# Quality-Aware Resource Model Discovery
## 3 Quality-Aware Resource Model Discovery
### Statistical Analysis on Resources  
As discussed in Section 1, the discovered integrated quality-aware resource model has a limitation in that it has a high complexity as the process gets more extended or more resources are engaged in activities. To overcome this limitation, we need to find out mainline states and transitions that have a statistical significance to quality and simplify it. This section suggests a series of procedures by exploiting a couple of statistical techniques. In a nutshell, all states engaged in the same activity are compared to each other, and the statistical significance of states is investigated. The followings are the details of the proposed method. We first testify the normality of distributions of all states involved in the same activity. As with Section 3.3, we employ the _Kolmogorov-Smirnov one sample test_[22], and the null (\(H_{0}\)) and the alternative (\(H_{1}\)) hypothesis are defined such that \(F_{n}(X)=F(X,\mu,\sigma)\) and \(F_{n}(X)\
eq F(X,\mu,\sigma)\), respectively. In accordance with the normality testing result, further steps are divided into two different directions. Though they exploit different statistical techniques, each step of them has common objectives. First, it is analyzed to identify whether the states engaged in the same activity have a discrepancy with each other. After that, if the difference exists, what states make the quality of products better or worse. When case distributions of all states follow the normal distribution, _Analysis of variance_, i.e., _ANOVA_[24], is applied to identify whether all states have the same quality value. ANOVA testing has the null (\(H_{0}\)) and the alternative (\(H_{1}\)) hypothesis such that; \(H_{0}\) is that the means of all groups under consideration are equal, and \(H_{1}\) is the means are not all equal. If the null hypothesis is accepted, all relevant states are classified as _moderate_, i.e., \(M\), whereas when rejected, it is connected to the next phase. The following step is _Linear contrast_[25] that is effective to testify whether the distribution of a particular group is different from other groups. Here, contrast signifies the linear combination of variable coefficients, and their sum becomes zero. Assume that we build a contrast using four sample groups, i.e., X\({}_{1}\), X\({}_{2}\), X\({}_{3}\), and X\({}_{4}\). Then, the coefficient of the target group (\(\overline{X_{1}}\)) becomes 3, whereas the others have minus 1 (\(\overline{X_{2}},\overline{X_{3}},\overline{X_{4}}\)) as coefficients. That is, the contrast L is defined such that \(3\overline{X_{1}}-\overline{X_{2}}-\overline{X_{3}}-\overline{X_{4}}\). The null hypothesis of linear contrast testing for the contrast L is defined as \(H_{0}:L=0\), while the alternative one, i.e., H\({}_{1}\), becomes \(L\
eq 0\). If the \(H_{0}\) is accepted, the corresponding state is classified as _moderate_, i.e., \(M\). In the opposite case, i.e., \(H_{0}\) is rejected, we conduct the further analysis for identifying whether the average of the state is greater or less than the average of the other states. As a result, if greater, the relevant state is classed as _superior_, i.e., \(S\), otherwise it is labeled as _inferior_, i.e., \(I\). If any of the states do not have normality, it is required that a _Kruskal-Wallis H-Test_[26] is employed instead of ANOVA. The overall approach is quite similar to ANOVA testing, and the null (\(H_{0}\)) and the alternative (\(H_{1}\)) hypothesis are defined as follows; \(H_{0}\) is that all populations have the same distribution, and \(H_{1}\) is that not all populations have the same distribution. After that, as a substitute for linear contrast, a _Mann-Whitney U Test_[26] is applied, which has the null and the alternative hypothesis as follows; \(H_{0}\) is that the two populations are equal, and \(H_{1}\) is that the two populations are not equal. Different from the linear contrast, the distribution of the target state and the others are compared with a rank-based approach.