Computing Euclidean distance for numpy in python

Question

I am new to Python so this question might look trivia. However, I did not find a similar case to mine. I have a matrix of coordinates for 20 nodes. I want to compute the euclidean distance between all pairs of nodes from this set and store them in a pairwise matrix. For example, If I have 20 nodes, I want the end result to be a matrix of (20,20) with values of euclidean distance between each pairs of nodes. I tried to used a for loop to go through each element of the coordinate set and compute euclidean distance as follows:

ncoord=numpy.matrix('3225   318;2387    989;1228    2335;57      1569;2288  8138;3514   2350;7936   314;9888    4683;6901   1834;7515   8231;709   3701;1321    8881;2290   2350;5687   5034;760    9868;2378   7521;9025   5385;4819   5943;2917   9418;3928   9770')
n=20 
c=numpy.zeros((n,n))
for i in range(0,n):
    for j in range(i+1,n):
        c[i][j]=math.sqrt((ncoord[i][0]-ncoord[j][0])**2+(ncoord[i][1]-ncoord[j][1])**2)

How ever, I am getting an error of "input must be a square array ". I wonder if anybody knows what is happening here. Thanks

Answer 1

There are much, much faster alternatives to using nested for loops for this. I'll show you two different approaches - the first will be a more general method that will introduce you to broadcasting and vectorization, and the second uses a more convenient scipy library function.

---

1. The general way, using broadcasting & vectorization

One of the first things I'd suggest doing is switching to using np.array rather than np.matrix. Arrays are preferred for a number of reasons, most importantly because they can have >2 dimensions, and they make element-wise multiplication much less awkward.

import numpy as np

ncoord = np.array(ncoord)

With an array, we can eliminate the nested for loops by inserting a new singleton dimension and broadcasting the subtraction over it:

# indexing with None (or np.newaxis) inserts a new dimension of size 1
print(ncoord[:, :, None].shape)
# (20, 2, 1)

# by making the 'inner' dimensions equal to 1, i.e. (20, 2, 1) - (1, 2, 20),
# the subtraction is 'broadcast' over every pair of rows in ncoord
xydiff = ncoord[:, :, None] - ncoord[:, :, None].T

print(xydiff.shape)
# (20, 2, 20)

This is equivalent to looping over every pair of rows using nested for loops, but much, much faster!

xydiff2 = np.zeros((20, 2, 20), dtype=xydiff.dtype)
for ii in range(20):
    for jj in range(20):
        for kk in range(2):
            xydiff[ii, kk, jj] = ncoords[ii, kk] - ncoords[jj, kk]

# check that these give the same result
print(np.all(xydiff == xydiff2))
# True

The rest we can also do using vectorized operations:

# we square the differences and sum over the 'middle' axis, equivalent to
# computing (x_i - x_j) ** 2 + (y_i - y_j) ** 2
ssdiff = (xydiff * xydiff).sum(1)

# finally we take the square root
D = np.sqrt(ssdiff)

The whole thing could be done in one line like this:

D = np.sqrt(((ncoord[:, :, None] - ncoord[:, :, None].T) ** 2).sum(1))

---

2. The lazy way, using pdist

It turns out that there's already a fast and convenient function for computing all pairwise distances: scipy.spatial.distance.pdist.

from scipy.spatial.distance import pdist, squareform

d = pdist(ncoord)

# pdist just returns the upper triangle of the pairwise distance matrix. to get
# the whole (20, 20) array we can use squareform:

print(d.shape)
# (190,)

D2 = squareform(d)
print(D2.shape)
# (20, 20)

# check that the two methods are equivalent
print np.all(D == D2)
# True

Answer 2

for i in range(0, n):
    for j in range(i+1, n):
        c[i, j] = math.sqrt((ncoord[i, 0] - ncoord[j, 0])**2 
        + (ncoord[i, 1] - ncoord[j, 1])**2)

Note: ncoord[i, j] is not the same as ncoord[i][j] for a Numpy matrix. This appears to be the source of confusion. If ncoord is a Numpy array then they will give the same result.

For a Numpy matrix, ncoord[i] returns the ith row of ncoord, which itself is a Numpy matrix object with shape 1 x 2 in your case. Therefore, ncoord[i][j] actually means: take the ith row of ncoord and take the jth row of that 1 x 2 matrix. This is where your indexing problems comes about when j > 0.

Regarding your comments on assigning to c[i][j] "working", it shouldn't. At least on my build of Numpy 1.9.1 it shouldn't work if your indices i and j iterates up to n.

As an aside, remember to add the transpose of the matrix c to itself.

It is recommended to use Numpy arrays instead of matrix. See this post.

If your coordinates are stored as a Numpy array, then pairwise distance can be computed as:

from scipy.spatial.distance import pdist

pairwise_distances = pdist(ncoord, metric="euclidean", p=2)

or simply

pairwise_distances = pdist(ncoord)

since the default metric is "euclidean", and default "p" is 2.

In a comment below I mistakenly mentioned that the result of pdist is a n x n matrix. To get a n x n matrix, you will need to do the following:

from scipy.spatial.distance import pdist, squareform

pairwise_distances = squareform(pdist(ncoord))

or

from scipy.spatial.distance import cdist

pairwise_distances = cdist(ncoord, ncoord)