Sum along axis in numpy array

Question

I want to understand how this ndarray.sum(axis=) works. I know that axis=0 is for columns and axis=1 is for rows. But in case of 3 dimensions(3 axes) its difficult to interpret below result.

arr = np.arange(0,30).reshape(2,3,5)

arr
Out[1]: 
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, 24],
        [25, 26, 27, 28, 29]]])

arr.sum(axis=0)
Out[2]: 
array([[15, 17, 19, 21, 23],
       [25, 27, 29, 31, 33],
       [35, 37, 39, 41, 43]])


arr.sum(axis=1)
Out[8]: 
array([[15, 18, 21, 24, 27],
       [60, 63, 66, 69, 72]])

arr.sum(axis=2)
Out[3]: 
array([[ 10,  35,  60],
       [ 85, 110, 135]])

Here in this example of 3 axes array of shape(2,3,5), there are 3 rows and 5 columns. But if i look at this array as whole, seems like only two rows (both with 3 array elements).

Can anyone please explain how this sum works on array of 3 or more axes(dimensions).

Answer 1

If you want to keep the dimensions you can specify keepdims:

>>> arr = np.arange(0,30).reshape(2,3,5)
>>> arr.sum(axis=0, keepdims=True)
array([[[15, 17, 19, 21, 23],
        [25, 27, 29, 31, 33],
        [35, 37, 39, 41, 43]]])

Otherwise the axis you sum along is removed from the shape. An easy way to keep track of this is using the numpy.ndarray.shape property:

>>> arr.shape
(2, 3, 5)

>>> arr.sum(axis=0).shape
(3, 5)  # the first entry (index = axis = 0) dimension was removed 

>>> arr.sum(axis=1).shape
(2, 5)  # the second entry (index = axis = 1) was removed

You can also sum along multiple axis if you want (reducing the dimensionality by the amount of specified axis):

>>> arr.sum(axis=(0, 1))
array([75, 81, 87, 93, 99])
>>> arr.sum(axis=(0, 1)).shape
(5, )  # first and second entry is removed

Answer 2

Here is another way to interpret this. You can consider a multi-dimensional array as a tensor, T[i][j][k], while i, j, k represents axis 0,1,2 respectively.

T.sum(axis = 0) mathematically will be equivalent to:

Similary, T.sum(axis = 1):

And, T.sum(axis = 2):

So in another word, the axis will be summed over, for instance, axis = 0, the first index will be summed over. If written in a for loop:

result[j][k] = sum(T[i][j][k] for i in range(T.shape[0])) for all j,k

for axis = 1:

result[i][k] = sum(T[i][j][k] for j in range(T.shape[1])) for all i,k

etc.