Hierarchical clustering (linkage) in SciPy

Introduction

Hierarchical clustering groups similar data points step-by-step. It helps find natural groups without knowing how many groups there are.

You want to organize customers into groups based on buying habits.

You need to find clusters in data without deciding the number of clusters first.

You want to visualize how data points group together using a tree diagram.

You have small to medium datasets and want to explore data structure.

You want to compare different ways of measuring distance between groups.

Syntax

SciPy

from scipy.cluster.hierarchy import linkage
Z = linkage(data, method='single', metric='euclidean')

data is your input data as a 2D array or matrix.

method chooses how to link clusters: 'single', 'complete', 'average', etc.

Examples

Uses the shortest distance between clusters to link them.

SciPy

Z = linkage(data, method='single')

Uses the longest distance between clusters to link them.

SciPy

Z = linkage(data, method='complete')

Uses the average distance between all points in clusters.

SciPy

Z = linkage(data, method='average')

Minimizes variance within clusters, good for compact clusters.

SciPy

Z = linkage(data, method='ward')

Sample Program

This code clusters 5 points using average linkage. It prints the linkage matrix showing how clusters merge step-by-step. Then it draws a dendrogram to visualize the cluster hierarchy.

SciPy

import numpy as np
from scipy.cluster.hierarchy import linkage, dendrogram
import matplotlib.pyplot as plt

# Sample data: 5 points with 2 features each
data = np.array([[1, 2], [2, 3], [5, 8], [6, 8], [7, 9]])

# Perform hierarchical clustering using average linkage
Z = linkage(data, method='average')

# Print linkage matrix
print(Z)

# Plot dendrogram to visualize clustering
plt.figure(figsize=(6, 4))
dendrogram(Z, labels=["A", "B", "C", "D", "E"])
plt.title('Hierarchical Clustering Dendrogram')
plt.xlabel('Sample')
plt.ylabel('Distance')
plt.tight_layout()
plt.show()

Important Notes

The linkage matrix has 4 columns: indices of clusters merged, distance between them, and number of original points in the new cluster.

Use dendrograms to understand cluster merging visually.

Different linkage methods can produce different cluster shapes.

Summary

Hierarchical clustering groups data step-by-step without preset cluster count.

Linkage methods control how distances between clusters are calculated.

Dendrograms help visualize the cluster structure and merging process.

Practice

1. What does the linkage function in scipy.cluster.hierarchy do in hierarchical clustering?

easy

A. It calculates distances between clusters step-by-step to form a hierarchy.

B. It assigns data points to fixed clusters before clustering.

C. It visualizes the final clusters using a scatter plot.

D. It normalizes the data before clustering.

Solution

1. Step 1: Understand hierarchical clustering process

   Hierarchical clustering builds clusters step-by-step by merging closest groups.

2. Step 2: Role of linkage function

   The linkage function calculates distances between clusters at each step to decide which to merge next.

3. Final Answer:

   It calculates distances between clusters step-by-step to form a hierarchy. -> Option A

4. Quick Check:

   Linkage = stepwise cluster distance calculation [OK]

Hint: Linkage = stepwise cluster distance calculation [OK]

Common Mistakes:

• Thinking linkage assigns fixed clusters first
• Confusing linkage with visualization functions
• Assuming linkage normalizes data

2. Which of the following is the correct way to import the linkage function from scipy.cluster.hierarchy?

easy

A. from scipy.cluster import linkage

B. import linkage from scipy.cluster.hierarchy

C. import linkage from scipy.cluster

D. from scipy.cluster.hierarchy import linkage

Solution

1. Step 1: Identify correct module path

   The linkage function is inside the hierarchy submodule of scipy.cluster.

2. Step 2: Use correct Python import syntax

   Python import syntax for functions is from module import function. So, from scipy.cluster.hierarchy import linkage is correct.

3. Final Answer:

   from scipy.cluster.hierarchy import linkage -> Option D

4. Quick Check:

   Correct import = from scipy.cluster.hierarchy import linkage [OK]

Hint: Use 'from scipy.cluster.hierarchy import linkage' [OK]

Common Mistakes:

• Using wrong module path
• Wrong import syntax like 'import linkage from ...'
• Importing from scipy.cluster directly

3. What is the output of this code snippet?

from scipy.cluster.hierarchy import linkage
import numpy as np

X = np.array([[1, 2], [3, 4], [5, 6]])
Z = linkage(X, method='single')
print(Z.shape)

medium

A. (2, 3)

B. (3, 4)

C. (2, 4)

D. (3, 3)

Solution

1. Step 1: Understand linkage output shape

   For n data points, linkage returns a matrix with n-1 rows and 4 columns.

2. Step 2: Calculate shape for 3 points

   Here, n=3, so output shape is (2, 4).

3. Final Answer:

   (2, 4) -> Option C

4. Quick Check:

   Linkage shape = (n-1, 4) = (2, 4) [OK]

Hint: Linkage output shape = (n-1, 4) for n points [OK]

Common Mistakes:

• Expecting shape (n, 4) instead of (n-1, 4)
• Confusing columns count
• Miscounting number of data points

4. Identify the error in this code snippet:

from scipy.cluster.hierarchy import linkage
import numpy as np

X = np.array([[1, 2], [3, 4], [5, 6]])
Z = linkage(X, method='fast')
print(Z)

medium

A. The method 'fast' is not a valid linkage method.

B. The input array X must be 1-dimensional.

C. The linkage function requires a distance matrix, not raw data.

D. The print statement is missing parentheses.

Solution

1. Step 1: Check valid linkage methods

   Valid methods include 'single', 'complete', 'average', 'ward', etc. 'fast' is not valid.

2. Step 2: Confirm input data and syntax

   Input can be raw data array; print statement syntax is correct in Python 3.

3. Final Answer:

   The method 'fast' is not a valid linkage method. -> Option A

4. Quick Check:

   Invalid method name causes error [OK]

Hint: Check method names carefully; 'fast' is invalid [OK]

Common Mistakes:

• Assuming 'fast' is a valid method
• Thinking input must be 1D array
• Confusing linkage input requirements

5. You have a dataset with 5 points and want to perform hierarchical clustering using the 'ward' method. After computing linkage, how many merges will be recorded in the linkage matrix, and why?

hard

A. 3 merges, because only the closest points are merged.

B. 4 merges, because each merge reduces clusters by one until one cluster remains.

C. 6 merges, because the 'ward' method adds an extra merge step.

D. 5 merges, because there are 5 points to merge individually.

Solution

1. Step 1: Understand merges in hierarchical clustering

   For n points, hierarchical clustering performs n-1 merges to combine all points into one cluster.

2. Step 2: Apply to 5 points with 'ward' method

   With 5 points, the linkage matrix records 4 merges regardless of method.

3. Final Answer:

   4 merges, because each merge reduces clusters by one until one cluster remains. -> Option B

4. Quick Check:

   Merges = n-1 = 4 for 5 points [OK]

Hint: Number of merges = number of points minus one [OK]

Common Mistakes:

• Thinking merges equal number of points
• Assuming method changes merge count
• Confusing merges with cluster count