# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 2 Materials and Methods
### Theoretical Background of Poisson Mixture Distribution  
As a statistical modeling tool, FMMs are widely utilized for analyzing diverse types of random phenomena and can effectively represent complex probability distributions. Mixture models are particularly suitable for data characterized by the following properties: the dataset is derived from a mixture of different populations, and the distribution of each underlying component within the mixture is unknown [18]. Mathematically, FMMs are often expressed as a weighted sum of probability density functions from individual components within the mixture. Assuming that a random variable x is obtained from the mixture distributions, the probability density function f can be defined as expressed as in the following Equation (1). \[f(x;\theta)=\sum_{k=1}^{K}w_{k}g_{k}(x,\lambda_{k}) \tag{1}\]  
This is called a  
In the context of MLE for Poisson mixture distributions, the estimate \(\hat{\theta}\) represents a sequence of roots for the log-likelihood equation, which is expressed mathematically as follows:  
\[\frac{\partial\mathbb{J}(\theta)}{\partial\theta}=0 \tag{6}\]  
Equation (6) indicates that the score function, which is the gradient of the log-likelihood function with respect to the parameters \(\theta\), must equal zero at the maximum likelihood estimates. The score functions derived from Equation (6) are as follows:  
\[\frac{\partial\mathbb{J}(\theta)}{\partial\lambda_{k}}=\sum_{i=1}^{n}\frac{w_ {k}g^{\prime}(x_{i},\lambda_{k})}{\sum_{k=1}^{K}w_{k}g_{k}(x_{i};\lambda_{k})} \qquad k=1,\ \ldots,\ K \tag{7}\]  
where \(g^{\prime}(x_{i},\lambda_{k})\) represents the derivative of the Poisson density function with respect to the parameter \(\lambda_{k}\). Equation (8) is as follows:  
\[\frac{\partial\mathbb{J}(\theta)}{\partial w_{k}}=\sum_{i=1}^{n}\frac{g_{k}(x _{i};\lambda_{k})-g_{k}(x_{i};\lambda_{K})}{\sum_{k=1}^{K}w_{k}g_{k}(x_{i}; \lambda_{k})} \tag{8}\]  
Equation (8) reflects the sensitivity of the log-likelihood to changes in the mixing proportions \((w_{k})\). It compares the contribution of the k-th component to the overall mixture likelihood against the contributions from the other components. Since a closed-form solution for the maximum likelihood estimation (MLE) of Poisson mixture distributions is not available, the expectation-maximization (EM) algorithm, originally developed by [PERSON] et al. [33], is employed to estimate the parameters of the mixture distribution. The EM algorithm is an iterative computational method used to determine the maximum likelihood or maximum a posteriori estimates of parameters in statistical models. It involves defining missing data, representing incomplete information that can be assumed to simplify the model and its likelihood function. The details of the EM algorithm for the Poisson distribution have been demonstrated by numerous researchers. Let \((x_{1},\ \ldots,\ x_{n})\) represent an independent and identically distributed (i.i.d.) random sample drawn from the Poisson mixture distribution defined in Equation (1). The likelihood function of \(f(x;\theta)\) is expressed as Equation (4). For computational convenience, it is more practical to utilize the following log-likelihood function (see Equation (9)), which was presented in [34]:  
\[\log L(x;\theta)=\mathbb{J}(\theta)=log\prod_{i=1}^{n}\sum_{k=1}^{K}w_{k}g_{k} (x_{i},\lambda_{k})=\sum_{i=1}^{n}log\left\{\sum_{k=1}^{K}w_{k}g_{k}(x_{i}; \lambda_{k})\right\} \tag{9}\]  
In the context of the Poisson mixture model, the missing data can be defined as follows:  
\[z_{ik}=\left\{ \begin{array}{cc}1,&\text{if }i\text{th observation is from }k\text{th Poisson distibution}\ \ \ \text{i}=1,\ \ldots,\ \text{n and j}=1,\ \ldots,\ \text{k}\\ 0,&\text{other wise}\\ z_{i}=(z_{i1},\ \ldots,\ z_{ik})^{T}\end{array} \right.\]  
By incorporating the missing data \(z_{ik}\), the augmented data can be defined as \(y=(y_{1},\ \ldots,\ y_{n})\), where \(y_{i}=(x_{i},z_{i}^{T})\). The augmented log-likelihood function is then expressed by Equation (10).