Distance metrics (euclidean, cosine, manhattan) in SciPy - Step-by-Step Execution

Concept Flow - Distance metrics (euclidean, cosine, manhattan)

Start with two points

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Choose distance metric

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Euclidean: sqrt(sum((x_i - y_i)^2))

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Cosine: 1 - (dot(x,y) / (||x||*||y||))

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Manhattan: sum(|x_i - y_i|)

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Calculate distance

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Output distance value

We start with two points, pick a distance type, calculate it, then get the distance value.

Execution Sample

SciPy

from scipy.spatial import distance

x = [1, 2, 3]
y = [4, 5, 6]

print(distance.euclidean(x, y))
print(distance.cosine(x, y))
print(distance.cityblock(x, y))

Calculate Euclidean, Cosine, and Manhattan distances between two 3D points.

Execution Table

Step	Metric	Calculation	Intermediate Values	Result
1	Euclidean	sqrt((1-4)^2 + (2-5)^2 + (3-6)^2)	Differences squared: 9, 9, 9	5.196152422706632
2	Cosine	1 - (dot(x,y) / (||x|| * ||y||))	dot=32, ||x||=3.7417, ||y||=8.7749	0.025368153802923787
3	Manhattan	sum(|1-4| + |2-5| + |3-6|)	Absolute differences: 3, 3, 3	9
4	End	All distances calculated	-	-

💡 All three distances computed for given points, execution ends.

Variable Tracker

Variable	Start	After Euclidean	After Cosine	After Manhattan	Final
x	[1, 2, 3]	[1, 2, 3]	[1, 2, 3]	[1, 2, 3]	[1, 2, 3]
y	[4, 5, 6]	[4, 5, 6]	[4, 5, 6]	[4, 5, 6]	[4, 5, 6]
euclidean_distance	None	5.196152422706632	5.196152422706632	5.196152422706632	5.196152422706632
cosine_distance	None	None	0.025368153802923787	0.025368153802923787	0.025368153802923787
manhattan_distance	None	None	None	9	9

Key Moments - 3 Insights

Why is cosine distance close to zero even though points look far apart?

Cosine distance measures angle difference, not length. See execution_table row 2: dot product and norms show vectors point similarly, so cosine distance is small.

Why do Euclidean and Manhattan distances give different values?

Euclidean uses squares and square root (row 1), Manhattan sums absolute differences (row 3). They measure distance differently, so results differ.

Are the original points changed during calculation?

No, variable_tracker shows x and y stay the same throughout all steps.

Visual Quiz - 3 Questions

Look at the execution_table at step 1, what is the Euclidean distance calculated?

A 5.196

B 9

C 0.025

D 3

💡 Hint

Check the 'Result' column in execution_table row 1.

At which step does the condition for cosine distance calculation appear?

A Step 1

B Step 3

C Step 2

D Step 4

💡 Hint

Look for cosine metric details in execution_table.

If we change point y to [1, 2, 3], what happens to the Manhattan distance?

A It becomes 5.196

B It becomes 0

C It becomes 9

D It becomes 1

💡 Hint

Check variable_tracker for Manhattan distance and think about absolute differences.

Concept Snapshot

Distance metrics measure how far points are.
Euclidean: straight line distance (sqrt of sum squares).
Cosine: angle difference (1 - cosine similarity).
Manhattan: sum of absolute differences.
Use scipy.spatial.distance for easy calculation.

Full Transcript

We start with two points x and y. We pick a distance metric: Euclidean, Cosine, or Manhattan. For Euclidean, we square differences, sum them, then take the square root. For Cosine, we calculate the dot product and norms, then find 1 minus their ratio. For Manhattan, we sum absolute differences. Variables x and y stay unchanged. The distances are calculated step-by-step and output. Beginners often confuse cosine distance with length; it measures angle difference. Euclidean and Manhattan differ because one uses squares and root, the other sums absolute values. The code example uses scipy to compute these distances easily.