Ward Linkage

Understanding Ward Linkage: A Simple Guide to Hierarchical Clustering

Hierarchical clustering is a powerful technique used in data analysis to group similar data points together. Imagine sorting a pile of mixed-colored marbles into groups based on their color. Hierarchical clustering does something similar with data, creating a hierarchy of clusters, visualized as a dendrogram (a tree-like diagram). One of the key methods used in hierarchical clustering is called Ward linkage. This article simplifies the complex ideas behind Ward linkage, explaining its mechanics and applications.

What is Ward Linkage?

Ward linkage is an agglomerative hierarchical clustering method. "Agglomerative" means it starts with each data point as its own cluster and progressively merges the closest clusters until all points belong to a single large cluster. The "linkage" refers to how the distance between clusters is measured. Ward linkage uniquely measures this distance based on the increase in within-cluster variance caused by merging two clusters. In simpler terms, it aims to minimize the total variance within each cluster at each step of the merging process. The less the variance increases after a merge, the better that merge is considered.

How Does Ward Linkage Work?

1. Initialization: Each data point begins as its own cluster.
2. Distance Calculation: Ward linkage calculates the distance between all pairs of clusters. The distance isn't a simple distance between two points, but rather a measure of how much the variance within the merged cluster would increase if those two clusters were combined.
3. Merging: The two clusters with the smallest increase in within-cluster variance are merged. This means that Ward linkage prefers merging clusters that are most similar in terms of their spread or distribution of data points.
4. Iteration: Steps 2 and 3 are repeated until all data points are in a single cluster. This process creates a hierarchy of clusters represented in a dendrogram.

Understanding Within-Cluster Variance

Within-cluster variance is a measure of how spread out the data points are within a single cluster. A low variance indicates that data points are clustered tightly together, while a high variance indicates more spread-out data. Ward linkage aims to keep this variance low throughout the clustering process, leading to compact and well-separated clusters.

Example: Imagine two clusters of exam scores: Cluster A (85, 88, 90) and Cluster B (82, 84, 86). Merging them would result in a new cluster (82, 84, 85, 86, 88, 90). Ward linkage calculates the variance within both the original clusters and the merged cluster. If the increase in variance is minimal, it indicates a good merge. If the increase is substantial, it suggests the clusters are dissimilar.

Visualizing with a Dendrogram

The results of Ward linkage are often displayed as a dendrogram. This is a tree-like diagram where each branch represents a cluster. The height of the branch connecting two clusters reflects the increase in within-cluster variance caused by their merger. Longer branches indicate a larger increase in variance, implying less similarity between the merged clusters. By cutting the dendrogram at different heights, you can obtain different numbers of clusters.

Practical Applications of Ward Linkage

Ward linkage finds applications in various fields:

Customer Segmentation: Grouping customers with similar purchasing behaviors.
Image Segmentation: Grouping similar pixels in an image for object recognition.
Document Clustering: Grouping documents with similar topics.
Biological Classification: Grouping species based on their characteristics.

Key Insights and Takeaways

Ward linkage is an agglomerative hierarchical clustering method that aims to minimize the within-cluster variance.
It's particularly useful when you want compact and well-separated clusters.
The resulting dendrogram provides a visual representation of the cluster hierarchy.
The choice of linkage method depends on the specific characteristics of the data and the research question.

Frequently Asked Questions (FAQs)

1. What are the advantages of Ward linkage? Ward linkage tends to produce relatively spherical clusters, which are often desirable. It's also relatively robust to outliers, though less so than some other methods.

2. What are the disadvantages of Ward linkage? It can be computationally expensive for large datasets, and it struggles with non-spherical clusters.

3. How do I choose the optimal number of clusters? There's no single answer. Techniques like examining the dendrogram for large jumps in branch lengths, using silhouette analysis, or the elbow method on the within-cluster variance can help determine the appropriate number of clusters.

4. How does Ward linkage differ from other linkage methods (e.g., single linkage, complete linkage)? Other methods use different distance measures. Single linkage uses the shortest distance between points in two clusters, complete linkage uses the longest distance, while Ward linkage focuses on minimizing the increase in variance.

5. Can Ward linkage handle datasets with missing values? Most implementations of Ward linkage require handling missing values beforehand, typically through imputation (filling in missing values) or removing rows or columns with missing data. The best approach depends on the specific dataset and the nature of the missing data.

Search Results:

Should we most of the time use Ward's method for hierarchical ... 11 Sep 2019 · In the third dataset, I see that Ward's method is clearly superior to the others. What I suspect is that people assume most of the time real data to follow the geometry of dataset 3, thus explaining the popularity of Ward's method over the others.

Hierarchical clustering, linkage methods and dynamic time warping 12 Feb 2015 · Ward linkage also may give reasonable results in practice, although I would be cautious of relying on it exclusively because of the ambiguity surrounding the meaning of a centroid in the context of DTW similarity measures. Check out the answers and links at this question for more info on Ward's linkage in non-Euclidean spaces.

Is there an advantage to squaring dissimilarities when using Ward ... Two different algorithms are found in the literature for Ward clustering. The one used by option " ward.D " (equivalent to the only Ward option " ward " in R versions <= 3.0.3) does not implement Ward's (1963) clustering criterion, whereas option " ward.D2 " implements that criterion (Murtagh and Legendre 2013).

Difference between Ward hierarchical clustering and K-Means for ... 11 Mar 2018 · The question doesn't say which exact algorithm was used and whether enough initialisations were used. Neither is it clear whether the WSS found by K-means is better than that of Ward's method here (if Ward's WSS is better, K-means can be improved running more initialisations). (2) There is no unique objectively best clustering of a data set.

Linkage method for hierarchical clustering of binary data 4 Mar 2018 · For example, complete linkage may be nice, because it means any two instances have at not h bits different at height h. Or you may want average linkage, so that the average number of bits is h. Or you may want minimax linkage, so that there exists one object, where all others are at most h bits different.

Using Ward's method on a dissimilarity matrix of Gower distances 3 Nov 2022 · Ward's linkage method (it is not a "variance" method, - it is the "increase of sum-of-squares" method) requires (squared) euclidean distances. See also . Gower distance sqrt(1-GS) is geometrically "euclidean", so it suits, but 1-GS distance won't suit as geometrically "euclidean".

Is it ok to use Manhattan distance with Ward's inter-cluster … Although Ward is meant to be used with Euclidean distances, this paper suggests that the clustering results using Ward and non-euclidean distances are essentially the same as if they had been used with Euclidean distances as it is meant to be.

Applying Ward's method for calculating linkage Having banged my head on the wall for the last 2 hours on this, I feel your pain. The result is the square root of the increase in within-cluster sum of squares (vs. cluster means), multiplied by $\sqrt{2}$ for some reason.

Choosing the right linkage method for hierarchical clustering 14 Feb 2016 · Methods which are most frequently used in studies where clusters are expected to be solid more or less round clouds, - are methods of average linkage, complete linkage method, and Ward's method. Ward's method is the closest, by it properties and efficiency, to K-means clustering; they share the same objective function - minimization of the pooled within-cluster …

Intuitive explanation of Ward's method - Cross Validated 21 Apr 2022 · I got this explanation of the Ward's method of hierarchical clustering from Malhotra et. al (2017), and I don't really get what it means: Ward’s procedure is a variance method which attempts to generate clusters to minimise the within-cluster variance. For each cluster, the means for all the variables are computed.