Hierarchical Clustering and Linkage Methods Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Hierarchical Clustering and Linkage Methods notes for GATE DA: 33 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Hierarchical Clustering

Chapter Journey: Hierarchical Clustering

Distance Metrics & First Merge

Euclidean, Manhattan, Minkowski. How do we measure similarity between two points?

Linkage Methods

Single, Complete, Average, Ward’s. How do we measure distance between clusters?

Dendrograms

Visualizing the hierarchy. Reading the tree.

Cutting the Tree

Deciding the optimal number of clusters

k

The Core Idea: Agglomerative Clustering

Agglomerative Hierarchical Clustering

Intuition: Start with

N

points, each as its own cluster. Merge the nearest pair. Repeat until one cluster remains.

The Process:

Compute pairwise distances between all points.
Find the pair with the minimum distance.
Merge them into a new cluster.
Update the distance matrix (this is where Linkage Methods come in).
Repeat steps 2-4.

Note: This topic focuses on Step 1 and 2: Distance Metrics and identifying the First Merge.

Minkowski Distance Family

Minkowski Distance

For two points $A = (a_{1}, a_{2}, \dots, a_{n})$ and $B = (b_{1}, b_{2}, \dots, b_{n})$ :

D_{p} (A, B) = (k = 1 \sum n ∣ a_{k} - b_{k} ∣^{p})^{1/ p}

Common Cases:

Value of $p$	Name	Formula	Use Case
$1$	Manhattan	$\sum ∥ a_{k} - b_{k} ∥$	Grid-like paths, sparse data
$2$	Euclidean	$\sum (a_{k} - b_{k})^{2}$	Standard geometric distance
$\infty$	Chebyshev	$max ∥ a_{k} - b_{k} ∥$	Chess king moves, worst-case

Calculating First Merge: Euclidean vs Manhattan

Example: Which pair merges first?

Points: $P_{1} (0, 0)$ , $P_{2} (1, 2)$ , $P_{3} (2, 1)$

1. Euclidean Distance (

p = 2

)

D (P_{1}, P_{2}) = (1 - 0)^{2} + (2 - 0)^{2} = 5 \approx 2.236

D (P_{1}, P_{3}) = (2 - 0)^{2} + (1 - 0)^{2} = 5 \approx 2.236

D (P_{2}, P_{3}) = (2 - 1)^{2} + (1 - 2)^{2} = 2 \approx 1.414

First Merge:

P_{2}

and

P_{3}

(Min distance

2

)

2. Manhattan Distance (

p = 1

)

D (P_{1}, P_{2}) = ∣1 - 0∣ + ∣2 - 0∣ = 3

D (P_{1}, P_{3}) = ∣2 - 0∣ + ∣1 - 0∣ = 3

D (P_{2}, P_{3}) = ∣2 - 1∣ + ∣1 - 2∣ = 2

First Merge:

P_{2}

and

P_{3}

(Min distance

2

)

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Hierarchical Clustering and Linkage Methods Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Hierarchical Clustering and Linkage Methods notes for GATE DA: 33 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

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