Euclidean distance of two matrices

Question

I am writing codes to have the Euclidean distance of two matrices and then storing them in a matrix. Could anyone please review this?

a <-matrix(1:2,ncol = 2,nrow = 2)
a
     [,1] [,2]
[1,]    1    1
[2,]    2    2

b <-matrix(3:6,ncol = 2,nrow = 2)
b
     [,1] [,2]
[1,]    3    5
[2,]    4    6

n <-matrix(data = NA)
 for (i in 1:nrow(a)) {
   for (j in 1:nrow(b)) {
    n[i]<-sum(abs(a[i,]-b[j,])^2)
  }
}
n

Answer 1

First let's take a quick look at what your code is doing. Your innermost line of code is:

n[i]<-sum(abs(a[i,]-b[j,])^2)

Due to how you've structured your for loops, this is run for times

1. with i=1 and j=1, setting n[1] to the squared distance between row 1 with a and row 1 of b,
2. with i=1 and j=2, overwriting n[1] to the squared distance between row 1 of a and row 2 of b,
3. with i=2 and j=1, setting n[2] to the squared distance between row 2 of a and row 1 of b, and
4. with i=2 and j=2, overwriting n[2] to the squared distance between row 2 of a and row 2 of b.

So we end up with n = c(34, 20), the squared distances between each row of a and the last row of b.

It seems most likely to me that you are trying to compute the distances between each pair of points (since your n is structured as a vector). In this case, check out what we accomplish with the following, much simpler code:

(a-b)^2
#      [,1] [,2]
# [1,]    4   16
# [2,]    4   16

The resulting matrix is the squared difference of each element in the two matrices. All we need to do is to sum up the rows:

n <- rowSums((a-b)^2)
n
# [1] 20 20

Or if we wanted the actual distance instead of the squared distance:

n <- sqrt(rowSums((a-b)^2))
n
# [1] 4.472136 4.472136

Note that we dramatically simplified the calculation; a nice side benefit is that this code is much faster than using a for loop in R.