The first method discussed here is one of the most commonly used – K-means. K-means clustering is a method that splits the initial set of points (objects) into groups, using distance measure, representing a distance from the given point of the group to the group's centre representing a group's prototype: centroid. The result of the clustering is N points grouped into K clusters, where each point has assigned a cluster index, which means that the distance from the point of the cluster centroid is closer than the distance to any other centroids of other clusters. Distance measure employs Euclidian distance, which requires scaled or normalised data to avoid the dominance of a single dimension over others. The algorithm steps schematically are represented in the following figure 1:

In the figure:

• STEP 1: Initial data set where points do not belong to any of the clusters.
• STEP 2: Cluster initial centres are selected randomly.
• STEP 3: For each point, the closest cluster centre, which is the point marker, is selected.
• STEP 4: Cluster mark is assigned to each point.
• STEP 5: The initial cluster centre is being refined to minimise the average distance to it from each cluster point. As a result, cluster centres might no longer be physical points; instead, they become imaginary.
• STEP 6: Cluster marks of the points are updated.

Steps 4-6 are repeated until changes in cluster positions do not change or changes are insignificant. The distance is measured using Euclidian distance:

where:

• Data points - points {x_i}, i = 1, … ,N in multi-dimensional Euclidian space i.e. each point is a vector.
• K – number of clusters set by the user.
• r_nk – an indicator variable with values {0,1} – indicates if data point x_n belongs to cluster k.
• m_k – centroid of kth cluster.
• D – Squared Sum of all distances di to their assigned cluster centroids.
• Goal is to find such values of variables r_nk and m k to minimise D.

Example of initial data and assigned cluster marks with cluster centres after running the K-means algorithm (figure 3):

Unfortunately, the K-means algorithm does not possess automatic mechanisms to select the number of clusters K, i.e., the user must set it. Example of setting different numbers of cluster centres (figure 4):

In K-means clustering, a practical method – the Elbow method is used to select a particular number of clusters. The elbow method is based on finding the point at which adding more clusters does not significantly improve the model's performance. As explained, K-means clustering optimises the sum of squared errors (SSE) or squared distances between each point and its corresponding cluster centroid. Since the optimal number of clusters (NC) is not known initially, it is wise to increase the NCs iteratively. The SSE decreases as the number of clusters increases because the distances to the cluster centres also decrease. However, there is a point where the improvement in SSE diminishes significantly. This point is referred to as the “elbow” ^[1].

Steps of the method:

1. Plot SSE against the number of clusters:
   • Computing the SSE for different values of NC, typically starting from NC=2 reasonable maximum value (e.g., 10 or 20).
   • Plotting the SSE values on the y-axis and the number of clusters NC on the x-axis.
2. Observe the plot:
   • As the number of clusters NC increases, the SSE will decrease because clusters become more specialised.
   • Initially, adding clusters will result in a significant drop in SSE.
   • After a certain point, the reduction in SSE will slow down, not showing a significant drop in the SSE.
3. The “elbow” point:
   • The point on the curve where the rate of decrease sharply levels off forms the “elbow.”
   • This is where adding more clusters beyond this point doesn't significantly reduce SSE, indicating that the clusters are likely well-formed.
4. Select optimal NC:
   • The value of NC at the elbow point is often considered the optimal number of clusters because it balances the trade-off between model complexity and performance.

Since the method requires iteratively running the K-means algorithm, which might be resource-demanding, a selection of data might be employed to determine the NC first and then used to run the K-means on the whole dataset.

Limitations:

• The elbow point is not always obvious; in some cases, the curve may not show a distinct “elbow.”
• The elbow method is heuristic and might not always lead to the perfect number of clusters, especially if the data structure is complex.
• Other methods, like the Silhouette score, can complement the elbow method to help determine the optimal NC.

The figure above (figure 5) demonstrates more and less obvious “elbows”, where users could select the number of clusters equal to 3 or 4.

The Silhouette Score is a metric used to evaluate the quality of a clustering result. It measures how similar an object (point) is to its own cluster (cohesion) compared to other clusters (separation). The score ranges from −1 to +1, where higher values indicate better-defined clusters ^[2].

The Silhouette score considers two main factors for each data point:

• Cohesion (a(i)) - The cohesion measure for the ith point is the average distance between the point and all other points in the same cluster. It measures the point's proximity to other points in its cluster. A low a(i) value indicates that the point is tightly grouped with other points in the same cluster.
• Separation (b(i)) – The separation measure for the ith point estimates the average distance between the point and points in the nearest neighbouring cluster - the cluster that is not its own but is closest to it. A large value for b(i) indicates that the point is far away from the closest other cluster, meaning it is well-separated.

The silhouette score for a point i is then calculated as:

where:

• s(i) is the silhouette score for point i.
• a(i) is the average distance from point i to all other points in the same cluster.
• b(i) is the average distance from point i to all points in the nearest other cluster.
• s(i) ≈+1 indicated that the point i is well clustered.
• s(i) around 0 indicates that the point lies close to the boundary between clusters.
• s(i) ≈-1 indicated that the point i was most probably wrongly assigned to the cluster.

Steps of the method:

1. Plot silhouette score (SC) against the number of clusters:
   • Computing the SC for different values of NC, typically starting from NC=2 reasonable maximum value (e.g., 10 or 20).
   • Plotting the SC values on the y-axis and the number of clusters NC on the x-axis.
2. Observe the plot:
   • As the number of clusters NC increases, the SC shows different score values, which may or may not gradually decrease, as in the case of the “elbow” method.
   • The main goal is to observe the maximum SC value and the corresponding NC value.
3. Select optimal NC:
   • The value of NC at the maximum SC value is often considered the optimal number of clusters because it balances the trade-off between model complexity and performance.

Limitations:

• It may not perform well if the data does not have a clear structure or if the clusters are of very different densities or sizes.
• The Silhouette score might not always match intuitive or domain-specific clustering insights.

An example is provided in the following figure 7:

The user should look for the highest score, which in this case is for the 3-cluster option.