One method that works fairly well although tends to underestimate the actual number of clusters is to look at the within cluster similarity at each stage. This was introduced rather amusingly in by R.
Thorndike Psychometrika , 18[4]; , and although in that treatise he didn't think he was that successful in determining a way to get at the right number of clusters the "Thorndike" method is used widely nonetheless. 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 weaknesses of hierarchical clustering or other terms, check out the Displayr blog.
Or even better, do your own hierarchical cluster analysis! Market research Social research commercial Customer feedback Academic research Polling Employee research I don't have survey data. R in Displayr Visualizations. Keep updated with the latest in data science. What is Hierarchical Clustering? Twitter Facebook LinkedIn Email. Using Displayr. Working faster with large data files 12 Nov by Andrew Kelly. As we can see from the dendrogram, the combination of points and the heights are the same than the ones obtained by hand.
Remember that hierarchical clustering is used to determine the optimal number of clusters. This optimal number of clusters can be determined thanks to the dendrogram. For this, we usually look at the largest difference of heights:. How to determine the number of clusters from a dendrogram? Take the largest difference of heights and count how many vertical lines you see. To determine the optimal number of clusters, simply count how many vertical lines you see within this largest difference.
In our case, the optimal number of clusters is thus 2. In R, we can even highlight these two clusters directly in the dendrogram with the rect. Note that determining the optimal number of clusters via the dendrogram is not specific to the single linkage, it can be applied to other linkage methods too! How to determine the optimal numbers of cluster in hierarchical clustering?
Source: Towards Data Science. See this hierarchical clustering cheatsheet for more visualizations like this. Again, look for the largest jump of heights. In our case, the largest jump is from 1 to 2 classes. Therefore, the optimal number of classes is 2. Note that determining the number of clusters using the dendrogram or barplot is not a strict rule. However, these methods are beyond the scope of this article and the method presented with the dendrogram is generally sufficient. See more detailed information in this article.
Note that hierarchical clustering is a bit different than k -means in the sense that is does not return a vector containing the information about which cluster the observations belong to. Instead, it creates a hierarchical structure a dendrogram , a tree from which we can cut branches to get a given number of clusters. This information can be found via the cutree function:.
This is exactly what was drawn in the dendrogram above, and if needed, this information can be added to the initial data:. In this case, the optimal number of clusters is thus 2. Clustering is rather a subjective statistical analysis and there can be more than one appropriate algorithm, depending on the dataset at hand or the type of problem to be solved.
So choosing between k -means and hierarchical clustering is not always easy. Moreover, no method is better than the other, each come with their sets of limitations and benefits. If you have a good reason to think that there is a specific number of clusters in your dataset for example if you would like to distinguish diseased and healthy patients depending on some characteristics but you do not know in which group patients belong to , you should probably opt for the k -means clustering as this technique is used when the number of groups is specified in advance.
If you do not have any reason to believe there is a certain number of groups in your dataset for instance in marketing when trying to distinguish clients without any prior belief on the number of different types of customers , then you should probably opt for the hierarchical clustering to determine in how many clusters your data should be divided.
In addition to this, if you are still undecided note that, on the one hand, with a large number of variables, k -means may be computationally faster than hierarchical clustering if the number of clusters is small. On the other hand, the result of a hierarchical clustering is a structure that is more informative and interpretable than the unstructured set of flat clusters returned by k -means. Therefore, it is easier to determine the optimal number of clusters by looking at the dendrogram of a hierarchical clustering than trying to predict this optimal number in advance in case of k -means.
We have seen that clustering is a nice tool to identify homogeneous observations with respect to the measured variables. Following your clustering, you may want to get a sense of the characteristics of each cluster. This can be done through some descriptive statistics for each cluster.
For example, for categorical variables, you could start to compute, for each cluster, the proportion of responses for each level of your categorical variables. You could also visualize this information via a barplot.
For quantitative variables, the mean or median response for each variable in each cluster is usually a good starting point. If your clustering performs well, you will see that within each cluster observations share a common pattern of responses across the variables. This is both a nice way to see what makes each cluster unique, and what makes clusters different.
Thanks for reading. I hope this article helped you understand the different clustering methods and how to compute them by hand and in R. As always, if you have a question or a suggestion related to the topic covered in this article, please add it as a comment so other readers can benefit from the discussion. The complete guide to clustering analysis: k-means and hierarchical clustering by hand and in R Antoine Soetewey 47 minute read R Statistics.
What is clustering analysis? The two most common types of classification are: k -means clustering Hierarchical clustering The first is generally used when the number of classes is fixed in advance, while the second is generally used for an unknown number of classes and helps to determine this optimal number. Application 2: k -means clustering Data For this exercise, the Eurojobs. Germany 6. Germany 4. Optimal number of clusters In order to find the optimal number of clusters for a k -means, it is recommended to choose it based on: the context of the problem at hand, for instance if you know that there is a specific number of groups in your data you have strong expectations or hypotheses, this is option is however subjective , or the following four approaches: Elbow method which uses the within cluster sums of squares Average silhouette method Gap statistic method Consensus-based algorithm We show the R code for these 4 methods below, more theoretical information can be found here.
Elbow method The Elbow method looks at the total within-cluster sum of square WSS as a function of the number of clusters. The Elbow method is sometimes ambiguous and an alternative is the average silhouette method. Silhouette method The Silhouette method measures the quality of a clustering and determines how well each point lies within its cluster. Gap statistic method Gap statistic set. Consensus-based algorithm Because no method is clearly better, a fourth alternative is to run many methods and take the number of clusters that is the most agreed upon i.
Visualizations To confirm that your number of classes is indeed optimal, there is a way to evaluate the quality of your clustering via the silhouette plot which shows the silhouette coefficient on the y axis.
A lower entanglement coefficient corresponds to a good alignment. The output of tanglegram can be customized using many other options as follow:. Similar to how we determined optimal clusters with k-means clustering , we can execute similar approaches for hierarchical clustering:.
To perform the average silhouette method we follow a similar process. And the process is quite similar to perform the gap statistic method.
Clustering can be a very useful tool for data analysis in the unsupervised setting. However, there are a number of issues that arise in performing clustering.
In the case of hierarchical clustering, we need to be concerned about:. Each of these decisions can have a strong impact on the results obtained. In practice, we try several different choices, and look for the one with the most useful or interpretable solution.
With these methods, there is no single right answer - any solution that exposes some interesting aspects of the data should be considered. So, keep in mind that although hierarchical clustering can be performed quickly in R, there are many important variables to consider.
However, this tutorial gets you started performing the hierarchical clustering approach and you can learn more with:.
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