How can I vectorize the averaging of 2x2 sub-array

Question

I have a very a very large 2D numpy array that contains 2x2 subsets that I need to take the average of. I am looking for a way to vectorize this operation. For example, given x:

#               |- col 0 -|   |- col 1 -|   |- col 2 -|       
x = np.array( [[ 0.0,   1.0,   2.0,   3.0,   4.0,   5.0],  # row 0
               [ 6.0,   7.0,   8.0,   9.0,  10.0,  11.0],  # row 0
               [12.0,  13.0,  14.0,  15.0,  16.0,  17.0],  # row 1
               [18.0,  19.0,  20.0,  21.0,  22.0,  23.0]]) # row 1

I need to end up with a 2x3 array which are the averages of each 2x2 sub array, i.e.:

result = np.array( [[ 3.5,  5.5,  7.5],
                    [15.5, 17.5, 19.5]])

so element [0,0] is calculated as the average of x[0:2,0:2], while element [0,1] would be the average of x[2:4, 0:2]. Does numpy have vectorized/efficient ways of doing aggregates on subsets like this?

Answer 1

If we form the reshaped matrix y = x.reshape(2,2,3,2), then the (i,j) 2x2 submatrix is given by y[i,:,j,:]. E.g.:

In [340]: x
Out[340]: 
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.]])

In [341]: y = x.reshape(2,2,3,2)

In [342]: y[0,:,0,:]
Out[342]: 
array([[ 0.,  1.],
       [ 6.,  7.]])

In [343]: y[1,:,2,:]
Out[343]: 
array([[ 16.,  17.],
       [ 22.,  23.]])

To get the mean of the 2x2 submatrices, use the mean method, with axis=(1,3):

In [344]: y.mean(axis=(1,3))
Out[344]: 
array([[  3.5,   5.5,   7.5],
       [ 15.5,  17.5,  19.5]])

If you are using an older version of numpy that doesn't support using a tuple for the axis, you could do:

In [345]: y.mean(axis=1).mean(axis=-1)
Out[345]: 
array([[  3.5,   5.5,   7.5],
       [ 15.5,  17.5,  19.5]])

See the link given by @dashesy in a comment for more background on the reshaping "trick".

---

To generalize this to a 2-d array with shape (m, n), where m and n are even, use

y = x.reshape(x.shape[0]/2, 2, x.shape[1], 2)

y can then be interpreted as an array of 2x2 arrays. The first and third index slots of the 4-d array act as the indices that select one of the 2x2 blocks. To get the upper left 2x2 block, use y[0, :, 0, :]; to the block in the second row and third column of blocks, use y[1, :, 2, :]; and in general, to acces block (j, k), use y[j, :, k, :].

To compute the reduced array of averages of these blocks, use the mean method, with axis=(1, 3) (i.e. average over axes 1 and 3):

avg = y.mean(axis=(1, 3))

Here's an example where x has shape (8, 10), so the array of averages of the 2x2 blocks has shape (4, 5):

In [10]: np.random.seed(123)

In [11]: x = np.random.randint(0, 4, size=(8, 10))

In [12]: x
Out[12]: 
array([[2, 1, 2, 2, 0, 2, 2, 1, 3, 2],
       [3, 1, 2, 1, 0, 1, 2, 3, 1, 0],
       [2, 0, 3, 1, 3, 2, 1, 0, 0, 0],
       [0, 1, 3, 3, 2, 0, 3, 2, 0, 3],
       [0, 1, 0, 3, 1, 3, 0, 0, 0, 2],
       [1, 1, 2, 2, 3, 2, 1, 0, 0, 3],
       [2, 1, 0, 3, 2, 2, 2, 2, 1, 2],
       [0, 3, 3, 3, 1, 0, 2, 0, 2, 1]])

In [13]: y = x.reshape(x.shape[0]/2, 2, x.shape[1]/2, 2)

Take a look at a couple of the 2x2 blocks:

In [14]: y[0, :, 0, :]
Out[14]: 
array([[2, 1],
       [3, 1]])

In [15]: y[1, :, 2, :]
Out[15]: 
array([[3, 2],
       [2, 0]])

Compute the averages of the blocks:

In [16]: avg = y.mean(axis=(1, 3))

In [17]: avg
Out[17]: 
array([[ 1.75,  1.75,  0.75,  2.  ,  1.5 ],
       [ 0.75,  2.5 ,  1.75,  1.5 ,  0.75],
       [ 0.75,  1.75,  2.25,  0.25,  1.25],
       [ 1.5 ,  2.25,  1.25,  1.5 ,  1.5 ]])