# What is Hierarchical Clustering?

* Hierarchical clustering, *also known as

*hierarchical cluster analysis,*is an algorithm that groups similar objects into groups called

*clusters*. The endpoint is a set of clusters

*,*where each cluster is distinct from each other cluster, and the objects within each cluster are broadly similar to each other.

If you want to do your own hierarchical cluster analysis, use the template below – just add your data!

## Required data

Hierarchical clustering can be performed with either a *distance matrix *or *raw data. *When raw data is provided, the software will automatically compute a distance matrix in the background. The distance matrix below shows the distance between six objects.

## How hierarchical clustering works

Hierarchical clustering starts by treating each observation as a separate cluster. Then, it repeatedly executes the following two steps: (1) identify the two clusters that are closest together, and (2) merge the two most similar clusters. This continues until all the clusters are merged together. This is illustrated in the diagrams below.

The main output of Hierarchical Clustering is a *dendrogram, *which shows the hierarchical relationship between the clusters:

## Measures of distance (similarity)

In the example above, the *distance *between two clusters has been computed based on length of the straight line drawn from one cluster to another. This is commonly referred to as the *Euclidean distance. *Many other *distance metrics *have been developed.

The choice of distance metric should be made based on theoretical concerns from the domain of study. That is, a distance metric needs to define similarity in a way that is sensible for the field of study. For example, if clustering crime sites in a city, city block distance may be appropriate (or, better yet, the time taken to travel between each location). Where there is no theoretical justification for an alternative, the Euclidean should generally be preferred, as it is usually the appropriate measure of distance in the physical world.

## Linkage Criteria

After selecting a distance metric, it is necessary to determine from where distance is computed. For example, it can be computed between the two most similar parts of a cluster (*single-linkage*), the two least similar bits of a cluster (*complete-linkage*), the center of the clusters (*mean *or *average-linkage*), or some other criterion. Many linkage criteria have been developed.

As with *distance metrics, *the choice of linkage criteria should be made based on theoretical considerations from the domain of application. A key theoretical issue is what causes variation. For example, in archeology, we expect variation to occur through innovation and natural resources, so working out if two groups of artifacts are similar may make sense based on identifying the most similar members of the cluster.

Where there are no clear theoretical justifications for choice of linkage criteria, *Ward’s method* is the sensible default. This method works out which observations to group based on reducing the sum of squared distances of each observation from the average observation in a cluster. This is often appropriate as this concept of distance matches the standard assumptions of how to compute differences between groups in statistics (e.g., *ANOVA*, *MANOVA*).

## Agglomerative versus divisive algorithms

Hierarchical clustering typically works by sequentially merging similar clusters, as shown above. This is known as *agglomerative hierarchical clustering. *In theory, it can also be done by initially grouping all the observations into one cluster, and then successively splitting these clusters. This is known as *divisive hierarchical clustering. *Divisive clustering is rarely done in practice.

**To find out more about the strengths and weakness of hierarchical clustering or other terms, check out the Displayr blog. Or even better, do your own hierarchical cluster analysis!**

#### About Tim Bock

Tim Bock is the founder of Displayr. Tim is a data scientist, who has consulted, published academic papers, and won awards, for problems/techniques as diverse as neural networks, mixture models, data fusion, market segmentation, IPO pricing, small sample research, and data visualization. He has conducted data science projects for numerous companies, including Pfizer, Coca Cola, ACNielsen, KFC, Weight Watchers, Unilever, and Nestle. He is also the founder of Q www.qresearchsoftware.com, a data science product designed for survey research, which is used by all the world’s seven largest market research consultancies. He studied econometrics, maths, and marketing, and has a University Medal and PhD from the University of New South Wales (Australia’s leading research university), where he was an adjunct member of staff for 15 years.