Hierarchical clustering (linkage) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of linkage matrix with single linkage

What is the output of the linkage matrix when using single linkage on the given data points?

SciPy

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)

A

[[0. 1. 2.82842712 2.]
 [2. 3. 2.82842712 2.]
 [4. 5. 5.65685425 3.]]

B

[[0. 1. 2.82842712 2.]
 [2. 3. 4.24264069 2.]
 [4. 5. 5.65685425 3.]]

C

[[0. 1. 2.82842712 2.]
 [2. 3. 2.82842712 3.]]

D

[[0. 1. 1.41421356 2.]
 [2. 3. 2.82842712 2.]
 [4. 5. 4.24264069 3.]]

💡 Hint

Remember single linkage merges clusters based on the smallest distance between points.

Single linkage merges clusters by the minimum distance between points. The distances between points are Euclidean. The linkage matrix shows the pairs merged, the distance, and the cluster size.

❓ data_output

intermediate

Number of clusters from dendrogram cut

Given the linkage matrix below, how many clusters remain if we cut the dendrogram at distance 3.0?

SciPy

import numpy as np
Z = np.array([[0, 1, 1.5, 2], [2, 3, 2.5, 2], [4, 5, 4.0, 4]])

A 4

B 2

C 3

D 1

💡 Hint

Clusters merge when linkage distance is less than the cut distance.

Cutting at distance 3.0 means merges with distance <= 3.0 are joined. The first two merges are below 3.0, so they form clusters. The last merge at 4.0 is above 3.0, so it is not merged. Thus, 2 clusters remain.

🔧 Debug

advanced

Identify the error in linkage method usage

What error will this code raise when running linkage with an invalid method name?

SciPy

from scipy.cluster.hierarchy import linkage
import numpy as np
X = np.array([[1, 2], [3, 4], [5, 6]])
Z = linkage(X, method='invalid_method')

A ValueError: Unknown linkage method invalid_method

B TypeError: linkage() missing 1 required positional argument

C AttributeError: 'numpy.ndarray' object has no attribute 'invalid_method'

D SyntaxError: invalid syntax

💡 Hint

Check the allowed method names for linkage.

The linkage function checks if the method name is valid. If not, it raises a ValueError with a message about unknown linkage method.

🚀 Application

advanced

Choosing linkage method for compact clusters

Which linkage method is best to produce compact, spherical clusters in hierarchical clustering?

A Ward linkage

B Complete linkage

C Average linkage

D Single linkage

💡 Hint

Ward linkage minimizes variance within clusters.

Ward linkage merges clusters to minimize the total within-cluster variance, producing compact and spherical clusters. Other methods may produce elongated or irregular clusters.

🧠 Conceptual

expert

Effect of linkage method on dendrogram shape

How does the choice of linkage method affect the shape of the dendrogram in hierarchical clustering?

A It only affects the color of the dendrogram branches, not the structure.

B It changes the number of data points in the dataset.

C It does not affect the dendrogram; all linkage methods produce identical dendrograms.

D It changes the order of merges and the height of branches, reflecting different cluster distances.

💡 Hint

Think about how distances between clusters are computed.

Different linkage methods compute distances between clusters differently, affecting which clusters merge first and at what distance. This changes the dendrogram's branch heights and merge order.