# A Statistical Approach to Discovering Process Regime Shifts and Their Determinants
## 4 Case Studies
### Case 1: Manufacturing Quality  
The first case involves a technological process. More precisely, we evaluate the output of a _turning process_, which originates from a simple _lathe machine system_. This system comprises several interacting components (or _things_) that include structural body parts, a cutting tool, a motor and a steel rod vised and rotated for cutting. Although several basic and emergent properties can be identified for this system, the _dependent property_ of interest is the rod diameter that changes when the turning process is enacted. The _aim_ of the process is to reduce the diameter of the rod to a specified level (Figure 4). This process is continuously repeated to manufacture several hundred rods. After a rod is produced, the reduced diameter is logged and checked to ensure its acceptability. As the process is prone to tool breakage (tool length being an _internal determinant property_), which causes a sudden increase in the diameters being produced. The intention is to catch such a regime shift in the diameter data, followed by the needed tool replacement. We ignore any external environmental determinants (e.g., room temperature) in this analysis for simplicity. This case is selected mainly for two reasons. First, it is a simple technological process that is easily visualizable, involving just one dependent property, i.e., the diameter of the rod, and one independent internal process determinant, i.e., the tool length. Second, as we exactly know when the tool was broken (nearly 5 mm tool tip lost when the 80 th rod was processed), causing a regime change in our diameter log at the corresponding time, we use this data to validate the efficacy of the methodology. The time series event log for this case is presented in Figure 5, which reports data for 120 steel rods produced in a sequence. The intended reduced diameter is 100 mm. After the occurrence of tool breakage, an increase in the diameter at the 80 th rod can be seen. As the tool continues to be used, an increase of approximately 5 mm is clearly visible in the diameter of the rest of the rods, showing the same breakage length as that which happened with the tool. For this problem, the determinant data was tool length, as recorded after every cutting run. To test and validate the proposed methodology, the criterion used is the _level_ basis, in which the minimal base model via Equation (1a) and Equation (1b) is simply reduced to \(p_{i}^{\prime}=\varepsilon_{i}\), where \(p_{i}^{\prime}\) represents the diameter of the rod. Using this minimal base model, we tested the rod diameter time series data for regime changes. The results from both statistical tests are shown in Figure 6a,b, respectively, which clearly show a deviation and a peak well outside the critical band (indicated by red lines) at around the 80 th data point, indicating at least one regime shift. We then moved to the steps for dating the shift(s) and producing the corresponding segmented base minimal models. The results for both these aspects are shown in Figure 7, Tables 1 and 2. First, it indicates that at a confidence level of 97.5%, the shift occurred between rod number 79 and 81 (most likely at the 80 th rod). Second, it shows that two data segments are created, i.e., the first time series segment covering rods 1 to 79 and the second segment covering rods 80 to 120. The fitted model segments (Table 2) are also graphically depicted in Table 1, which shows the average diameter in the first segment to be around 100.05 mm and 105.12 in the second segment. A single level-model fitting the whole data is also shown (grey dashed line) in the figure to demonstrate why it is important to identify various regimes and have separate models for each. Figure 4: Rod diameter reduction via turning manufacturing process. Figure 5: Steel rod diameter data. Figure 6: Statistical test results (minimal base model). (**a**) Empirical Fluctuation (**b**) F-Statistics. Figure 7: Break date and segmented model.