# Overlap in Automatic Root Cause Analysis in Manufacturing: An Information Theory-Based Approach
## 4 Proposed Methodology
### Proposed Method  
Given the measure defined above, we propose a new factor ranking algorithm to evaluate the overlap between tuples and detect the ones that are most likely to be root causes. Mutual information has already been used for factor ranking in previous studies (e.g., [45; 46; 47]). The proposed factor ranking algorithm is integrated into a method illustrated in Figure 2. As the manufacturing process is operating, data are extracted from it and are analyzed by a problematic-moment-identification algorithm, which tries to identify parts of data relating to periods where there were problems in the manufacturing process. It then selects those parts of the data, which are analyzed by a factor-ranking algorithm that outputs a list of the most likely root causes. The algorithm selected for the problematic moment identification stage was Exponentially Weighted Moving Averages (EWMA) [48]. In the proposed solution, EWMA is used to control the proportion of problematic products in a lot. This algorithm was selected after a thorough study analysing the performance of three control charts (EWMA, Cumulative Sum (CUSUM) control charts, and Shewhart charts). The variance to be used in EWMA was estimated using the proportion of problematic products per lot, modeled using the beta distribution. We estimated the \(\alpha\) and \(\beta\) parameters and then used them to compute the variance. The choice of using the beta distribution was based on its flexibility, which enables it to fit into the proportion of problematic products that appeared in the real case-study. In what concerns the novel factor-ranking algorithm proposed, the intuition is straightforward: the goal is to find the tuples that are overlapped with the tuple of the factor representing the final quality (e.g., the label) of a problematic product. In addition, this procedure makes it possible to determine the tuples that are overlapped among themselves and the label, finding clusters of machines that are related to problematic products. After the PMI is computed for all the interactions, the tuples presenting a PMI value in relation to a \"Label High\" tuple greater than the threshold are kept. These values are then  
Figure 2: Illustration of the proposed ARCA method. sorted in decreasing order and presented as a list to the analyst. Algorithm 1 summarises the novel factor ranking algorithm. ```
Input: Dataset of problematic moment; Threshold Output: Ordered list of most likely root causes begin Output \(\leftarrow\) [\(\varnothing\)];  To-use-dataset \(\leftarrow\) OneHotEncoding(Input);  Compute PMI Matrix of To-use-dataset; Output \(\leftarrow\) PMI Matrix's column related to problem w/ tuples' names;  Remove from Output values below Threshold;  Sort Output;  end while
```  
**Algorithm 1**PMI algorithm  
Considering the example in Table 3, the only tuples that overlap (i.e., have a PMI value above the threshold) with the problematic product tuple (\"Problem 1\") are the tuples \"A1\" and \"B3\" (excluding the self overlap of \"Problem 1\"). Of these two tuples, we can see that they also overlap each other, and that there exist no other overlap clusters among other tuples. Therefore, the tuples \"A1\" and \"B3\" would be presented to the analyst as the most likely root causes. To ease the detection of root causes by practitioners, and in addition to the list, we propose a visualization that depicts tuples as nodes and overlap between them as edges between the nodes. We opted to show directed edges to represent the order in which the steps occurred. The nodes are colored according to their steps, e.g., one color is used for Step04 and another for Step06. Figure 3 is an example of the visualization on a small part of the stochastic simulation experiment (different from the previous example). The visualization tool displays all the nodes and overlaps as a grid of nodes, but these can be moved to ease the perception of clusters. Specifically, we focus on the cluster that contains the node \"Label High\", which indicates a problematic product.