keita-dc / dsc-distance-metrics-lab-dc-ds-060319

Distance Metrics - Lab

Introduction

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

Objectives

You will be able to:

• Calculate Euclidean Distance between 2 points
• Calculate Manhattan Distance between 2 points
• Compare and Contrast Manhattan, Euclidean, and Minkowski Distance

Getting Started

To begin this lab, 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:

How These Distance Metrics Are Related

Recall from the previous lesson that Manhattan Distance and Euclidean Distance are both just 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 just 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.
• This function should take in 3 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)
• 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.
• Include a parameter called verbose which is 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.
• This function should implement the Minkowski distance equation above, and return the result.

NOTE: Remember that for Manhattan Distance, you need to make use of np.abs() to get the absolute value of the distance for each dimension, since we don't have the squaring function to make this positive for us!

HINT: 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, see 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 so solve some practice problems.

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

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 how to calculate distance metrics, you can easily apply this to writing a K-Nearest Neighbors classifier from scratch!