Distance Metrics - Lab

Introduction

In this lab, you'll calculate various distances between multiple points using the distance metrics you learned about!

Objectives

In this lab you will:

• Calculate Manhattan distance between two points
• Calculate Euclidean distance between two points
• Calculate Minkowski distance between two points

Getting Started

You'll start by writing a generalized function to calculate any of the three distance metrics you've learned about. Let's review what you know so far:

The Manhattan distance and Euclidean distance are both special cases of Minkowski distance.

Take a look at the formula for Minkowski distance below:

$$\large d(x,y) = \left(\sum_{i=1}^{n}|x_i - y_i|^c\right)^\frac{1}{c}$$

Manhattan distance is a special case where $c=1$ in the equation above (which means that you can remove the root operation and just keep the summation).

Euclidean distance is a special case where $c=2$ in the equation above.

Knowing this, you can create a generalized distance() function that calculates Minkowski distance, and takes in c as a parameter. That way, you can use the same function for every problem, and still calculate Manhattan and Euclidean distance metrics by just passing in the appropriate values for the c parameter!

In the cell below:

• Complete the distance() function which should implement the Minkowski distance equation above to return the distance, a single number

• This function should take in 4 arguments:

  • a: a tuple or array that describes a vector in n-dimensional space
  • b: a tuple or array that describes a vector in n-dimensional space (this must be the same length as a!)
  • c: which tells us the norm to calculate the vector space (if set to 1, the result will be Manhattan, while 2 will calculate Euclidean distance)
  • verbose: set to True by default. If true, the function should print out if the distance metric returned is a measurement of Manhattan, Euclidean, or Minkowski distance

• Since euclidean distance is the most common distance metric used, this function should default to using c=2 if no value is set for c

HINT:

1. You can avoid using a for loop like we did in the previous lesson by simply converting the tuples to NumPy arrays

2. Use np.power() as an easy way to implement both squares and square roots. np.power(a, 3) will return the cube of a, while np.power(a, 1/3) will return the cube root of 3. For more information on this function, refer the NumPy documentation!

import numpy as np

# Complete this function! 
def distance():
    pass



test_point_1 = (1, 2)
test_point_2 = (4, 6)
print(distance(test_point_1, test_point_2)) # Expected Output: 5.0
print(distance(test_point_1, test_point_2, c=1)) # Expected Output: 7.0
print(distance(test_point_1, test_point_2, c=3)) # Expected Output: 4.497941445275415

Great job!

Now, use your function to calculate distances between points:

Problem 1

Calculate the Euclidean distance between the following points in 5-dimensional space:

Point 1: (-2, -3.4, 4, 15, 7)

Point 2: (3, -1.2, -2, -1, 7)

# Expected Output: 17.939899665271266

Problem 2

Calculate the Manhattan distance between the following points in 10-dimensional space:

Point 1: [0, 0, 0, 7, 16, 2, 0, 1, 2, 1]
Point 2: [1, -1, 5, 7, 14, 3, -2, 3, 3, 6]

# Expected Output: 20.0

Problem 3

Calculate the Minkowski distance with a norm of 3.5 between the following points:

Point 1: (-2, 7, 3.4)
Point 2: (3, 4, 1.5)

# Expected Output: 5.268789659188307

Summary

Great job! Now that you know about the various distance metrics, you can use them to writing a K-Nearest Neighbors classifier from scratch!