Cluster analysis is a technique that groups similar data points into clusters based on some criteria, such as distance, density, or connectivity. But how do you decide how many clusters to use for your data set? Choosing the wrong number can lead to overfitting, underfitting, or misleading results. Learn methods and tips to find the optimal number of clusters for your data set.
The elbow method is one of the most common and simple methods to determine the optimal number of clusters. It involves plotting the sum of squared errors (SSE) or the within-cluster variation against the number of clusters. The SSE measures how close the data points are to their assigned cluster centers. Ideally, you want to minimize the SSE, but as you increase the number of clusters, the SSE will decrease until it reaches zero when each data point is its own cluster. Therefore, you need to look for the point where the SSE curve bends or forms an elbow, indicating that adding more clusters does not improve the fit significantly. The number of clusters at the elbow point is considered the optimal number.
The silhouette method is another popular and intuitive method to determine the optimal number of clusters. It involves calculating the silhouette coefficient for each data point and averaging them over all data points. The silhouette coefficient measures how well a data point fits in its assigned cluster compared to other clusters. It ranges from -1 to 1, where 1 means the data point is very similar to its own cluster and very dissimilar to other clusters, and -1 means the opposite. A high average silhouette coefficient indicates a good clustering quality. To find the optimal number of clusters, you need to plot the average silhouette coefficient against the number of clusters and look for the peak value.
The gap statistic method is a more sophisticated and rigorous method to determine the optimal number of clusters. It involves comparing the within-cluster variation of the actual data set with that of a reference data set generated by sampling from a uniform distribution. The idea is that the reference data set has no inherent clustering structure, so any clustering of it will have a higher within-cluster variation than the actual data set. The gap statistic measures the difference between the log of the within-cluster variation of the actual data set and the reference data set. To find the optimal number of clusters, you need to plot the gap statistic against the number of clusters and look for the point where the gap statistic is the largest or does not increase much.
While these methods can help you find the optimal number of clusters for your data set, they are not foolproof or definitive. They may give different results depending on the data set, the clustering algorithm, the distance metric, and the random seed. Therefore, you should always use your domain knowledge, visual inspection, and common sense to validate and interpret the results. Consider the purpose and context of your cluster analysis, such as whether you want to explore, describe, or predict your data, and what are the implications and limitations of your clustering choices.
Spectral clustering provides a fully unsupervised effective and interpretable method to determine the number of clusters by looking at gaps between the eigenvalues. Since the eigenvalues are being computed together with the eigenvectors needed for spectral clustering, this method doesn't require new computations. The gaps can determine not only the number of clusters but also a hierarchy of clusters. A lack of a noticable gap can serve as an indication that no high quality clusters can be found, suggesting changes in the setup of the matrix used for spectral clustering.
After running these methods, I'd recommend visualizing the clusters through plots. This may uncover that what the method recommended as "best" may not be optimal due to the sparsity and variance of your data. I'd also consider exploring other clustering algorithms that don't require a cluster number as an input parameter such as dbscan.