Establishment and application of inferential equations for glass types

: In order to analyze the classification rules of high-potassium glass and lead-barium glass, this paper performs gray correlation analysis on the two types of glass, selects the appropriate chemical composition for each category to divide them into subcategories, gives specific division methods and division results according to the results of principal component analysis, and derives the inferred equation for each category of glass. The inferred equation is used to analyze the chemical composition of the unknown category of glass artifacts, calculate the central distance, and determine which glass category it is by comparing the size of the central distance Identify the type to which it belongs.


Introduction
Glass was an important item in the early trade between China and the West, and because of the scarcity of ancient glass products [1][2][3], it has a very high collector's value and artistic value. The chemical composition of ancient glass differs from that of Western glass due to the geographical and material conditions in China [4][5][6]. Therefore, the analysis of the chemical composition of ancient glass objects is an important task, and archaeologists can infer and identify the region, age, production method [7], and degree of weathering based on the chemical composition of glass. The analysis of the composition of ancient glass objects is of great significance for the study of ancient history and the conservation of cultural relics [8][9][10].

Modeling by principal component analysis
Assuming a total of n samples and p indicators in the sample, a sample matrix x of size n × p can be formed.
Standardize the sample and calculate the correlation coefficient matrix Calculate mean values by column: Standard deviation: The standardized data is: After standardization the sample matrix changes to: Calculate the correlation coefficients for each row and column: After processing the above analysis, the correlation coefficient matrix can be obtained as: Calculate the eigenvalues and eigenvectors: Eigenvalues of the correlation coefficient matrix R 1 ≥ 2 ≥ ⋯ ≥ ≥ 0.

Analysis of the classification law of high potassium glass and lead-barium glass
The combined table was split into a high potassium glass table and a lead-barium glass table to analyze the gray correlations of high potassium glass and lead-barium glass separately. The chemical components with the top four gray correlations (the reasons for selecting the four components are given below in the principal component analysis) were selected for the analysis. The results are shown in the table 1 and 2. 0.954 4 According to the above two figures, it can be seen that the chemical composition of silicon dioxide (SiO2) is higher when the glass type is high potassium glass, and the content of lead oxide (PbO) is higher when the glass type is lead-barium glass. High potassium glass is mainly related to the content of silicon dioxide (SiO2), aluminum oxide (Al2O3), copper oxide (CuO), and phosphorus pentoxide (P2O5). Lead-barium glass is mainly related to the content of lead oxide (PbO), silicon dioxide (SiO2), barium oxide (BaO), aluminum oxide (Al2O3).

Perform subcategorization
According to the above gray correlation analysis of high potassium glass and lead-barium glass respectively, it can be seen. High potassium glass is mainly related to silica, alumina, copper oxide and phosphorus pentoxide. The lead-barium glass is mainly related to lead oxide, silica, barium oxide, and alumina.
The specific classification of each glass type is given by the following principal component analysis of the data in the combined table.
Firstly, the principal component analysis was carried out by the principal component of 1 to obtain the following fragmentation diagram. By observing the fragmentation diagram, we can see that the image tends to be flat when the number of principal components k=4, so the number of principal components can be determined as 4. The results are shown in the figure 1  By the ratio it is found that the effects are the same so it can be assumed that the principal component fraction is 4.

Analysis of the reasonableness and sensitivity of the classification results
The regression equation obtained by the principal component analysis is inferred from the main chemical components in the combined table for the glass types. The rate of change is 0.18, which can be regarded as reasonable because of the small rate of change. The results are shown in the table 4. Processed data 30 37 Figure 2: Effect of ±5% data perturbation on silica on center distance Sensitivity analysis can analyze the robustness of the model by perturbing the data for any of the principal components. The perturbed principal component content is then used to infer the glass type based on the regression equation of the principal component analysis (the data are obtained as the center distance, if the distance to 1 is close, it is high potassium glass, if not, it is lead-barium glass, if the distance to 2 is closer, it is lead-barium glass, otherwise, it is high potassium glass). In this paper, the silica content is perturbed by ±5%, and the type of glass does not change when the data is perturbed, as can be seen from the line graph of center-to-center distance below. The results are shown in the figure 2.

Analysis of chemical composition of glass artifacts of unknown category
Using the regression equation for the glass type established by the principal component analysis in Problem 2, the center distance is calculated by combining the four principal components of lead oxide (PbO), silicon dioxide (SiO2), barium oxide (BaO), and aluminum oxide (Al2O3) in Form 3 and determining whether the center distance is close to 1 or close to 2. If it is close to 1, it is a high potassium glass, and if it is close to 2, it is a lead-barium glass. Therefore, the glass artifacts of unknown category in Form 3 belong to the following glass types. The results are shown in the table 5.

Analysis of the sensitivity of the classification results
Based on the robustness, the data is perturbed for any of the main components and the center distance is calculated to determine whether the glass type changes. In this paper, the glass type does not change when the data is perturbed by ±5% for silica content, as can be seen from the following line graph.The results are shown in the figure 3.

Conclusions
In this paper, gray correlation analysis was performed for high potassium or lead-barium glasses, and the classification pattern was analyzed according to the correlation magnitude of each chemical composition. The inference equation of each glass category was obtained by using Matlab to determine the number of principal components in the gravel diagram and to find the major components according to the correlation magnitude. The results were checked for reasonableness and sensitivity analysis for data perturbation based on robustness. The correlation magnitudes were compared to show that high potassium glasses were used when the SiO2 content was high, and leadbarium glasses were used when the PbO content was high. The number of principal components was 4. The four principal components with higher correlation were selected to classify them into subclasses. The inferred equation of the glass type was derived from the model by principal component analysis, which is the result of the classification. The inferred model is based on the chemical composition of each glass type, and the error tolerance of the glass type is 0.18, which is reasonable. The regression equation of the principal component analysis did not change the glass type when the data was perturbed by ±5% for silica, which has a large influence on the principal component. The results do not change when the data are perturbed.
In identifying the glass type, the inferred equation for each glass type was obtained by using principal component analysis, and the distance to the center of each glass number was obtained by combining the data with cluster analysis. The sensitivity analysis of the data perturbation is also based on the robustness. When the data are perturbed, the results do not changes.