A More Precise Elbow Method for Optimum K-means Clustering

TL;DR

This paper introduces a mathematically precise and universally applicable enhancement to the elbow method for determining the optimal number of clusters in K-means clustering, replacing the heuristic with an objective algorithm.

Contribution

It develops an analytical geometric and derivative-based algorithm to accurately identify the elbow point, improving reliability and applicability over traditional visual methods.

Findings

The new method precisely measures the elbow point without complex calculations.

It is universally applicable to different graph behaviors.

The approach enhances the reliability of K-means cluster number determination.

Abstract

K-means clustering is an unsupervised clustering method that requires an initial decision of number of clusters. One method to determine the number of clusters is the elbow method, a heuristic method that relies on visual representation. The method uses the number based on the elbow point, the point closest to 90 degrees that indicates the most optimum number of clusters. This research improves the elbow method such that it becomes an objective method. We use the analytical geometric formula to calculate an angle between lines and real analysis principle of derivative to simplify the elbow point determination. We also consider every possibility of the elbow method graph behaviour such that the algorithm is universally applicable. The result is that the elbow point can be measured precisely with a simple algorithm that does not involve complex functions or calculations. This improved method gives an alternative of more reliable cluster determination method that contributes to more optimum k-means clustering.