# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 3 Results and Discussion  
### Distribution Fitting of Yarn Imperfection Parameters  
The results of the chi-square (\(\chi^{2}\)) goodness-of-fit test revealed that the observed data for thin places, thick places, and neps in the yarn samples did not conform to standard single distributions, such as binomial, geometric, or Poisson distributions. This finding strongly suggests that these parameters follow a more complex, heterogeneous pattern that can be better described using mixture models. The histograms of the observed data further supported this conclusion by showing multiple peaks, indicative of an underlying mixture of distributions (Figures 1 and 2). Poisson mixture distribution was applied to model each dataset, with parameters estimated using the EM algorithm. The results of these estimations are summarized in Table 3. The proportions and parameters of each Poisson mixture distribution were calculated using the Flexmix (version 2.3-19) package in the R (version 4.1.0) programming environment. These results indicate that a two-component Poisson mixture distribution is sufficient for modeling thin places and reps (200%), while a three-component distribution better fits the data for thick places and reps (140%). The estimated proportions and distribution parameters provide valuable insights into the occurrence of imperfections in the yarn. * Thin places: The analysis identifies a high-proportion component (0.79) with a mean of 11.64. This indicates that most thin places are relatively minor deviations, with occasional severe irregularities (mean of 35.68). Figure 1: Histograms of thin and thick places. Figure 2: Histograms of reps (140%, 200%). * Thick places: The three components represent varying degrees of thickness irregularities, with the largest proportion (0.72) attributed to minor variations. The presence of more extreme thick places (means of 25.76 and 11.80) highlights potential process inefficiencies, potentially attributable to such factors as inappropriate tension settings or irregular fiber feeding. * Neps: The data indicate that across both the 140% and 200% sensitivity levels for neps, there is a mixture of mild, moderate, and severe instances. The extreme values identified could be indicative of inconsistencies in fiber preparation or during spinning processes. Such irregularities are often linked to fiber contamination or processing errors that can lead to significant adverse effects on the overall fabric quality. The validity of the Poisson mixture models was evaluated through both visual inspection and statistical testing using the Mann-Whitney U test. The same dataset used for parameter estimation was also used for this evaluation. A close agreement between the observed data and the fitted models was observed, as evidenced by the overlay of histograms of the observed data with the probability density functions derived from the estimated models. These visual comparisons are presented for thin places, thick places, and both neps (140% and 200%) in Figures 3-6, respectively. An even more reliable method for determining whether the observed data and the data generated from the distribution come from the same population is to perform the Mann-Whitney U test. The null hypothesis (H\({}_{0}\)) of the Mann-Whitney U test suggests that the two groups are drawn from the same population. In other words, it proposes that the two independent groups are homogeneous and follow the same distribution. The alternative hypothesis (H\({}_{1}\)), which is tested against the null hypothesis, suggests that the distribution of data in the first group differs from that of the second group [37]. Thus, the hypotheses are as follows:  
**H\({}_{0}\). ** _There is no significant difference between the distribution of the observed dataset and the Poisson mixture distribution._  
**H\({}_{1}\). ** _There is a significant difference between the distribution of the observed dataset and the Poisson mixture distribution._  
Figure 3: Thin places: histogram of observed data and fitted Poisson mixture data.