# How to compare two clustering solutions when their labelling differs

I am planning to test the reliability of a clustering approach for some data. My plan is to repeatedly (with replacement) draw a number of random subsample pairs (e.g. 2x 10% of the total data), run the clustering on both individually, and then compare the results. The issue is that I am using HDBSCAN, which not only creates a non-fixed number of clusters (for different sets of data but same params), but it also therefore labels clusters differently since k is not defined, and the input data will always have slightly different structure due to variability.

I tested this by using the same HDBSCAN parameters on two subsamples (A, B) of my data, and my issue is quite easy to see. The cluster labels with corresponding samples for A were:
{-1: 4306, 0: 1737, 1: 2999, 2: 72068, 3: 20628, 4: 3120}

while for B they were:
{-1: 4478, 0: 1711, 1: 3048, 2: 72089, 3: 3123, 4: 20408}.

From this, it seems that the solution is very close until we compare label 3. It looks like label 3 of A corresponds to label 4 of B.

My initial thought was that I could just relabel them both in order of each cluster's sample size. But this assumes that the two solutions will be similar across many tests (which is ultimately the whole point of the testing in the first place). So my next thought is I could set the constraints that (1) there should be a "similar" number of samples in the noise group, and (2) there should be the same number of clusters found. If these two conditions are met then I could relabel the clusters by order of their sample size, and then make my comparison using ARI or AMI.

I am doubtful that this is good, because I don't believe it is necessarily true that (even given the two constraints) two clusters labelled the same on the basis of their sample should necessarily correspond to the same "global" cluster. It therefore seems problematic to me but I can't think of an alternative.

Is the above approach generally reasonable? If not, is there something else I could do to assess the reliability/stability of HDBSCAN solutions? As an alternative, would it be better to just compute the DBCV score, %noise, and the number of clusters, and then use this as an indication of the quality of the clustering?

• It looks like you need a matching/alignment method between clsuters before being able to compare the clusterings. I didn't understand whether the instances are the same for A and B? If they are the simple matching method is to maximize the number of instances in common. – Erwan Jul 19 at 18:45
• A and B will be completely different subsamples from the total dataset, while the clustering algorithm and parameters will be identical each time. Is this what you meant? – fffrost Jul 19 at 18:56
• yes that's what I meant. As far as I understand you need to find which cluster in A corresponds to which cluster in B, i.e. an alignment between the clusters labels of A and of B. It's not recommended to match based only on the size, since it could happen by chance that a cluster in A has the same size as a cluster in B. Since the instances are different, you would have to rely on the features. – Erwan Jul 19 at 20:59
• I see, that was my worry. Could you please elaborate on what you mean by relying on the features? – fffrost Jul 19 at 22:14
• actually I said features because I'm more familiar with probabilistic clustering methods, which represent each cluster as a distribution over the features. I'm not familiar with HDBSCAN so I don't know how the clusters are represented inside the model: whatever this is, the idea would be to compare each internal representation of a cluster in A vs. the same in B. For example with k-means one would compare the centroids and match the pairs of clusters for which the distance is the shortest. – Erwan Jul 19 at 23:04