of the spectrum shape, and the precision of the spectrum decomposition is ensured.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a graph showing the results of spectral decomposition and the difference between before and after updating of the parameters as a function of the number of iterations, where (a) is the result of spectral decomposition when the phenol absorption spectrum and the sub-component number are 2, (B) is the result of spectral decomposition when the phenol absorption spectrum and the sub-component number are 3, (C) is the change of the difference between before and after updating of the parameters as a function of the number of iterations when the sub-component number is 2, and (D) is the change of the difference between before and after updating of the parameters as a function of the number of iterations when the sub-component number is 3.
FIG. 3 is a graph showing the results of absorption spectrum and spectral decomposition of tryptophan, (A) shows the results of absorption spectrum and spectral decomposition of tyrosine, (B) shows the results of absorption spectrum and spectral decomposition of sodium benzoate, and (D) shows the results of absorption spectrum and spectral decomposition of 1-naphthylamine.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 3, a spectral decomposition method for water pollution analysis is used for qualitative and quantitative analysis of water pollution, and the method firstly normalizes an absorption spectrum to be decomposed to obtain a probability density function f (lambda) of observation data lambda; taking a double Gaussian function bi-Gaussian as a subcomponent model, and giving the number and initial parameters of the subcomponent model; iteratively updating the model parameters of each subcomponent, judging whether iteration under the condition of all preset subcomponent numbers is finished or not when a convergence condition is met, and if not, adding 1 to the subcomponent numbers to continue iterative operation; after the iteration work under the condition of all preset sub-component numbers is finished, selecting the optimal sub-component number and the model parameters thereof; converted to a spectral decomposition result, see fig. 1.
Taking the absorption spectrum of a 10mg/L phenol aqueous solution as an example for decomposition, the method comprises the following steps:
a: performing probabilistic observation data conversion on the spectrum to be decomposed;
referring to fig. 2(a) in which the solid gray line represents the absorption spectrum of phenol, it can be seen that two absorption peaks are included. The absorption spectrum data were normalized as follows:
Figure BDA0002875141450000061
where SP (λ) is the absorbance of phenol at each wavelength and the denominator is the integral of the spectral data with respect to wavelength. Then, f (λ) is regarded as a probability density function of the observation data λ.
B, regarding the observation data as the mixture of K bi-Gaussian subcomponents;
probability density function phi (lambda | theta) of kth sub-componentk) Comprises the following steps:
Figure BDA0002875141450000062
wherein beta iskIs the normalized coefficient of the sub-component function model, whose value is:
Figure BDA0002875141450000063
setting a sub-composition number upper limit KmaxTo 3, the sub-component fraction K is selected as 2, and the initial values of the model parameters are respectively theta1＝(220,49,100,0.5),θ2The first iteration is performed (272,144,81, 0.5).
Iteratively updating the parameters of the subcomponent model;
a. calculating the probability that the observation data come from each subcomponent;
the probability that the observed data λ comes from the kth sub-component is:
Figure BDA0002875141450000071
b. updating parameters of each subcomponent model;
updated k-th sub-component akAnd mukRespectively as follows:
ak new＝∑p(k|λ,θk)f(λ)δλ (5)
Figure BDA0002875141450000072
using updated muk newCalculating updated sigma in sequence1kSum sigma2k：
Figure BDA0002875141450000073
Figure BDA0002875141450000074
Then, the updated kth sub-component parameter is obtained as θk new＝(μk new,Σ1k new,Σ2k new,ak new)
c. Judging a convergence condition;
calculating the difference between the parameters of each subcomponent model before and after updating, and judging whether the difference is smaller than a threshold epsilon (taking 0.01 in the process):
Figure BDA0002875141450000075
when the difference value is smaller than epsilon, the iteration is ended, and each sub-model parameter obtained at the moment is listed in the table 1. It can be seen that the decomposed subcomponents are not axisymmetric, and the left and right widths of the subcomponents are greatly different, so that the double-Gaussian subcomponent model used in the invention can better meet the actual needs of spectral decomposition. In addition, the number of iterations in the current round is 10, and the change of the difference value before and after updating the parameter along with the number is as shown in fig. 2(C), so that the iteration convergence is fast and the local minimum does not occur.
The number of components K is then increased by 1, i.e. 3 subformsThe initial value of the model parameter is set as theta1＝(220,49,100,1/3),θ2＝(272,144,81,1/3)，θ3A new iteration is performed (300,169,169, 1/3). Similarly, when the difference is less than ε, the iteration is terminated, and the sub-model parameters obtained at this time are listed in Table 2. The iteration frequency is 35 times, and the change of the difference value before and after updating the parameters along with the frequency is as shown in fig. 2(D), so that the convergence is relatively slow, the local minimum exists in the process, and the difference value has a certain fluctuation rather than a stable descending trend. Table 1 shows the respective subcomponent model parameters after iterative convergence when the subcomponent number is 2, and table 2 shows the respective subcomponent model parameters after iterative convergence when the subcomponent number is 3.
Figure BDA0002875141450000081
TABLE 1
Figure BDA0002875141450000082
TABLE 2
D, comparing the decomposition errors and selecting the optimal subcomponent number K;
the decomposition error for each sub-component number K is calculated, and K with the smallest error is selected as the best component number.
Figure BDA0002875141450000083
Through calculation, in the present example, the error is 0.082 when K is 2 and 0.512 when K is 3. Therefore, the optimum subcomponent number K is 2.
E, converting the form of the spectral data;
the probability density function phi (lambda | theta) of each componentk) Conversion to spectral dataform:
spk(λ)＝akφ(λ|θk)·∑SP(λ)δλ (11)
as shown with reference to fig. 2(a) and with reference to fig. 2(B), the broken line is a spectrum curve of each sub-component. It can be seen from the figure that when the sub-component number is 2, the decomposed sub-components are closer to the original spectrum.
The method can be used for decomposing the overlapped spectral peaks, and intensively and finely analyzing the absorption peak generated by a functional group in the subsequent water pollutant analysis, thereby being beneficial to obtaining more information and improving the detection precision.
Referring to FIG. 3, the absorption spectra of tryptophan, tyrosine, sodium benzoate, and 1-naphthylamine and their decomposed subcomponents are shown. By this method, an absorption peak due to an absorption band of benzene ring B can be separated from each substance. Due to the action of different substituents on a benzene ring, a large amount of substance information is contained in a B absorption band, and the B absorption band can play a greater role in subsequent analysis. For example, in qualitative analysis, the characteristic extraction of the decomposed B absorption band can be used for fully reflecting the characteristics of the substances contained in the B absorption band; in quantitative analysis, the peak value of the B absorption band can be dynamically tracked by using the spectral decomposition method, and a corresponding inversion calculation model is established.
The embodiments described in this specification are merely illustrative of implementations of the inventive concept and the scope of the present invention should not be considered limited to the specific forms set forth in the embodiments but rather by the equivalents thereof as may occur to those skilled in the art upon consideration of the present inventive concept.

Claims (4)

1. A method of spectral decomposition for analysis of water contaminants, the method comprising the steps of:
1) carrying out probabilistic observation data conversion on the spectrum to be decomposed, wherein the process is as follows:
normalizing the spectral data according to a formula (1) so as to convert the spectral data into a probability density function of observation data lambda;
Figure FDA0002875141440000011
wherein SP (λ) is spectral data, the denominator is the integral of the spectral data with respect to wavelength, the value is equal to the area of the region enclosed by the spectrum and the wavelength axis, since the spectral data is discrete, a summation sign is used, so that f (λ) is taken as a probability density function of the observed data λ, the integral thereof is equal to 1, indicating that the sum of the probabilities of occurrence of all wavelengths λ is 1;
2) the observation data was considered as a mixture of K bi-Gaussian subcomponents as follows:
selecting a bigaussian function bi-Gaussian as the subcomponent model, with parameters including θ ═ μ, Σ1,Σ2A) respectively representing the mean value, the variance of the left Gaussian function and the variance of the right Gaussian function of the left and right Gaussian functions, and the proportion of the subcomponents in the total; suppose the observed data comes from K subcomponents (K2, 3, …, K)max) The initial parameters are respectively (theta)1，θ2，θ3，...，θk，...，θK) Probability density function phi (lambda | theta) of the kth sub-componentk) Comprises the following steps:
Figure FDA0002875141440000012
wherein beta iskIs the normalized coefficient of the sub-component function model, whose value is:
Figure FDA0002875141440000013
when sigma1kSum-sigma2kWhen the values are equal, the values are the same as the common Gaussian function, and the subcomponent models are axisymmetric; on the contrary, the left and right widths are different;
3) iteratively updating each subcomponent parameter, wherein the process is as follows;
31) calculating the probability that the observed data comes from each subcomponent, comprising:
the probability that the observed data λ is from the kth sub-component is expressed as:
Figure FDA0002875141440000014
here, phi (lambda | theta) in step 2)k) Substituting into the formula (4) for calculation;
32) updating each subcomponent parameter, including:
updated k-th sub-component akAnd mukRespectively as follows:
ak new＝∑p(k|λ,θk)f(λ)δλ (5)
Figure FDA0002875141440000015
here f (λ) in step 1) and p (k | λ, θ) in step 31) are comparedk) Substituting into formulas (5) and (6) for calculation; f (lambda) is introduced to indicate that the parameters are adjusted by utilizing the actual distribution of the spectral data to be decomposed;
reuse of f (λ), p (k | λ, θ)k) And updated μk newCalculating updated sigma in sequence1kSum sigma2k：
Figure FDA0002875141440000021
Figure FDA0002875141440000022
Then, the updated kth sub-component parameter is obtained as θk new＝(μk new,Σ1k new,Σ2k new,ak new)
33) Judging a convergence condition, comprising:
determining whether the deviation between the updated subcomponent parameter and the original parameter is less than a small positive number epsilon, i.e.
Figure FDA0002875141440000023
If the deviation is larger than or equal to epsilon, the sub-component parameter theta updated in the step 32) is addedk newRepeating the calculation of p (k | λ, θ) in the sequential steps 31) and 32)k) And thetak newUntil the updating deviation is less than epsilon; when the updating deviation is smaller than epsilon, the parameter change is very small after one iteration, the iteration is ended, and whether the current sub-component number K reaches the preset upper limit K or not is continuously judgedmaxIf K is reachedmaxStep 4) is entered, otherwise, K is added by 1, and the step 2) is returned;
4) and comparing the decomposition errors, and selecting the optimal subcomponent number K according to the following process:
calculating the decomposition error of each sub-component K according to a formula (10), and selecting K with the minimum error as the optimal component;
Figure FDA0002875141440000024
5) the parameter theta of iterative convergence under the condition of the optimal sub-component number K obtained in the step 4)k newConverting into spectral data to obtain a spectral decomposition result, wherein the process is as follows:
the iterative convergence parameter theta when the best sub-component number K is givenk newSubstituting the formula (2) to obtain the probability density function phi (lambda | theta) of each componentk) Then, the data is converted back to a spectrum data form by using a formula (11), and each decomposed sub-component is as follows:
spk(λ)＝akφ(λ|θk)·∑SP(λ)δλ (11)。
2. the method of claim 1, wherein in step 2), the upper K sub-component count limit is setmaxIs 3.
3. A use as claimed in claim 1 or 2The spectral decomposition method for water pollutant analysis is characterized in that in the step 2), the initial parameter of the subcomponents is theta1＝(220,49,100,1/K),θ2＝(272,144,81,1/K),θ3＝(300,169,169,1/K)。
4. A method of spectral decomposition for analysis of water pollutants according to claim 1 or 2, wherein in step 33) said update deviation threshold e is 0.01.
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