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Alternatively, this distance can be seen as the difference in k-means cost between the new cluster and the two old clusters. Ward's distance is also reducible, as can be seen more easily from a different formula for calculating the distance of a merged cluster from the distances of the clusters it was merged from: [9] [11]
We then proceed to update the initial distance matrix into a new distance matrix (see below), reduced in size by one row and one column because of the clustering of with . Bold values in D 2 {\displaystyle D_{2}} correspond to the new distances, calculated by averaging distances between each element of the first cluster ( a , b ) {\displaystyle ...
The clusters are then sequentially combined into larger clusters until all elements end up being in the same cluster. The method is also known as farthest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence of cluster fusion and the distance at which each fusion took place. [1] [2] [3]
It is defined as the ratio between the minimal inter-cluster distance to maximal intra-cluster distance. For each cluster partition, the Dunn index can be calculated by the following formula: [40] = < (,) ′ (), where d(i,j) represents the distance between clusters i and j, and d '(k) measures the intra-cluster distance of cluster k.
The method is also known as nearest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence in which clusters were merged and the distance at which each merge took place. [3] Mathematically, the linkage function – the distance D(X,Y) between clusters X and Y – is described by the expression
The Dunn index, introduced by Joseph C. Dunn in 1974, is a metric for evaluating clustering algorithms. [1] [2] This is part of a group of validity indices including the Davies–Bouldin index or Silhouette index, in that it is an internal evaluation scheme, where the result is based on the clustered data itself.
Ward's minimum variance method can be defined and implemented recursively by a Lance–Williams algorithm. The Lance–Williams algorithms are an infinite family of agglomerative hierarchical clustering algorithms which are represented by a recursive formula for updating cluster distances at each step (each time a pair of clusters is merged).
At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters and , each of size (i.e., cardinality) | | and | |, is taken to be the average of all distances (,) between pairs of objects in and in , that is, the mean distance between elements of each cluster: