After consulting relevant information, it is found that the degree of weathering of glass is affected by both the type of glass itself and other external factors, such as the burial environment. An important result of weathering is the change of glass color: some areas of glass that were originally green may become purple or gray-yellow. The decoration of the glass does not affect weathering. However, there is a close correspondence between different types of decoration and weathering results, which is contrary to the known theory that ornamentation has little effect on weathering. Therefore, this paper speculates that the decoration of glass, as a symbol of regional culture, represents the geographical environment of different regions, and the natural environment of different regions is an important factor influencing the weathering results. A graph of the relationship between weathering and glass type, decoration, and color is shown in Figure 3.

The glass type and decoration were first analyzed in relation to weathering, and the statistical results are shown in Table 2.

Table 2 suggests the following conclusions: three types of ornamentation, A, B, and C, are present in high-potassium glass. Of these, the Type B samples are all weathered, while the Type A and Type C samples are unweathered. In the lead-barium glass, the Type A samples are weathered, and the Type C samples are unweathered.

When analyzing the relationship between glass type and weathering, this article uses ornamentation as a control variable. Because lead-barium glass does not contain samples with Type B ornamentation, this paper analyzes Types A and C. Among them, all of the high-potassium glass is not weathered, while the total weathering rate of the Types A and C lead-barium glass is 70%. Therefore, it is concluded that high-potassium glass is less susceptible to weathering than lead-barium glass.

Analyzing the relationship between ornamentation and weathering began with the analysis of Type B ornamentation. In the given sample, all of the glass with Type B ornamentation was weathered, probably because the geographic environment in which the Type B ornamentation was found made it more susceptible to weathering. Therefore, it is speculated that all glass with Type B motifs would be weathered.

The statistical analysis of the chemical composition content is divided into two parts: high-potassium glass and lead-barium glass.

(1) In high-potassium glass, the means, standard deviations, and standard errors of the mean of each component in the weathered and unweathered cases were calculated, and the variance was tested by an independent sample test.

Hypothesis

testing of the variance: Consider the statistical hypothesis H 0 : σ 1 2 = σ 2 2 ; H 1 : σ 1 2 ≠ σ 2 2 , and μ 1 , μ 2 are unknown. This hypothesis can be transformed: H 0 : σ 1 2 σ 2 2 = 1 , (2) H 1 : σ 1 2 σ 2 2 ≠ 1 . (3)

Because S X 1 , S X 2 is the minimum variance unbiased statistic for the variance σ 1 2 , σ 2 2 , respectively, S X 1 S X 2 should be in the neighborhood of 1 in case H 0 holds, and therefore, the rejection domain of H 0 can be chosen to be of the form: K 0 = S X 2 S Y 2 < c 1 , or  S X 2 S Y 2 > c 2 , c 1 < c 2 (4)

By: P S X 2 S Y 2 < c 1 ∪ S X 2 S Y 2 > c 2 ∣ H 0  is established  = α (5)

Order: P S X 2 S Y 2 < c 1 = P S X 2 S Y 2 > c 2 = α 2 , (6) P S X 2 S Y 2 < F α 2 n − 1 , m − 1 = P S X 2 S Y 2 > F 1 − α 2 n − 1 , m − 1 = α 2 . (7)

Get the denial field of H 0 as follows: S X 2 S Y 2 < F α 2 n − 1 , m − 1  or  S X 2 S Y 2 > F 1 − α 2 n − 1 , m − 1 (8)

The test revealed that the p-value > 0.05 for the differences between the relative amounts of potassium oxide, aluminum oxide, copper oxide, phosphorus pentoxide, and tin oxide in the weathered and unweathered cases. Therefore, the original hypothesis could not be rejected. It was concluded that the relative amount of these five compounds did not change due to weathering.

At this point, comparisons of the means of these five components will be made with equal variances. A hypothesis test is performed on the means of two components, at which point the σ 1 2 , σ 2 2 of the two components is known.

When H 0 holds, there is as follows: X ¯ ∼ N μ 1 , σ 1 2 n , (9) Y ¯ ∼ N μ 2 , σ 2 2 m . (10)

So, there is as follows: X ¯ − Y ¯ ∼ N 0 , σ 1 2 n + σ 2 2 m , (11) u = X ¯ − Y ¯ σ 1 2 n + σ 2 2 m ∼ N 0 , 1 , (12) P X ¯ − Y ¯ σ 1 2 n + σ 2 2 m > u 1 − α 2 = α , (13) P X ¯ − Y ¯ > u 1 − α 2 σ 1 2 n + σ 2 2 m = α . (14)

In this paper, R software was used to test the results obtained. The test results show that the p-value > 0.05 for the different amounts of copper oxide and tin oxide with and without weathering, which does not reject the original hypothesis. That is to say, the amounts of copper oxide and tin oxide with and without weathering are equal. The p-value < 0.05 for the different amounts of potassium oxide, alumina, and phosphorus pentoxide, which rejects the original hypothesis, that is to say, the relative amounts of potassium oxide, alumina, and phosphorus pentoxide change with weathering.

This results in unequal variances for the residual components of the weathered and unweathered cases. R software is still utilized to realize the hypothesis test for the comparison of two overall means.

From the test results, it can be seen that the test results show that the p-value > 0.05 for the different relative amounts of silicon oxide, sodium oxide, lead oxide, barium oxide, and sulfur dioxide with and without weathering, which does not reject the original hypothesis. That is, the relative amounts of silicon oxide, sodium oxide, lead oxide, barium oxide, and sulfur dioxide do not change with weathering.

The p-value < 0.05 for the different amounts of calcium oxide, magnesium oxide, iron oxide, and strontium oxide, which rejects the original hypothesis. That is, the relative amounts of calcium oxide, magnesium oxide, iron oxide, and strontium oxide do not change with weathering. However, the total relative amount of silicon oxide is larger before weathering, and the difference between the silicon oxide content in the weathered and unweathered samples is larger; this paper considers that the mean values are unequal.

Therefore, the following equations can be established to determine the pattern of chemical content with and without weathering. y i = x i , i = 2 , 8 , 9 , 10 , 13 , 14 (15) y i = a i x i , i = 1 , 3 , 4 , 5 , 6 , 7 , 11 (16) y i = x i + a i , i = 12 (17)

Eq. 15, Eq. 16, and Eq. 17 express the relationship between the components with and without weathering.

Eq. 15 expresses the relationship between the content of the six constituents, sodium oxide, copper oxide, lead oxide, barium oxide, tin oxide, and sulfur dioxide, with or without weathering when the mean values are equal, that is, the content of the unweathered constituents is equal to the content of the weathered constituents.

Eq. 16 represents the relationship between the content of silica, potassium oxide, calcium oxide, magnesium oxide, alumina, iron oxide, and phosphorus pentoxide with and without weathering when the mean values are not equal, that is, the content of the weathered component multiplied by the coefficient a i equals the content of the non-weathered component, of which the values of a 1 , a 3 , a 4 , a 5 , a 6 , , a 7 , a 11 are 0.74, 17.98, 6.69, 5.35, 3.36, 6.67, and 4.29, respectively. The vector can be expressed as a 1 , a 3 , a 4 , a 5 , a 6 , , a 7 , a 11 = 0.74 , 17.98 , 6.69 , 5.35 , 3 , 36 , 6.67 , 4.29 .

Eq. 17 expresses the relationship between strontium oxide content with and without weathering when the mean values are unequal, that is, the content of the weathered component multiplied by the coefficient a i is equal to the content of the non-weathered component, where the value of a 12 is 0.03, which can be expressed by the vector a 12 = 0.03 .

(2) In lead-barium glass, the mean, standard deviation, and standard error of the mean were calculated for each component in the weathered and unweathered cases, and the independent sample test was conducted to determine the variance. The test revealed that the p-values < 0.05 for the difference in the relative amounts of sodium oxide and phosphorus pentoxide with and without weathering. The original hypothesis was rejected. The relative amounts of sodium oxide and phosphorus pentoxide change with weathering. The other components of lead-barium glass did not change with weathering.

Sodium oxide and phosphorus pentoxide have unequal variances with and without weathering, in which case the hypothesis test for the comparison of the two overall means is carried out. From the test results, the p-value > 0.05 for the amounts of sodium oxide with and without weathering, which does not reject the original hypothesis that the means with and without weathering are equal. The p-value < 0.05 for the relative amounts of phosphorus pentoxide with and without weathering, which rejects the original hypothesis.

The variance of other elements in the weathered and unweathered cases are equal, and the hypothesis test of comparing the two overall means is still carried out. From the test results, it can be seen that the p-value > 0.05 for nine components (potassium oxide, magnesium oxide, aluminum oxide, iron oxide, copper oxide, barium oxide, strontium oxide, tin oxide, and sulfur dioxide), which does not reject the original hypothesis, that is, the means with and without weathering are equal. The p-value < 0.05 for the amounts of silicon dioxide, calcium oxide, and lead oxide with and without weathering, and the hypothesis is rejected. The relative amounts of silicon dioxide, calcium oxide, and lead oxide change with weathering. The following equation was established to determine the chemical content pattern with and without weathering. y i = x i , i = 2 , 3 , 5 , 6 , 7 , 8 , 10 , 12 , 13 , 14 (18) y i = b i x i , i = 1 , 4 , 9 , 11 (19)

Eq. 18 expresses the relationship between the relative amounts of nine constituents (sodium oxide, potassium oxide, magnesium oxide, aluminum oxide, iron oxide, copper oxide, barium oxide, strontium oxide, tin oxide, and sulfur dioxide), with or without weathering when the mean values are equal, that is, the relative amounts of the unweathered constituents is equal to the relative amounts of the weathered constituents.

Eq. 19 expresses the relationship between the relative amounts of silica, calcium oxide, lead oxide, and phosphorus pentoxide with and without weathering when the mean values are not equal, that is, the content of the weathered component multiplied by the factor a i equals the content of the unweathered component, where b 1 , b 4 , b 9 , b 11 = 1.97 , 0.38 , 0.51 , 0.20 .

Due to the small amount of sample data explored in this article, a gray prediction model was used to predict the chemical composition content of artifacts before weathering. Gray prediction is a method for predicting systems that contain uncertainties. Gray prediction analyzes the degree of correlation between the various systems and the original data and does a cumulative generation of the sequence and the mean of the sequence to generate the sequence, resulting in a strong regularity of the data sequence. Then, gray prediction establishes a corresponding differential equation model to predict the status of future development trends. Its solving steps are shown in Figure 4.

After solving, it can be concluded that the data series of the silica content of high-potassium glass is as follows: x 0 = x 0 1 , x 0 2 , … x 0 12 = 69.33 , 87.05 , 61.71 , 65.88 , 61.58 , 67.65 , 59.81 , 62.47 , 65.18 , 79.46 , 76.68 (37)

Its class ratio is as follows: λ k = 0.7964 , 1.4106 , 0.9367 , 1.0698 , 0.9103 , 1.1311 , 1.0136 , 0.9446 , 0.9584 , 0.8203 , 1.0363 . (38)

Because all the λ k ∈ 0.7964 , 1.4106 , k = 2 , 3 , ⋯ , 12 , this can be used as the model of the sequence of data for gray prediction; the original data at a time cumulative obtained: x 1 = 69.33 , 156.38 , 218.09 , 283.97 , 345.55 , 413.20 , 473.01 , 532.02 , 594.49659.67 , 739.13 , 815.81 (39)

Construct data matrices and data vectors: B = − 112.855 1 − 187.235 1 − 251.03 1 − 314.76 1 − 379.375 1 − 443.105 1 − 502.515 1 − 563.255 1 − 627.08 1 − 699.4 1 − 777.47 1 (40) Y = 87.05 61.71 65.88 61.58 67.65 59.81 59.01 62.47 65.18 79.46 76.68 (41)

Calculated through the matrix: u ^ = a ^ , b ^ T = B T B − 1 B T Y = − 0.0015 67.1997 (42)

This can be obtained from a ^ = − 0.0015 , b ^ = 67.1997 . Thus, the differential equation is established: d x 1 t d t + a x 1 t = b (43)

Eventually available: x ^ 1 k + 1 = x 0 1 − b ^ a ^ e − a ^ k + b ^ a ^ = 44869.13 e 0.0015 k − 44799.8 (44)

Representative chemical components of high-potassium glass and lead-barium glass, such as silica, potassium oxide, and lead oxide, are selected for analysis. The predicted content map of the main elements in the high-potassium category is shown in Figure 5.

The data analysis was carried out by MATLAB, and it can be seen from Figure 5 that the predicted silica content of weathered high-potassium glass increases continuously with the amount of weathering, while the relative amounts of potassium oxide and calcium oxide decrease. The silica content of unweathered high-potassium glass varies. The prediction of the main components of the lead-barium glass is shown in Figure 6.

Figure 6 shows that the predicted values of silica content in lead-barium glass before weathering are 57.534%, 58.171%, and 60.787% for the three severe weathering sample points of 08, 26, and 54, respectively. That is to say, the silica content decreases while the percentages of lead oxide and barium oxide increase with increased weathering.

The results of the calculations are shown in Table 3 and Table 4 as an example. Due to space limitations in the main text, only some of the detailed data are shown.

This article will further analyze the correlations between chemical compositions for different categories of glass artifact samples and compare the differences in the correlations between different categories.

In this article, the generalized Shapley function based on a better is selected to analyze the correlation relationship between indicators, where λ denotes the fuzzy measure that can analyze the interactions between individual indicators relative to other remaining arbitrary indicator sets and is also able to analyze the interactions between arbitrary indicator sets relative to the remaining arbitrary indicator sets. This fuzzy measure can make the analysis results more accurate and more in line with the reality.

The use of this analysis requires data standardization, and when the indicator C j is numerical, the data are processed and standardized to obtain it: y i j = a i j ∑ i = 1 n a i j 2 (45)

Let X = x 1 , x 2 , … , x n be a nonempty set and P(X) be a power set on X λ ∈ − 1 , ∞ , μ : P X → 0 , 1 , if P(X) satisfies the following three points: (i) μ 0 = 0 , μ X = 1 ; ② ∀ A , B ∈ P X , A ⊂ B , then μ A ≤ μ B ; ③ μ A ∪ B = μ A + μ B + λ μ A λ μ B , then μ is said to be the λ - fuzzy measure on X can be found: ① When λ = 0 , which can be obtained from μ A ∪ B = μ A + μ B , then A and B are independent of each other; ② When λ < 0 , μ A ∪ B < μ A + μ B , then A and B have a cross-relationship; ③ When λ > 0 , μ A ∪ B > μ A + μ B , then A and B have a complementary relationship.

Based on the correlation relationship between indicators of λ - fuzzy measure, let μ x i denote the fuzzy measure. Then, the fuzzy measure formula of A is as follows: μ A = 1 λ ∏ x j ∈ A 1 + λ μ x j − 1 , λ ≠ 0 ∑ x j ∈ A μ x j , λ = 0 (46)

Let A = X, then μ A = μ X = 1 can be obtained: λ + 1 = ∏ j = 1 m 1 + λ μ x j , − 1 < λ < ∞ , λ ≠ 0 (47)

The Shapley function formula based on the λ - fuzzy measure is as follows: g s g , X = ∑ T ⊆ X − S n − s − t ! t ! n − s + 1 ! μ A ∪ T − μ T (48) where X denotes the full set of indicators, S denotes a subset in X, X-S is the difference set between the set X and the set S, T denotes a subset in X-S, and n, T, S denotes the base of X, T, S.

The above study makes it clear that studying the relationship between the chemical composition of different glass artifact samples requires consideration of weathering. Hence, we address four categories.

(1) High-potassium glass in a weathered background

First, for weathered high-potassium glass, the standard values of individual chemical components are shown in Table 5, and the importance of each component is shown in Table 6. The calculated fuzzy measurements are shown in Table 7.

According to the fuzzy measurements in Table 12, the chemical composition of weathered high-potassium glass is not highly correlated, and its correlation is weak.

(2) Lead-barium glass in a weathered context

For weathered lead-barium glass, the standard values of the individual chemical compositions are shown in Table 8. The resulting calculated chemical compositional importance is shown in Table 9. The calculated fuzzy measurements are shown in Table 10.

According to the fuzzy measure, there is a medium degree of correlation for the chemical composition of weathered lead-barium glass: there is a correlation between the chemical content of silica, barium oxide, and lead oxide.

(3) High-potassium glass in an unweathered background

For unweathered high-potassium glass, the standard values for individual chemical compositions are shown in Table 11. The resulting calculated chemical compositional importance is shown in Table 12. The calculated fuzzy measurements are detailed in Table 13.

According to the fuzzy measurements in Table 13, it can be seen that the degree of correlation is not high for the chemical composition of unweathered high-potassium glass, but there is a correlation between the chemical content of silica and calcium oxide.

(4) Lead-barium glass in an unweathered background

For unweathered lead-barium glass, the standard values for the individual chemical constituents are shown in Table 14. The resulting calculated chemical compositional importance is shown in Table 15. The calculated fuzzy measurements are shown in Table 16.

Table 16 shows the degree of correlation for the chemical composition of unweathered lead-barium glass is high, and there is a certain correlation consistency relationship.

The Shapley function values for each category are then calculated according to Eq. 48. For weathered high-potassium glass, g 1 g , X = 0.305 ; for weathered lead-barium glass, g 2 g , X = 0.431 , for unweathered high-potassium glass, g 3 g , X = 0.394 , and for unweathered lead-barium glass, g 4 g , X = 0.497 .

It is clear that there is a difference between these correlations. The Shapley function values progressively decrease, and the correlations become weaker from unweathered lead-barium glass to weathered lead-barium glass to unweathered high-potassium glass and finally to weathered high-potassium glass.