# Hierarchical Clustering: What is the difference between linkages and distance measures?

Clustering algorithm defines a particular distance (correlation or euclidean) and a linkage (which, strangely some books call distance - single, complete, average or centroid). Conceptually, correlation or euclidean distance measure distance between two points (but not clusters, perhaps); linkages measure distance between one cluster and other clusters (or points).

So, when the algorithm is applied, how does it matter what distance (correlation/euclidean) I choose, if the dissimilarity and subsequent grouping done only on the basis of linkages?

I know the distance choice matters because it gave me a different answer and dendograms for both measures.

• Distance measure is a formula how to define, compute adistance between two points. Linkage method in HCA is how to compute the distance (whichever selected) in case where one or both of the two points are actually not singleton points but clusters, group of points. – ttnphns May 9 '18 at 15:30
• Note that many linkage methods (Ward, centroid, etc) require euclidean distance for geometric validity of the clustering. – ttnphns May 9 '18 at 15:34

The linkage defines how the distances are aggregated.

Without the underlying distance, there would be nothing to aggregate. Nothing to take the minimum/maximum/average of. The definitions of linkages require a distance.

The linkage criterion is a function of the distance metric that you choose.

For instance, let's consider the average-linkage criterion. Here's the formula: Notice that the d in the above formula is the chosen distance metric.

If you had chosen Euclidean distance, then that could provide a different value for the linkage function as compared to say, the Mahalanobis distance.