Minkowski distance in Python

The Minkowski distance is a metric in a normed vector space that measures the distance between two or more vectors. This metric is widely used in machine learning algorithms for measuring similarity between data points. The formula for calculating Minkowski distance is:

Where:

• r_i and s_i: Corresponding elements of two vectors in n-dimensional space
• p: Order parameter that determines the distance type. When p=1, it becomes Manhattan distance; when p=2, it represents Euclidean distance
• n: Number of dimensions

Example Calculation

For vectors x = (0, 3, 1, 4) and y = (2, 9, 3, 7) with p = 5, the Minkowski distance is approximately 6.047.

Method 1: Using Custom Implementation

We can implement Minkowski distance using Python's built-in math functions ?

from math import pow, fabs

def minkowski_distance(x, y, p):
    """Calculate Minkowski distance between two vectors"""
    if len(x) != len(y):
        raise ValueError("Vectors must have the same length")
    
    # Calculate sum of |xi - yi|^p
    distance_sum = sum(pow(fabs(a - b), p) for a, b in zip(x, y))
    
    # Return pth root
    return pow(distance_sum, 1/p)

# Example vectors
x = (0, 3, 1, 4)
y = (2, 9, 3, 7)
p = 5

result = minkowski_distance(x, y, p)
print(f"Minkowski distance: {result:.3f}")

Minkowski distance: 6.047

Method 2: Using SciPy

SciPy provides a built-in minkowski() function in the scipy.spatial.distance module ?

from scipy.spatial import distance

# Define vectors
x = (0, 3, 1, 4)
y = (2, 9, 3, 7)

# Calculate Minkowski distance with p=5
minkowski_dist = distance.minkowski(x, y, p=5)
print(f"Minkowski distance (p=5): {minkowski_dist:.3f}")

Minkowski distance (p=5): 6.047

Special Cases of Minkowski Distance

Different values of p produce well-known distance metrics ?

from scipy.spatial import distance

x = (0, 3, 1, 4)
y = (2, 9, 3, 7)

# Manhattan distance (p=1)
manhattan = distance.minkowski(x, y, p=1)
print(f"Manhattan distance (p=1): {manhattan}")

# Euclidean distance (p=2) 
euclidean = distance.minkowski(x, y, p=2)
print(f"Euclidean distance (p=2): {euclidean:.3f}")

# Higher order distance (p=3)
higher_order = distance.minkowski(x, y, p=3)
print(f"Minkowski distance (p=3): {higher_order:.3f}")

Manhattan distance (p=1): 13.0
Euclidean distance (p=2): 7.280
Minkowski distance (p=3): 6.498

Comparison of Methods

Conclusion

Minkowski distance is a versatile metric that generalizes Manhattan and Euclidean distances. Use SciPy's implementation for production code, as it's optimized and well-tested. The parameter p controls the sensitivity to outliers ? higher values make the distance less sensitive to individual large differences.