Distance metrics (euclidean, cosine, manhattan) in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Euclidean distance between two points?

Euclidean distance is the straight-line distance between two points in space. It is calculated as the square root of the sum of squared differences between corresponding coordinates.

beginner

How does the Cosine distance measure similarity between two vectors?

Cosine distance measures the angle between two vectors. It calculates 1 minus the cosine of the angle, focusing on direction rather than magnitude.

beginner

What is the Manhattan distance and when is it useful?

Manhattan distance is the sum of absolute differences between coordinates of two points. It is useful in grid-like path problems, like city blocks.

intermediate

How to calculate Euclidean distance using scipy?

Use scipy.spatial.distance.euclidean(point1, point2) to get the Euclidean distance between two points.

intermediate

What is the difference between Euclidean and Manhattan distances?

Euclidean distance measures straight-line distance, while Manhattan distance sums absolute differences along each axis, like moving along grid lines.

Which distance metric measures the straight-line distance between two points?

A Cosine distance

B Euclidean distance

C Manhattan distance

D Hamming distance

Euclidean distance is the straight-line distance between two points.

Which distance metric focuses on the angle between two vectors?

A Cosine distance

B Euclidean distance

C Manhattan distance

D Jaccard distance

Cosine distance measures similarity based on the angle between vectors.

Manhattan distance is best described as:

A Sum of squared differences

B Difference of angles

C Minimum distance

D Sum of absolute differences

Manhattan distance sums the absolute differences of coordinates.

Which scipy function calculates Euclidean distance?

A scipy.spatial.distance.cosine()

B scipy.spatial.distance.manhattan()

C scipy.spatial.distance.euclidean()

D scipy.spatial.distance.jaccard()

scipy.spatial.distance.euclidean() calculates Euclidean distance.

If you want to measure distance like walking city blocks, which metric is best?

A Manhattan distance

B Cosine distance

C Hamming distance

D Euclidean distance

Manhattan distance sums absolute differences, like moving along grid paths.

Explain the differences between Euclidean, Cosine, and Manhattan distances with examples.

Think about how you measure distance walking straight, turning angles, or moving on a grid.

Describe how to use scipy to calculate Euclidean, Cosine, and Manhattan distances between two points.

Recall the scipy distance functions and their names.

Practice

1. Which distance metric measures the straight-line distance between two points in space?

easy

A. Jaccard distance

B. Cosine distance

C. Manhattan distance

D. Euclidean distance

Solution

1. Step 1: Understand distance types

   Euclidean distance is the direct straight-line distance between two points.

2. Step 2: Compare with other distances

   Cosine measures angle difference, Manhattan sums absolute differences, Jaccard measures set similarity.

3. Final Answer:

   Euclidean distance -> Option D

4. Quick Check:

   Straight-line distance = Euclidean distance [OK]

Hint: Straight line means Euclidean distance [OK]

Common Mistakes:

• Confusing cosine with Euclidean
• Thinking Manhattan is straight-line
• Mixing Jaccard with numeric distances

2. Which of the following is the correct way to import the Euclidean distance function from scipy?

easy

A. from scipy.spatial.distance import euclidean

B. import scipy.distance.euclidean

C. from scipy.distance import euclidean

D. import euclidean from scipy.spatial

Solution

1. Step 1: Recall scipy import syntax

   The correct import path for distance functions is scipy.spatial.distance.

2. Step 2: Check each option

   from scipy.spatial.distance import euclidean uses correct syntax: from scipy.spatial.distance import euclidean. Others are invalid or incorrect syntax.

3. Final Answer:

   from scipy.spatial.distance import euclidean -> Option A

4. Quick Check:

   Correct import path = from scipy.spatial.distance import euclidean [OK]

Hint: Use 'from scipy.spatial.distance import' for distance functions [OK]

Common Mistakes:

• Using wrong module path
• Incorrect import syntax
• Trying to import from scipy.distance

3. What is the output of this code?

from scipy.spatial.distance import cosine
v1 = [1, 0, 1]
v2 = [0, 1, 0]
result = cosine(v1, v2)
print(round(result, 2))

medium

A. 1.0

B. 0.5

C. 0.0

D. 0.71

Solution

1. Step 1: Calculate cosine similarity basics

   Vectors v1 and v2 are orthogonal (no overlap), so cosine similarity is 0.

2. Step 2: Calculate cosine distance

   Cosine distance = 1 - cosine similarity = 1 - 0 = 1.0.

3. Final Answer:

   1.0 -> Option A

4. Quick Check:

   Orthogonal vectors cosine distance = 1.0 [OK]

Hint: Cosine distance = 1 - cosine similarity [OK]

Common Mistakes:

• Confusing cosine similarity with distance
• Rounding errors
• Assuming zero distance for orthogonal vectors

4. Find the error in this code snippet calculating Manhattan distance:

from scipy.spatial.distance import manhattan
a = [1, 2, 3]
b = [4, 5]
print(manhattan(a, b))

medium

A. Function name should be cityblock, not manhattan

B. Import statement is incorrect

C. Lists a and b must be the same length

D. Manhattan distance requires numpy arrays, not lists

Solution

1. Step 1: Check input vector lengths

   Vectors a and b have different lengths (3 and 2), which is invalid for distance calculation.

2. Step 2: Validate function and import

   Import and function name are correct; scipy.spatial.distance.manhattan is valid and accepts lists.

3. Final Answer:

   Lists a and b must be the same length -> Option C

4. Quick Check:

   Equal length vectors needed for distance [OK]

Hint: Vectors must be same length for distance functions [OK]

Common Mistakes:

• Assuming different length lists work
• Confusing function name with cityblock
• Thinking lists are invalid inputs

5. You have two points: p1 = [3, 4, 5] and p2 = [1, 1, 1]. Which distance metric will give the largest value?
Options:
A) Euclidean
B) Cosine
C) Manhattan
D) All equal

hard

A. Euclidean

B. Manhattan

C. Cosine

D. All equal

Solution

1. Step 1: Calculate Euclidean distance

   Euclidean = sqrt((3-1)^2 + (4-1)^2 + (5-1)^2) = sqrt(4 + 9 + 16) = sqrt(29) ≈ 5.39.

2. Step 2: Calculate Manhattan distance

   Manhattan = |3-1| + |4-1| + |5-1| = 2 + 3 + 4 = 9.

3. Step 3: Calculate Cosine distance

   Cosine similarity = (3*1 + 4*1 + 5*1) / (sqrt(3^2+4^2+5^2)*sqrt(1^2+1^2+1^2)) = 12 / (7.07*1.73) ≈ 0.98; Cosine distance = 1 - 0.98 = 0.02.

4. Final Answer:

   Manhattan -> Option B

5. Quick Check:

   Manhattan distance > Euclidean > Cosine distance [OK]

Hint: Manhattan sums absolute differences, usually largest [OK]

Common Mistakes:

• Assuming Euclidean is always largest
• Confusing cosine distance scale
• Ignoring absolute values in Manhattan