# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 1 Introduction  
Effective data mining methods are needed to access valuable information and knowledge hidden within large heterogeneous data groups encountered in various fields, such as business, government, engineering, and health. These heterogeneous populations are typically composed of subpopulations, often referred to as components [1, 2]. Finite mixture models (FMMs), which provide a flexible and powerful probabilistic approach, have been used by researchers in many fields to analyze heterogeneous populations due to their well-principled approach to clustering. Known as unsupervised learning models, they can also be used in latent class analysis, image analysis, survival analysis, disease mapping, and meta-analysis. Fields in which mixture models have been successfully applied include, but are not limited to, chemistry [3], engineering [4], criminology [5], psycho-oncology [6], psychiatry [7], education [8], health and medical sciences [9, 10, 11, 12], and acoustic signal processing and sonar data analysis [13]. However, despite their wide range of applications in the aforementioned fields, the use of mixture distributions in the textile industry remains relatively limited. In view of thetextile literature, it is observed that the FMM is mostly applied to generate the cotton fiber length probability density function. Notably, an FMM based on two Weibull distributions has been proposed as an effective method for this purpose [14]. This approach was further validated by [15; 16], both of which applied a two-component Weibull mixture model to represent fiber length distribution. Additionally, the normal distribution and the power function were introduced as part of the FMM framework to derive the probability density functions for carded and combed cotton fibers, utilizing parameters derived from fiber length measurements [17]. The characterization of cotton fiber length distribution using an FMM is also addressed in [18]. Based on the characteristics of fiber length histograms, a mixture of Weibull distributions was used to generate a parametric continuous function. This model was applied to cotton fiber samples collected from different spinning processes (bale, card mat, carded sliver, combed sliver, and finished sliver) to describe the fiber length distribution. Collectively, these studies highlight the FMM as a reliable and robust method for generating cotton fiber length probability density functions. Building on these findings, in this study, we investigate the effectiveness of FMMs in modeling the probability distributions of yarn quality parameters, including yarn imperfection and mechanical parameters. Yarn imperfection parameters include the total number of neps and thick and thin places within a specified length of yarn. In yarn manufacturing, \"thick and thin places\" are irregularities where the yarn's diameter varies significantly. Thin places indicate reductions in mass, while thick places signify increases in mass [19]. More specifically, thick places are yarn defects identified by a diameter larger than that of adjacent sections, extending approximately 6 mm (1/4 inch) in length, and are primarily due to inadequate fiber drafting during yarn formation. Conversely, thin places are defects marked by segments at least 25% smaller in diameter than the average yarn diameter, usually resulting from inconsistencies in sliver or yarn count [20]. Another imperfection parameter is called neps, and is defined as small fiber clusters that become entangled within the yarn and enlarge its diameter, which can negatively influence the fabric's visual appeal and tactile quality [21]. The presence of neps adversely affects both the appearance of the yarn and the fabrics produced from it, impacting their smoothness and the evenness of dye application [22]. The reason for the negative impact on the appearance of dyed or printed fabrics is typically the absorption of dye by neps. The low cellulose content of the immature fibers reflects light differently, resulting in white spots or flecks on finished fabrics [23].