12.5: Hierarchical Clustering
- Page ID
- 138499
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Hierarchical clustering is a foundational method in unsupervised learning that builds a multilevel hierarchy of clusters by either merging smaller clusters into larger ones (agglomerative clustering) or splitting larger clusters into smaller ones (divisive clustering). In business analytics, the agglomerative approach is most common due to its intuitive, bottom-up structure, making it particularly valuable when the number of clusters is unknown or when an interpretable visual summary of group relationships is desired.
Mathematical Foundation
The agglomerative hierarchical clustering process begins by treating each observation as its own cluster. It then repeatedly merges the two closest clusters based on a linkage criterion—a mathematical rule that defines how distances between clusters are computed. The process continues until all data points are merged into a single cluster, forming a tree-like structure called a dendrogram.
The three most commonly used linkage methods are:
- Single Linkage: Distance between the closest pair of observations from two cluste
- Complete Linkage: Distance between the farthest pair of observations.
- Average Linkage: Average distance between all pairs of observations between two clusters.
The choice of linkage method affects the shape and sensitivity of the resulting dendrogram. For example, single linkage can result in 'chained' clusters, while complete linkage tends to form more compact, spherical clusters.
This produces a dendrogram that visually displays how clusters merge at different distance thresholds. Analysts can 'cut' the dendrogram at a chosen height to define the desired number of clusters.
The dendrogram generated using average linkage. The vertical lines represent the distances at which clusters merge. A larger vertical space suggests a natural separation in the data.
Strengths of Hierarchical Clustering
- No predefined k required
- Visual interpretation through dendrograms
- Deterministic output (no random initialization)
Limitations of Hierarchical Clustering
- Computational complexity for large datasets
- Rigid structure (no re-clustering once merged)
- Sensitive to noise and scaling
Outputs and Interpretation
The primary output is a dendrogram. Cutting the dendrogram at a specific height yields a set number of clusters. Other outputs include merge heights and cluster assignments. These visuals are useful for communicating segmentation strategies to stakeholders.
Evaluation Metrics
- Cophenetic Correlation Coefficient: Measures how well the dendrogram represents the true distances.
- Dendrogram Height Comparison: Large vertical merges suggest distinct clusters.
Summary
In summary, hierarchical clustering offers a flexible and visually intuitive approach to data segmentation, particularly useful when the number of clusters is not known in advance. Its strength lies in its transparency and the simplicity of its visual outputs, making it ideal for exploratory analysis.