The missing value is mainly generated because the content of one or several elements in the sample is very low and does not reach the detection limit, or the absence of artificial data entry cannot be excluded. There are three commonly used data missing value processing methods: 1) Directly delete the sample or variable containing the missing value; 2) Assign an arbitrary value below the detection limit to the missing value; The missing value is estimated based on the component analysis of the associated sample (e.g., likelihood estimation). In view of the fact that the original data is the component data, the missing values are too numerous and too scattered, and it is not suitable to directly delete or directly assign arbitrary values below the detection limit. Therefore, we use the component analysis based on the associated samples to estimate the missing values. Specifically, the number of missing components is calculated for each sample, and the missing data is filled by the expected maximum likelihood estimation method combined with the ideal condition of 100% of the proportion of each component.

The absence of an element in the sample or the amount of an element that does not reach the detection limit may result in a zero value in the recorded data. Considering the small number of zero values of the original data in the sample and the scattered distribution, we choose to modify the zero values recorded in the original data to a small positive number (0.001), which is convenient for subsequent data processing and modeling calculation.

The component data is non-negative, and the sum of each individual element content in sample is 1 [13] , that is W N × D = { W i j ≥ 0 | i = 1 , 2 , ⋯ , N ; j = 1 , 2 , ⋯ , D } , and W i 1 + W i 2 + ⋯ + W i D = 1 ( i = 1 , 2 , ⋯ , N ) . Therefore, if the cumulative sum of component data is biased, the correlation analysis results between elements will be biased, which will further affect the covariance and correlation matrix between components [14] .

The direct data studied in this paper is the content proportion of each chemical component, that is, the component data, which should theoretically meet the constraint of cumulative sum of 1. However, due to the accumulation of the proportion of each chemical component and non-100% deviation caused by the detection means, it is necessary to correct the deviation. If the cumulative sample component content of the original data is between 85% and 105%, it is considered that the fixed sum deviation of the data is not large, and the sample data is valid and can be corrected. If the component content of the original data accumulates and exceeds the range of 85% to 105%, the sample is considered invalid data and cannot be corrected. For valid sample data that can be corrected, the sum is fixed according to formula (1):

x ′ i j = x i j ∑ j = 1 m x i j (1)

where x i j ( i = 1 , 2 , ⋯ , n ; j = 1 , 2 , ⋯ , m ) represents the original data of the indicator j of the sample i, x ′ i j represents the converted data. After conversion, the proportion of each component of each sample is 100%.

The descriptive statistical results of the data after modification are shown in Table 2. It can be seen from Table 1 that the skewness coefficients of component indicators in the sample are all greater than 0, indicating that there are more data on the right side of the mean value. The skewness coefficient of SiO₂ is close to 0, indicating that the data of SiO₂ is close to symmetric distribution. The kurtosis coefficient of 2 component indices in the sample is less than 0, indicating that the distribution is flatter at the top or thinner at the tail than the normal distribution. The kurtosis coefficients of the remaining components are greater than 0, indicating that the distribution is sharper at the top or thicker at the tail than the normal distribution. In general, the content of most components in the data does not conform to the normal distribution.

Since the ratio of component data variables is not restricted by the “constant sum” restriction, and the logarithm of the ratio usually follows the characteristics of normal distribution, this paper adopts the methods of additive log-ratio transformation (ALR) and central log-ratio transformation (CLR) [15] , so that the component data presents the situation of multivariate normal distribution after transformation. Considering that the vector after the additive log-ratio transform (ALR) cannot correspond to the original vector one-to-one, the central log-ratio transform [16] is adopted:

Let component vector x = ( x 1 , x 2 , ⋯ , x D ) , ( x i > 0 ) , and ∑ i = 1 D x i = 1 , let

y i = ln x i ∏ i = 1 D x i D , i = 1 , 2 , ⋯ , D . (2)

This transformation is called the central log-ratio transformation (CLR).

The corresponding inverse transformation is:

x i = exp ( y i ) ∑ i = 1 D exp ( y i ) , i = 1 , 2 , ⋯ , D . (3)

The transformed component corresponds to the original component one by one, and further enhances the interpretability of the variable. In addition, the central log-ratio transformation can make the data more stable, reduce the influence of outliers, and reflect the real situation of the data more accurately.

Use x 1 , x 2 , ⋯ , x 14 to indicate the chemical composition of glass relics SiO₂, Na₂O…, SO₂. According to formula (2), for the component vector x = ( x 1 , x 2 , ⋯ , x 14 ) ∈ S 14 ( x i > 0 ) , the data transformed by CLR is:

y = ( y 1 , y 2 , ⋯ , y 14 ) = ( log x 1 ∏ i = 1 14 x i 14 , ⋯ , log x 14 ∏ i = 1 14 x i 14 ) (4)

According to formula (4), the data after logarithm transformation of the content center of high potassium and lead barium glass can be calculated. In view of the rationality of the hypothesis of normal distribution of data, it is necessary to test whether the data distribution pattern conforms to normal distribution. There are many testing methods for normal distribution, including Shapiro-Wilk test, Kolmogrov-Smirnov test, skewness test and kurtosis test [17] . In this paper, the normal distribution of data is tested by skewness and kurtosis test. In statistics, skewness is used to describe symmetry [18] , and kurtosis represents the characteristic number of the height of kurtosis at the mean of the probability density distribution curve. The skewness and kurtosis of the data can be calculated through the software. Limited by space, only the skewness and kurtosis of the lead-barium glass data are shown (Table 3). When the skewness and kurtosis values are closer to 0, the data tends to be more normally distributed. In order to intuitively understand the influence of the center log-ratio transformation on the normal distribution of data, the original data frequency histogram and the data frequency histogram after the center log-ratio transformation were made for the data in this paper (Figure 2 and Figure 3). Limited by space, only some data frequency histograms of lead-barium glass are shown.

As can be seen from Table 3 and Figure 2 and Figure 3, the skewness and kurtosis of the composition content data of lead-barium glass are high, so most of the composition content in the original data does not conform to the normal distribution. After the central logarithmic ratio transformation, skewness and kurtosis of most component content data are reduced, and the processed data tend to be more normal distribution.

The “closure effect” caused by the constant value of sum will lead to the deviation of the analysis results of the correlation between variables. Therefore, the Pearson correlation coefficient between the variables of high potassium and lead barium glass is calculated using the data after the transformation of the central logarithm ratio. The calculation formula is as follows:

ρ x y = ∑ i = 1 n ( x i − x ¯ ) ( y i − y ¯ ) ∑ i = 1 n ( x i − x ¯ ) 2 ∑ i = 1 n ( y i − y ¯ ) 2 (5)

where, x i , y i respectively represents the chemical composition variables of glass relics, x ¯ , y ¯ respectively represents the average value of the two variables, and n is the sample size. The closer the calculation result is to 1, the stronger the positive correlation between the two variables is, and the closer the result is, the stronger the negative correlation is. According to the calculation results of equation (5), we obtained the heat map of the correlation coefficient of high-potassium and lead-barium glasses, and the results are shown in Figure 4 and Figure 5. The results show that there is a high correlation between the glass type and the proportion of some chemical components, and the correlation between different types of glass and their chemical components is different. For example, for high potassium glass, SiO₂ is positively correlated with K₂O, CaO, MgO, Al₂O₃, CuO, P₂O₅ and SnO₂, and negatively correlated with Na₂O and SrO₂. SiO₂ has a very significant positive correlation with Al₂O₃ and a very significant negative correlation with Na₂O. For lead-barium glass, SiO₂ has a very significant positive correlation with Na₂O and Al₂O₃, and a very significant negative correlation with P₂O₅ and SrO. It can also be concluded that for high potassium glass, SiO₂ has a positive correlation with P₂O₅, and a positive correlation with Na₂O, while the lead barium glass is the opposite. Therefore, the correlation between variables of different glass types is quite different.

Since the weathering state is divided into weathering and unweathering, which belong to categorical variables, and the chemical composition content of glass relics belongs to continuous variables, linear regression analysis can be carried out by setting dummy variables. With no weathering as a reference variable and weathering state as a dummy variable, linear regression models of weathering state of high-potassium glass and lead-barium glass on the content of each chemical component were established according to the data after CLR transformation. The results are shown in Table 4.

The results showed that weathering had different effects on the contents of chemical components of different glass types. 1) Weathering can cause significant changes in the proportion of SiO₂ and Na₂O components of high potassium glass, p value is 0.000 (<0.05), and can cause changes in the proportion of CuO components to a certain extent. According to the linear model primary term coefficient, surface weathering causes the proportion of SiO₂ and CuO components to increase, and the proportion of Na₂O components to decrease significantly. 2) Weathering can cause significant changes in the proportions of SiO₂, Na₂O, K₂O, CaO, Al₂O₃, PbO and P₂O₅ of lead-barium glass to a certain extent, and the p value is less than 0.05. According to the linear model primary term coefficient, the surface weathering causes the proportion of SiO₂, Na₂O, K₂O and Al₂O₃ to decrease, while the proportion of CaO, PbO and P₂O₅ to increase.

Due to the lack of detection data before and after weathering, the problem of predicting the chemical composition content of glass before weathering from the detection data of weathering points and unweathered cultural relics needs to analyze the statistical laws of weathering and unweathered populations, and make prediction through the relationship between the two populations. Therefore, the idea of distributed matching can be used for prediction. According to the fact that the ratio of the component data variables is not restricted by the “fixed sum” and the logarithm of the ratio usually follows the characteristics of normal distribution, it is assumed that the data after the transformation of the central logarithm follows the normal distribution. Considering two populations of weathering and unweathering, namely weathered glass population X ~ N ( μ x , σ x 2 ) and unweathered glass population Y ~ N ( μ y , σ y 2 ) , the data after the transformation of the central logarithm ratio were classified and processed to obtain four types of data, namely, high potassium weathering (6), high potassium unweathering (14), lead barium weathering (26) and lead barium unweathering (23). Here, the detected data at its unweathered point in the weathering sample is regarded as unweathered, such as 49 unweathered points in the weathering sample of lead barium is regarded as unweathered category. The mean and standard deviation of each chemical component content of weathered and unweathered glass relics were analyzed respectively.

According to the weathered data C L R ( X ) , the pre-weathering data of high potassium and lead barium glass were calculated by the formula C L R ( Y ) = μ y + σ y σ x ( C L R ( X ) − μ x ) . The inverse transformation formula (3) of the central log-ratio transformation is then used to restore the unweathered component data (the result is multiplied by 100). The predicted contents of each chemical component of high-potassium glass and lead-barium glass before weathering were calculated, as shown in Table 5 and Table 6.

In order to intuitively understand the change trend of the predicted data, the calculation results are visualized as shown in Figure 6 and Figure 7. As can be seen from Figure 6, weathering will cause the proportion of SiO₂, CuO and P₂O₅

components to increase, and the proportion of Na₂O, CaO and K₂O components to decrease. As can be seen from Figure 7, weathering causes the proportion of SiO₂ and Al₂O₃ in lead-barium glass to decrease, while the proportion of CuO, CaO and PbO increases, while BaO does not change significantly before and after weathering. It is highly consistent with the above linear regression analysis results of weathering state and chemical composition, indicating that the prediction model constructed has a good prediction effect.