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The Manhattan distance between two vectors, A and B, is calculated as:

Σ|a_i – b_i|

where i is the i^th element in each vector.

This distance is used to measure the dissimilarity between any two vectors and is commonly used in many different machine learning algorithms.

This tutorial provides a couple examples of how to calculate Manhattan distance in R.

Example 1: Manhattan Distance Between Two Vectors

The following code shows how to create a custom function to calculate the Manhattan distance between two vectors in R:

#create function to calculate Manhattan distance
manhattan_dist <- function(a, b){
     dist <- abs(a-b)
     dist <- sum(dist)
     return(dist)
}

#define two vectors
a <- c(2, 4, 4, 6)

b <- c(5, 5, 7, 8)

#calculate Manhattan distance between vectors
manhattan_dist(a, b)
[1] 9

The Manhattan distance between these two vectors turns out to be 9.

We can confirm this is correct by quickly calculating the Manhattan distance by hand:

Σ|a_i – b_i| = |2-5| + |4-5| + |4-7| + |6-8| = 3 + 1 + 3 + 2 = 9.

Example 2: Manhattan Distance Between Vectors in a Matrix

To calculate the Manhattan distance between several vectors in a matrix, we can use the built-in dist() function in R:

#create four vectors
a <- c(2, 4, 4, 6)

b <- c(5, 5, 7, 8)

c <- c(9, 9, 9, 8)

d <- c(1, 2, 3, 3)

#bind vectors into one matrix
mat <- rbind(a, b, c, d)

#calculate Manhattan distance between each vector in the matrix
dist(mat, method = "manhattan")

   a  b  c
b  9      
c 19 10   
d  7 16 26

The way to interpret this output is as follows:

• The Manhattan distance between vector a and b is 9.
• The Manhattan distance between vector a and c is 19.
• The Manhattan distance between vector a and d is 7.
• The Manhattan distance between vector b and c is 10.
• The Manhattan distance between vector b and d is 16.
• The Manhattan distance between vector c and d is 26.

Note that each vector in the matrix should be the same length.

Additional Resources

How to Calculate Euclidean Distance in R
How to Calculate Mahalanobis Distance in R
How to Calculate Minkowski Distance in R