Distance metrics (euclidean, cosine, manhattan) in SciPy

Introduction

Distance metrics help us measure how far apart two points or objects are. They are useful to compare things in many areas like shopping, sports, or health.

To find the closest store to your home based on location coordinates.

To compare how similar two songs are by their features.

To measure the difference between two people's health data.

To group customers with similar buying habits.

To check how different two images are by their pixel values.

Syntax

SciPy

from scipy.spatial import distance

# Euclidean distance
distance.euclidean(point1, point2)

# Cosine distance
distance.cosine(point1, point2)

# Manhattan distance
distance.cityblock(point1, point2)

point1 and point2 are lists or arrays of numbers representing coordinates or features.

Euclidean is straight-line distance, Cosine measures angle difference, Manhattan sums absolute differences.

Examples

Calculates straight-line distance between points (1,2) and (4,6).

SciPy

distance.euclidean([1, 2], [4, 6])

Measures how different two vectors are in direction.

SciPy

distance.cosine([1, 0, 0], [0, 1, 0])

Calculates total absolute difference along each dimension.

SciPy

distance.cityblock([1, 2, 3], [4, 0, 6])

Sample Program

This program calculates three types of distances between two points and prints them with two decimals for clarity.

SciPy

from scipy.spatial import distance

pointA = [3, 4]
pointB = [7, 1]

# Calculate Euclidean distance
euclid = distance.euclidean(pointA, pointB)

# Calculate Cosine distance
cosine = distance.cosine(pointA, pointB)

# Calculate Manhattan distance
manhattan = distance.cityblock(pointA, pointB)

print(f"Euclidean distance: {euclid:.2f}")
print(f"Cosine distance: {cosine:.2f}")
print(f"Manhattan distance: {manhattan}")

Important Notes

Euclidean distance is like measuring with a ruler in straight line.

Cosine distance is good when you care about direction, not size.

Manhattan distance is like walking city blocks, adding up all steps.

Summary

Distance metrics measure how far or different two points are.

Euclidean is straight-line, Cosine is angle-based, Manhattan sums absolute differences.

Use scipy.spatial.distance functions to calculate these easily in Python.

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