Complete linkage

Complete Linkage, also known as maximum linkage, is a method used in cluster analysis that evaluates the distance between sets of observations as the maximum distance between any single observation in one cluster to any single observation in another cluster. It is one of the main hierarchical clustering methods, which are techniques for building a hierarchy of clusters.

Overview

In statistics and data analysis, clustering is a method used to group a set of objects in such a way that objects in the same group (called a cluster) are more similar to each other than to those in other groups. Hierarchical clustering is a strategy of cluster analysis which seeks to build a hierarchy of clusters. Complete linkage clustering is a specific approach within this strategy that defines the distance between two clusters as the maximum distance between any single member of one cluster and any single member of another cluster.

Algorithm

The complete linkage clustering algorithm involves the following steps:

1. Start by treating each observation as a separate cluster.
2. Find the pair of clusters that are closest together based on the maximum distance between their observations.
3. Merge these two clusters into one cluster.
4. Recalculate distances between the new cluster and each of the old clusters.
5. Repeat steps 2 through 4 until all observations are clustered into a single cluster.

The distance between clusters can be measured in various ways, though the most common metrics are Euclidean distance, Manhattan distance, and Minkowski distance.

Advantages and Disadvantages

Advantages

• Tends to create more compact clusters than other methods.
• Can be useful for identifying outliers, as the method focuses on the maximum distances.

Disadvantages

• Can be sensitive to noise and outliers, as these can significantly affect the maximum distance between clusters.
• May not perform well if clusters are of varying densities.

Applications

Complete linkage clustering is used in various fields such as bioinformatics for genetic and protein sequence analysis, market research for understanding consumer behavior, and image analysis for object recognition and classification.

Comparison with Other Methods

Complete linkage is often compared with other hierarchical clustering methods such as single linkage clustering (which considers the minimum distance between clusters) and average linkage clustering (which considers the average distance between clusters). Each method has its own strengths and weaknesses, and the choice of method can significantly affect the results of the cluster analysis.

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