## 6 Methodology
### Cauterization Analysis and Discussion
The reachability distance is the minimum distance required to reach a point from a core point. OPTICS creates a hierarchical cluster ordering based on the reachability distances, allowing the algorithm to identify clusters of varying densities and shapes. One advantage of DBSCAN is that it is relatively fast and efficient for small to medium-sized datasets with well-defined clusters. It is also easy to implement and does not require setting many parameters. However, it can struggle with datasets that have clusters of varying densities, shapes or sizes. OPTICS, on the other hand, can handle datasets with varying densities and shapes more effectively than DBSCAN. It also provides a hierarchical clustering ordering that can be useful for data exploration and visualization. However, OPTICS can be more computationally expensive than DBSCAN and may require more parameter tuning. The imagery provided in Figure 22 expresses the properties of the hourly status vectors in OP3, downsampled by 20. It shows the dependence of the number of DBSCAN clusters on the normalized radius of DBSCAN. Downsampling refers to reducing the number of data points in a dataset. In this case, the hourly status vectors in OP3 have been downsampled by a factor of 20. This means that only one data point out of every twenty is included in the analysis. Figure 22: Properties of the hourly status vectors in OP3, downsampled by 20: dependence of the no. of DBSCAN clusters vs. normalized radius. The normalized radius refers to the epsilon (\(\epsilon\)) value used in the DBSCAN algorithm, but scaled to a range between 0 and 1. This makes it easier to compare the results across different datasets or experiments. Dependence of the number of clusters on the radius: The graph shows how the number of clusters identified by DBSCAN changes as the normalized radius (\(\epsilon\)) increases. With a smaller \(\epsilon\) (represented by a lower value on the x-axis), the data points must be very close together to be considered in the same cluster. This typically leads to a larger number of smaller clusters. As \(\epsilon\) increases (higher values on the x-axis), more distant points can be included in a cluster, potentially resulting in fewer but larger clusters. As we can see from the imagery, the x-axis represents the normalized radius, which controls how tightly or loosely clusters are formed. The y-axis represents the number of clusters identified by DBSCAN. As the normalized radius increases, the clusters become more relaxed and fewer, larger clusters are formed. This is because more points are considered neighbors, leading to larger cluster mergers. Conversely, with a smaller normalized radius, only very close points are considered neighbors, resulting in more numerous, smaller clusters. The work point in the image refers to a specific operating condition or state of interest in OP3. The location of the work point on the curve indicates the number of clusters obtained for that specific operating condition using a chosen normalized radius. Figure 23 depicts the relationship between the normalized radius and the mean number of DBSCAN core points in a dataset. The normalized radius is obtained by dividing Eps by the characteristic distance scale of the data. This allows for comparing clustering results across datasets with different units or scales. As the normalized radius increases, the clusters become more relaxed and fewer, larger clusters are formed. This is because more data points are considered neighbors of a given point, potentially causing smaller, denser clusters to merge into larger ones. Consequently, the average number of core points generally decreases. Conversely, with a smaller normalized radius, only very close points are considered neighbors. This can lead to the separation of data points into more numerous, smaller clusters, potentially increasing the number of core points (since more data points might be central to smaller clusters). This behavior aligns with the expected characteristics of DBSCAN clustering.