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The function used to determine the distance between two clusters, known as the linkage function, is what differentiates the agglomerative clustering methods. In single-linkage clustering, the distance between two clusters is determined by a single pair of elements: those two elements (one in each cluster) that are closest to each other.
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]
The distance between any two clusters ... Implementing a different linkage is simply a matter of using a different formula to calculate inter-cluster distances during ...
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.
To do that, we need to take the distance between {a} and {b c}, and therefore define the distance between two clusters. Usually the distance between two clusters and is one of the following: The maximum distance between elements of each cluster (also called complete-linkage clustering):
Lloyd's algorithm is the standard approach for this problem. However, it spends a lot of processing time computing the distances between each of the k cluster centers and the n data points. Since points usually stay in the same clusters after a few iterations, much of this work is unnecessary, making the naïve implementation very inefficient.
At each step, the nearest two clusters, say and , are combined into a higher-level cluster . Then, its distance to another cluster k {\displaystyle k} is simply the arithmetic mean of the average distances between members of k {\displaystyle k} and i {\displaystyle i} and k {\displaystyle k} and j {\displaystyle j} :
With the above notation, if there are m clusters, then the Dunn Index for the set is defined as: = < (,) where (,) is the inter-cluster distance between the clusters and while is the within cluster distance, e.g. the maximum distance within one cluster when following Dunn's original definition.
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