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.

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