Array reduction operations

Operations like sum and mean are array reduction operations.

We call these “reduction operations” because operations like sum have the effect of slicing out 1D arrays from the input array, and reducing these 1D arrays to scalars.

For example, if we have a 2D array, we might take the sum over the first axis:

import numpy as np
a = np.reshape(np.arange(6), (2, 3))
a

array([[0, 1, 2],
       [3, 4, 5]])

np.sum(a, axis=0)

array([3, 5, 7])

What has happened here is that numpy takes each column (slice over the first axis), then reduces the column to a scalar (that is the sum of the column):

print('Sum over column 0:', np.sum(a[:, 0]))
print('Sum over column 1:', np.sum(a[:, 1]))
print('Sum over column 2:', np.sum(a[:, 2]))

Sum over column 0: 3
Sum over column 1: 5
Sum over column 2: 7

Notice that the new summed array has one fewer dimension than the input array. The dimension over which we have done the sum has gone (numpy “reduced” it):

np.sum(a, axis=0).shape

(3,)

Similarly, when we sum across the second axis, we reduce that second axis:

# Sum over second axis
np.sum(a, axis=1)

array([ 3, 12])

print('Sum over row 0:', np.sum(a[0, :]))
print('Sum over row 1:', np.sum(a[1, :]))

Sum over row 0: 3
Sum over row 1: 12

Now imagine a 3D array:

b = np.reshape(np.arange(24), (2, 3, 4))
b

array([[[ 0,  1,  2,  3],
        [ 4,  5,  6,  7],
        [ 8,  9, 10, 11]],

       [[12, 13, 14, 15],
        [16, 17, 18, 19],
        [20, 21, 22, 23]]])

Let’s say we want to sum across axis 0 (the first axis). How do we get 1D arrays from this first axis?

b[:, 0, 0]

array([ 0, 12])

b[:, 0, 1]

array([ 1, 13])

So, we can think of this 3D array as a 2D array shape (3, 4) where each element is a 1D array of length 2.

Sum then operates over this array of 1D arrays to reduce the first axis:

np.sum(b, axis=0)

array([[12, 14, 16, 18],
       [20, 22, 24, 26],
       [28, 30, 32, 34]])

print('Sum over column 0, plane 0:', np.sum(b[:, 0, 0]))
print('Sum over column 0, plane 1:', np.sum(b[:, 0, 1]))

Sum over column 0, plane 0: 12
Sum over column 0, plane 1: 14

You could imagine doing what Numpy does with something like this:

sum_over_axis_0 = np.zeros((b.shape[1:]))
for j in range(b.shape[1]):
    for k in range(b.shape[2]):
        arr_1d = b[:, j, k]
        sum_over_axis_0[j, k] = np.sum(arr_1d)
sum_over_axis_0

array([[12., 14., 16., 18.],
       [20., 22., 24., 26.],
       [28., 30., 32., 34.]])

It is the same for reducing over the second axis. Now the 1D arrays are slices over the second axis.

b[0, :, 0]

array([0, 4, 8])

b[0, :, 1]

array([1, 5, 9])

np.sum(b, axis=1)

array([[12, 15, 18, 21],
       [48, 51, 54, 57]])

print('Sum over row 0, plane 0:', np.sum(b[0, :, 0]))
print('Sum over row 0, plane 1:', np.sum(b[0, :, 1]))

Sum over row 0, plane 0: 12
Sum over row 0, plane 1: 15

It is the same idea over the third axis (axis=2):

b[0, 0, :]

array([0, 1, 2, 3])

b[0, 1, :]

array([4, 5, 6, 7])

np.sum(b, axis=2)

array([[ 6, 22, 38],
       [54, 70, 86]])

print('Sum over row 0, column 0:', np.sum(b[0, 0, :]))
print('Sum over row 0, column 1:', np.sum(b[0, 1, :]))

Sum over row 0, column 0: 6
Sum over row 0, column 1: 22