Explain this 4D numpy array indexing intuitively

问题:

x = np.random.randn(4, 3, 3, 2)
print(x[1,1])

output:
[[ 1.68158825 -0.03701415]
[ 1.0907524  -1.94530359]
[ 0.25659178  0.00475093]]

I am python newbie. I can't really understand 4-D array index like above. What does x[1,1] mean?

For example, for vector

a = [[2][3][8][9]], a[0 = 2, a[3] = 9. 

I get this but I don't know what x[1,1] refers to.

Please explain in detail. Thank you.

回答1:

A 2D array is a matrix : an array of arrays.

A 4D array is basically a matrix of matrices:

Specifying one index gives you an array of matrices:

>>> x[1]
array([[[-0.37387191, -0.19582887],
        [-2.88810217, -0.8249608 ],
        [-0.46763329,  1.18628611]],

       [[-1.52766397, -0.2922034 ],
        [ 0.27643125, -0.87816021],
        [-0.49936658,  0.84011388]],

       [[ 0.41885001,  0.16037164],
        [ 1.21510322,  0.01923682],
        [ 0.96039904, -0.22761806]]])

Specifying two indices gives you a matrix:

>>> x[1, 1]
array([[-1.52766397, -0.2922034 ],
       [ 0.27643125, -0.87816021],
       [-0.49936658,  0.84011388]])

Specifying three indices gives you an array:

>>> x[1, 1, 1]
array([ 0.27643125, -0.87816021])

Specifying four indices gives you a single element:

>>> x[1, 1, 1, 1]
-0.87816021212791107

x[1,1] gives you the small matrix that was saved in the 2nd column of the 2nd row of the large matrix.

回答2:

A 4d numpy array is an array nested 4 layers deep, so at the top level it would look like this:

[ # 1st level Array (Outer)
    [ # 2nd level Array
        [[1, 2], [3, 4]], # 3rd level arrays, containing 2 4th level arrays
        [[5, 6], [7, 8]]
    ], 
    [ # 2nd Level array
        [[9, 10], [11, 12]], 
        [[13, 14], [15, 16]]
    ]
]

x[1,1] expands to x[1][1], Let's unpack this one expression at a time, the first expression x[1] selects the first element from the global array which is the following object from the earlier array:

[
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
]

The next expression now looks like this:

[
    [[1, 2], [3, 4]],
    [[5, 6], [7, 8]]
][1]

So evaluating that (selecting the first element in the array) gives us the following result:

[[1, 2], [3, 4]]

As you can see selecting an element in a 4d array gives us a 3d array, selecting an element from a 3d array gives a 2d array and selecting an element from a 2d array gives us a 1d array.

回答3:

This page explains the concept of higher dimensional arrays with a very clear analogy together with some illustrations. I recommend to those people looking for an answer to this question

https://cognitiveclass.ai/blog/nested-lists-multidimensional-numpy-arrays