# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 3 Results and Discussion
### Distribution Fitting of Yarn Mechanical Properties  
Following the comprehensive evaluation of yarn imperfection parameters, the subsequent analysis focused on investigating the mechanical properties of the yarn, particularly tensile strength (tenacity) and elongation. This investigation was conducted under the assumption that the underlying data might follow a specific statistical distribution. However, the application of the chi-square goodness-of-fit test indicated that the datasets related to these mechanical properties did not conform to any single distribution, suggesting a more complex underlying structure in the observed data. Figures 7 and 8 illustrate the histogram of the respective datasets, revealing multiple peaks, which suggest that the data may follow a mixed distribution. Each dataset was modeled using gamma mixture distributions, with parameters estimated through the EM algorithm. A summary of these estimation results is provided in Table 5. The proportions and parameters of each gamma mixture distribution were determined using the Flexmix package in the R programming environment. A comparison of the histograms representing the observed data with the probability density functions derived from the fitted gamma mixture models is illustrated in Figures 9 and 10, which clearly demonstrates a strong degree of agreement between the empirical data and the models. Such visual consistency underscores the assertion that the proposed gamma mixture models adequately capture the underlying distribution of the observed data for both tensile strength and elongation. Figure 7: Histogram of elongation (%). In addition to visual assessments, the Mann-Whitney U test was employed to statistically analyze the differences between the observed and modeled distributions. The results of this non-parametric test indicated that all examined datasets produced \(p\)-values exceeding the conventional threshold of 0.05. Specifically, the \(p\)-value for the tensile strength dataset was 0.69, while the elongation dataset produced a \(p\)-value of 0.40. These findings imply that there are no significant differences between the distributions of the observed data and those modeled by the gamma mixture framework. The parameters of the gamma mixture models, detailed in Table 5, were employed alongside Equations (64) and (66) to calculate the mean and standard deviation for both elongation and tensile strength. The computed values from these equations indicate a close alignment with the observed data, which are documented in Table 6. This congruence  
Figure 8: Histogram of tensile strength (cN/tex). Figure 9: Tensile strength observed data and fitted gamma mixture data. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multirow{2}{*}{**k**} & \multicolumn{2}{c}{**Proportions**} & \multicolumn{2}{c}{**Gamma Mixture Distribution**} & \multirow{2}{*}{**Selection Criteria**} \\  & & & \multicolumn{2}{c}{**Parameters**} & & & \\ \cline{3-8}  & & \(\pi_{1}\) & \(\pi_{2}\) & \((\pi_{1},\beta_{1})\) & \((\pi_{2},\beta_{2})\) & **AIC** & **BIC** & **LL** \\ \hline Elongation & 2 & 0.33 & 0.67 & (320, 0.04) & (110, 0.146) & 3121.86 & 3145.06 & \(-\)1555.93 \\ \hline Tensile Str. & 2 & 0.52 & 0.48 & (120, 0.17) & (280, 0.09) & 3825.70 & 3848.90 & \(-\)1907.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Gamma mixture distribution parameters for each dataset. reinforces the validity of the gamma mixture models in accurately capturing the statistical characteristics of the underlying datasets. Following the validation of the mixed distribution, the corresponding probability functions for elongation and tensile strength were derived by substituting the gamma mixture distribution parameters from Table 5 into Equation (35). The resulting probability density functions are given in Equations (72) and (73).