# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 3 Results and Discussion
### Distribution Fitting of Yarn Imperfection Parameters
Figure 4: Thick places: histogram of observed data and fitted Poisson mixture data. Figure 5: Neps (%140): histogram of observed data and fitted Poisson mixture data. Figure 6: Neps (%200): histogram of observed data and fitted Poisson mixture data. According to the Mann-Whitney U test results, all examined datasets produced \(p\)-values greater than 0.05. Specifically, the \(p\)-value was 0.26 for the thin places dataset, 0.49 for the thick places dataset, 0.75 for the reps 200% dataset, and 0.30 for the reps 140% dataset. These findings confirm that the Poisson mixture distributions effectively represent each dataset. The Poisson mixture parameters provided in Table 3 were applied in conjunction with Equations (30) and (34) to compute the mean and standard deviation for thin places, thick places, and reps. The computed values exhibit strong concordance with the observed data for elongation and tensile strength, as presented in Table 4. Thus, the strong agreement between the observed and modeled mean and standard deviation values further strengthens the reliability of the proposed Poisson mixture models. After establishing the validity of the mixed distribution using the Mann-Whitney U test, the probability functions were obtained by substituting the Poisson mixture distribution parameters provided in Table 3 into Equation (2). \[f(x)_{thin}=0.79\frac{11.64^{x}e^{-11.64}}{x!}+0.21\ \frac{35.68^{x}e^{-35.68}}{x!} \tag{68}\]  
\[f(x)_{thick}=0.073\frac{25.76^{x}e^{-25.76}}{x!}+0.21\frac{11.8^{x}e^{-11.8}}{ x!}+0.72\frac{4.42^{x}e^{-4.42}}{x!} \tag{69}\]  
Figure 1:0  
Figure 1:1  
Determining the probability distribution function of thin places (Equation (68)), thick places (Equation (69)), and reps (Equations (70) and (71)) provides significant benefits for optimizing quality control and enhancing production efficiency. The derived probability functions offer predictive insight into the likelihood of yarn imperfections, enabling more proactive and data-driven quality control practices. These models can be directly compared against specification limits defined for each imperfection type to estimate the probability of conformance. This capability can significantly enhance quality control mechanisms in production processes. To further evaluate the practical applicability of the proposed Poisson mixture models in quality control, conformance probabilities were calculated based on upper specification limits provided by the yarn manufacturer. These limits reflect the quality thresholds expected by the company's customers for each defect type--namely, thin places (\(\leq\)35), thick places (\(\leq\)10), reps at 140% sensitivity (\(\leq\)127), and reps at 200% sensitivity (\(\leq\)26). The probability of conformance was computed as the cumulative probability \(P(X\leq spec. )\) under the respective mixture distribution functions (Equations (69)-(72)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Quality** & \multicolumn{2}{c}{**Observed Data**} & \multicolumn{2}{c}{**PMM Data**} \\ \cline{2-5}
**Parameters** & **Mean** & **Standard Dev.** & **Mean** & **Standard Dev.** \\ \hline Thin Places (40\%) & 16.63 & 14.83 & 16.68 & 10.60 \\ Thick Places (50\%) & 7.52 & 6.67 & 7.54 & 6.51 \\ Neps (140\%) & 29.70 & 23.61 & 30.68 & 22.92 \\ Neps (200\%) & 11.55 & 9.61 & 11.63 & 8.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean and standard deviation of yarn imperfection parameters: observed data vs. Poisson mixture model data. The results reveal that the estimated conformance probabilities are 0.923 for thin places, 0.793 for thick places, 0.998 for reps at 140%, and 0.892 for reps at 200%. These values indicate that, while reps at 140% and thin place imperfections are largely within acceptable bounds, reps at 200% and thick places exhibit relatively lower conformance levels. This insight can help quality engineers prioritize improvement efforts and implement corrective measures more effectively for specific defect categories.