# AutoElbow: An Automatic Elbow Detection Method for Estimating the Number of Clusters in a Dataset
## 2 Related Work  
In this section, we discuss notable methods from the prior art for automatically determining the ideal elbow point of an elbow-based graph. We classify these approaches according to the properties of the elbow-based graph from which they were derived, namely the line fitting methods [12; 14; 16], the angle-based [15; 17], derivative-based [11], and the distance-based methods [13].  
The L-method is an early line fitting method that was proposed in [16] to detect the elbow-point automatically. It works by fitting two best lines to the graph points on either side of the point of interest. Then, the best lines that produce the smallest root mean square error (RMSE) are fitted to the graph, and the point where the two lines intersect is deemed to be the optimal knee point. On a wide variety of datasets, the L-method outperformed the gap statistics and permutation tests approaches.  
Additionally, the L-method has inspired other variations, such as the MATLAB implementation version in [12] and the modified method in [14]. In particular, the research in [PERSON] et al. [14] applied the method of line fitting for elbow point detection to the capacity fade curves of lithium-ion cells. All of these implementations have demonstrated the line-fitting method's efficacy and its capacity to improve performance in various application domains.  
The angle-based technique of [PERSON] et al. [17] computes the Bayesian Information Criterion (BIC) measure of any clustering algorithm under consideration and then computes the first order difference of the BIC measure. The point with the greatest BIC angle is estimated to be the knee point. By conducting experiments with the K-means algorithm, the authors demonstrated that their technique outperformed other existing index metrics as well as using the BIC measure alone.  
Similarly, the authors of the more recent article in [15] developed an angle-based discriminant technique. Unlike the method of [PERSON] et al. [17], the discriminant method computes the angles created by each point of the elbow-based graph instead of the BIC graph. Multiple datasets were then used to demonstrate that the discriminant method outperformed the gap statistics approach.  
On the other hand, due to the variances that may occur along the elbow-based graph, derivative-based approaches are not widely utilized in the literature. Nonetheless, one notable example was proposed in [11] using an elbow strength algorithm (ESA). Here, the ESA determines the elbow point by computing the difference between the first- and second-order differences of the elbow-based graph.  
The algorithm then considers the point with the largest difference (referred to as the elbow strength) to be the ideal location of the elbow. In contrast, the Kneedle algorithm proposed in [13] was used to implement a distance-based strategy. In this instance, the Kneedle algorithm computes the squared Euclidean distance between each point of the elbow-based graph and a straight line that encloses the graph's endpoints. The point with the largest distance is considered to be the ideal elbow point.  
Despite the fact that the approaches described above have yielded significant results in a variety of application areas, there is always room for improvement or a better approach. Improved approaches are needed, for example, when the elbow graph returns as a smooth and extremely curved graph, when the graph is significantly perturbed, or when the tail of the graph is either too short or too long. These are some of the challenging circumstances that prompted our proposed approach, which we will describe in the next section.