# Finite Mixture Model-Based Analysis of Yarn Quality Parameters
## 2 Materials and Methods
### Theoretical Background of Poisson Mixture Distribution
\[logL_{a}(\theta,y)=\sum_{i=1}^{n}\sum_{k=1}^{K}z_{ik}log\left\{w_{k}g_{k}(x_{ i};\lambda_{k})\right\} \tag{10}\]Since the missing data \(z_{ik}\) are unobserved, a closed-form solution for the MLE of \(\lambda\) does not exist. Instead of directly obtaining the optimal likelihood estimate, the EM algorithm employs two main steps: the Expectation (E) step and the Maximization (M) step. In the E-step, the expectation of the log-likelihood function is calculated with respect to the missing data to provide an estimated mean. In the E-step of the EM algorithm, it is necessary to compute the expected value of the augmented log-likelihood function, conditioned on the observed data and the current parameter estimates. This expectation is expressed as follows. In the E-step, it is necessary to compute the following Equation (11). \[Q\Big{(}\theta\Big{|}\theta^{(r)}\Big{)}=E\Big{\{}logL_{\text{a}}(\theta;y) \Big{|}x;\theta^{(r)}\Big{\}} \tag{11}\]  
where \(\theta^{(r)}\) represents the parameter estimates at the r-th iteration. The conditional expectation of the missing data \(z_{ik}\) is then calculated. Given that \(logL_{\text{a}}(\theta,y)\) is a linear function of the missing data \(z_{ik}\), the expected value is then substituted into \(logL_{\text{a}}(\theta;y)\). These expected values are given in Equation (12). \[p_{ik}^{(r+1)}=E\Big{(}z_{ik}\Big{|}x_{i},\theta^{(r)}\Big{)}=P\Big{(}z_{ik}=1 \Big{|}x_{i},\theta^{(r)}\Big{)}=\frac{w_{k}^{(r)}g_{k}(x_{i};\lambda_{k}^{( r)})}{\sum_{y=1}^{K}w_{y}g_{v}(x_{i};\lambda_{v}^{(r)})} \tag{12}\]  
where \(p_{ik}^{(r+1)}\) represents the posterior probability that the i-th observation belongs to the k-th Poisson component, based on the current parameter estimates \(\theta^{(r)}\). These probabilities are used in the M-step to update the parameter estimates, ensuring the iterative convergence of the algorithm. Table 1 presents the steps of the EM algorithm applied to the Poisson mixture distribution. In the E-step of the EM algorithm, the probability density function of the Poisson mixture distribution (as defined in Equation (1)) is substituted into Equation (13) to compute the value of \(\hat{p}_{ik}\). The resulting expression is given in Equation (16). \[\hat{p}_{ik}=\frac{\hat{w}_{k}(\frac{e^{-\lambda_{k}\lambda_{k}^{z_{k}}}}{x_{ i}!})}{\sum_{y=1}^{K}\hat{w}_{y}(\frac{e^{-\lambda_{0}\lambda_{k}^{z_{k}}}}{x_{ i}!})} \tag{16}\]  
where \(\hat{p}_{ik}\) represents the probability that the i-th observation belongs to the k-th Poisson component, given the current parameter estimates. \begin{table}
\begin{tabular}{c} i. Start with initial parameter values of \(\hat{\lambda}_{k}^{(r)}\), \(\hat{w}_{k}^{(r)}\) (\(r=0\)) \(k=1,\ \dots,\ K\) and iterate the following steps until convergence. \\ ii. E-step: Find the classification probability that the observation \(x_{i}\) comes from the \(k^{th}\) Poisson distribution based on the following estimate: \\ \(p_{ik}^{(r+1)}=\frac{\frac{w_{k}^{(r)}g_{k}(x_{i};\lambda_{k}^{(r)})}{\sum_{z=1 }^{K}w_{y}g_{v}(x_{i};\lambda_{v}^{(r)})}}{\sum_{y=1}^{K}w_{y}g_{v}(x_{i}; \lambda_{v}^{(r)})}\) \\   where \(i=1,\dots\ n\) and \(k=1,\ \dots,\ K\) \\ iii. M-step: Update the Poisson mixture distribution parameters as follows: \\ \(\lambda_{k}^{(r+1)}=arg\underset{\lambda_{k}}{\max}\ \underset{i=1}{\overset{n}{ \sum}}\ p_{ik}^{(r+1)}\text{log}\,g_{k}(x_{i}|\lambda_{k})\) \\ \(w_{k}^{(r+1)}=\frac{\sum_{y=1}^{n}p_{ik}^{(r+1)}}{n}\) \\ \end{tabular}
\end{table} 
Table 1: EM algorithm for the Poisson mixture distribution. In the M-step of the EM algorithm, Equations (14) and (15) represent a constrained optimization problem and can be expressed in the form of Equations (17) and (18). This optimization problem can be transformed into Equation (19) using the Lagrange multiplier (\(\delta\)). \[\hat{\lambda}_{k},\hat{w}_{k}=\underset{\lambda_{k},w_{k}}{\text{argmax}}Q( \lambda_{1},\,\ldots,\,\lambda_{k},w_{1},\,\ldots,\,w_{K}) \tag{17}\]  
s.t.