Setup

consider the numpy array a

a = np.arange(30).reshape(2, 3, 5)
print(a)

[[[ 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]]]

Where are the dimensions?

The dimensions and positions are highlighted by the following

            p  p  p  p  p
            o  o  o  o  o
            s  s  s  s  s

     dim 2  0  1  2  3  4

            |  |  |  |  |
  dim 0     โ†“  โ†“  โ†“  โ†“  โ†“
  ----> [[[ 0  1  2  3  4]   <---- dim 1, pos 0
  pos 0   [ 5  6  7  8  9]   <---- dim 1, pos 1
          [10 11 12 13 14]]  <---- dim 1, pos 2
  dim 0
  ---->  [[15 16 17 18 19]   <---- dim 1, pos 0
  pos 1   [20 21 22 23 24]   <---- dim 1, pos 1
          [25 26 27 28 29]]] <---- dim 1, pos 2
            โ†‘  โ†‘  โ†‘  โ†‘  โ†‘
            |  |  |  |  |

     dim 2  p  p  p  p  p
            o  o  o  o  o
            s  s  s  s  s

            0  1  2  3  4

Dimension examples:

This becomes more clear with a few examples

a[0, :, :] # dim 0, pos 0

[[ 0  1  2  3  4]
 [ 5  6  7  8  9]
 [10 11 12 13 14]]

a[:, 1, :] # dim 1, pos 1

[[ 5  6  7  8  9]
 [20 21 22 23 24]]

a[:, :, 3] # dim 2, pos 3

[[ 3  8 13]
 [18 23 28]]

sum

explanation of sum and axis
a.sum(0) is the sum of all slices along dim 0

a.sum(0)

[[15 17 19 21 23]
 [25 27 29 31 33]
 [35 37 39 41 43]]

same as

a[0, :, :] + \
a[1, :, :]

[[15 17 19 21 23]
 [25 27 29 31 33]
 [35 37 39 41 43]]

a.sum(1) is the sum of all slices along dim 1

a.sum(1)

[[15 18 21 24 27]
 [60 63 66 69 72]]

same as 

a[:, 0, :] + \
a[:, 1, :] + \
a[:, 2, :]

[[15 18 21 24 27]
 [60 63 66 69 72]]

a.sum(2) is the sum of all slices along dim 2

a.sum(2)

[[ 10  35  60]
 [ 85 110 135]]

same as

a[:, :, 0] + \
a[:, :, 1] + \
a[:, :, 2] + \
a[:, :, 3] + \
a[:, :, 4]

[[ 10  35  60]
 [ 85 110 135]]

default axis is -1
this means all axes. or sum all numbers.

a.sum()

435
Answer from piRSquared on Stack Overflow
๐ŸŒ
Stack Overflow
stackoverflow.com โ€บ questions โ€บ 40857930 โ€บ how-does-numpy-sum-with-axis-work
python - How does NumPy Sum (with axis) work? - Stack Overflow
Top answer
1 of 3
95

Setup

consider the numpy array a

a = np.arange(30).reshape(2, 3, 5)
print(a)

[[[ 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]]]

Where are the dimensions?

The dimensions and positions are highlighted by the following

            p  p  p  p  p
            o  o  o  o  o
            s  s  s  s  s

     dim 2  0  1  2  3  4

            |  |  |  |  |
  dim 0     โ†“  โ†“  โ†“  โ†“  โ†“
  ----> [[[ 0  1  2  3  4]   <---- dim 1, pos 0
  pos 0   [ 5  6  7  8  9]   <---- dim 1, pos 1
          [10 11 12 13 14]]  <---- dim 1, pos 2
  dim 0
  ---->  [[15 16 17 18 19]   <---- dim 1, pos 0
  pos 1   [20 21 22 23 24]   <---- dim 1, pos 1
          [25 26 27 28 29]]] <---- dim 1, pos 2
            โ†‘  โ†‘  โ†‘  โ†‘  โ†‘
            |  |  |  |  |

     dim 2  p  p  p  p  p
            o  o  o  o  o
            s  s  s  s  s

            0  1  2  3  4

Dimension examples:

This becomes more clear with a few examples

a[0, :, :] # dim 0, pos 0

[[ 0  1  2  3  4]
 [ 5  6  7  8  9]
 [10 11 12 13 14]]

a[:, 1, :] # dim 1, pos 1

[[ 5  6  7  8  9]
 [20 21 22 23 24]]

a[:, :, 3] # dim 2, pos 3

[[ 3  8 13]
 [18 23 28]]

sum

explanation of sum and axis
a.sum(0) is the sum of all slices along dim 0

a.sum(0)

[[15 17 19 21 23]
 [25 27 29 31 33]
 [35 37 39 41 43]]

same as

a[0, :, :] + \
a[1, :, :]

[[15 17 19 21 23]
 [25 27 29 31 33]
 [35 37 39 41 43]]

a.sum(1) is the sum of all slices along dim 1

a.sum(1)

[[15 18 21 24 27]
 [60 63 66 69 72]]

same as 

a[:, 0, :] + \
a[:, 1, :] + \
a[:, 2, :]

[[15 18 21 24 27]
 [60 63 66 69 72]]

a.sum(2) is the sum of all slices along dim 2

a.sum(2)

[[ 10  35  60]
 [ 85 110 135]]

same as

a[:, :, 0] + \
a[:, :, 1] + \
a[:, :, 2] + \
a[:, :, 3] + \
a[:, :, 4]

[[ 10  35  60]
 [ 85 110 135]]

default axis is -1
this means all axes. or sum all numbers.

a.sum()

435
2 of 3
4

I use a nested loop operation to explain it.

import numpy as np

n = np.array(
[[[1, 2, 3],
 [4, 5, 6],
 [7, 8, 9]],

 [[2, 4, 6],
 [8, 10, 12],
 [14, 16, 18]],

 [[1, 3, 5],
 [7, 9, 11],
 [13, 15, 17]]])

print(n)

print("============ sum axis=None=============")

sum = 0
for i in range(3):
  for j in range(3): 
    for k in range(3):
      sum += n[k][i][j]
print(sum) # 216

print('------------------')
print(np.sum(n))  # 216
print("============ sum axis=0 =============") 
for i in range(3):
  for j in range(3):
    sum = 0
    for axis in range(3):
      sum += n[axis][i][j]
    print(sum,end=' ')
  print()

print('------------------')
print("sum[0][0] = %d" % (n[0][0][0] + n[1][0][0] + n[2][0][0]))
print("sum[1][1] = %d" % (n[0][1][1] + n[1][1][1] + n[2][1][1]))
print("sum[2][2] = %d" % (n[0][2][2] + n[1][2][2] + n[2][2][2]))
print('------------------')
print(np.sum(n, axis=0)) 
print("============ sum axis=1 =============") 
for i in range(3):
  for j in range(3):
    sum = 0
    for axis in range(3):
      sum += n[i][axis][j]
    print(sum,end=' ')
  print()
print('------------------')
print("sum[0][0] = %d" % (n[0][0][0] + n[0][1][0] + n[0][2][0]))
print("sum[1][1] = %d" % (n[1][0][1] + n[1][1][1] + n[1][2][1]))
print("sum[2][2] = %d" % (n[2][0][2] + n[2][1][2] + n[2][2][2]))
print('------------------')
print(np.sum(n, axis=1))  
print("============ sum axis=2 =============") 
for i in range(3):
  for j in range(3):
    sum = 0
    for axis in range(3):
      sum += n[i][j][axis]
    print(sum,end=' ')
  print()
print('------------------')
print("sum[0][0] = %d" % (n[0][0][0] + n[0][0][1] + n[0][0][2]))
print("sum[1][1] = %d" % (n[1][1][0] + n[1][1][1] + n[1][1][2]))
print("sum[2][2] = %d" % (n[2][2][0] + n[2][2][1] + n[2][2][2]))
print('------------------')
print(np.sum(n, axis=2))
print("============ sum axis=(0,1)) =============") 
for i in range(3):
  sum = 0
  for axis1 in range(3):   
    for axis2 in range(3):
      sum += n[axis1][axis2][i]
  print(sum,end=' ')

print()
print('------------------')
print("sum[1] = %d" % (n[0][0][1] + n[0][1][1] + n[0][2][1] +
              n[1][0][1] + n[1][1][1] + n[1][2][1] +
              n[2][0][1] + n[2][1][1] + n[2][2][1] ))
print('------------------')
print(np.sum(n, axis=(0,1)))

result๏ผš

[[[ 1  2  3]
  [ 4  5  6]
  [ 7  8  9]]

 [[ 2  4  6]
  [ 8 10 12]
  [14 16 18]]

 [[ 1  3  5]
  [ 7  9 11]
  [13 15 17]]]
============ sum axis=None=============
216
------------------
216
============ sum axis=0 =============
4 9 14 
19 24 29 
34 39 44 
------------------
sum[0][0] = 4
sum[1][1] = 24
sum[2][2] = 44
------------------
[[ 4  9 14]
 [19 24 29]
 [34 39 44]]
============ sum axis=1 =============
12 15 18 
24 30 36 
21 27 33 
------------------
sum[0][0] = 12
sum[1][1] = 30
sum[2][2] = 33
------------------
[[12 15 18]
 [24 30 36]
 [21 27 33]]
============ sum axis=2 =============
6 15 24 
12 30 48 
9 27 45 
------------------
sum[0][0] = 6
sum[1][1] = 30
sum[2][2] = 45
------------------
[[ 6 15 24]
 [12 30 48]
 [ 9 27 45]]
============ sum axis=(0,1)) =============
57 72 87 
------------------
sum[1] = 72
------------------
[57 72 87]
python - Make numpy.sum() return a sum of matrices instead of a single number
python - numpy sum along axis
python - Numpy sum elements in a multi-dimensional array according to indices
python
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๐ŸŒ
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numpy.org โ€บ doc โ€บ stable โ€บ reference โ€บ generated โ€บ numpy.sum.html
numpy.sum โ€” NumPy v2.5 Manual
Elements to sum. ... Axis or axes along which a sum is performed. The default, axis=None, will sum all of the elements of the input array. If axis is negative it counts from the last to the first axis.
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Elements to sum. ... Axis or axes along which a sum is performed. The default, axis=None, will sum all of the elements of the input array. If axis is negative it counts from the last to the first axis.
๐ŸŒ
Reddit
reddit.com โ€บ r/learnpython โ€บ numpy axis confusion
r/learnpython on Reddit: NumPy Axis Confusion
December 19, 2024 -

Not terribly new to Python or programming in general, but I'm looking at some initial NumPy exercises for Introduction to Statistical Learning with Applications in Python and I'm rather confused about the axis parameter to some methods.

Here's the relevant supporting code from the ISLP Lab Notebook: 

    rng = np.random.default_rng(3)
    X = rng.standard_normal((10, 3))
    
    # array([[ 0.22578661, -0.35263079, -0.28128742],
    #        [-0.66804635, -1.05515055, -0.39080098],
    #        [ 0.48194539, -0.23855361,  0.9577587 ],
    #        [-0.19980213,  0.02425957,  1.54582085],
    #        [ 0.54510552, -0.50522874, -0.18283897],
    #        [ 0.54052513,  1.93508803, -0.26962033],
    #        [-0.24355868,  1.0023136 , -0.88645994],
    #        [-0.29172023,  0.88253897,  0.58035002],
    #        [ 0.0915167 ,  0.67010435, -2.82816231],
    #        [ 1.02130682, -0.95964476, -1.66861984]])
    
    X.mean(axis=0)
    
    # array([ 0.15030588,  0.14030961, -0.34238602])

The accompanying MD text here says:

The np.mean(), np.var(), and np.std() functions can also be applied to the rows and columns of a matrix. To see this, we construct a matrix of random variables, and consider computing its row sums.

Since arrays are row-major ordered, the first axis, i.e. axis=0, refers to its rows. We pass this argument into the mean() method for the object X.

I know axis=0 is rows and axis=1 is columns (for this example, at least). Since the example is passing axis 0 (rows) to the mean method, I would expect an output array of length 10 that calculates the mean of 3 elements in each row.

But the example appears to be calculating the mean of 10 elements in each of 3 columns, given the output array of length 3.

Further, when I calculate the column mean using X.mean(axis=1), I get the result I expected for axis=0, so the outputs are being switched.

The NumPy documentation didn't provide any additional clarity.

Hoping someone can provide an explanation for what's going on here. Is there possibly a setting (perhaps embedded in the notebook, that I can't see) that allows axes to be switched, where rows are axis=1 and columns are axis=0?

Thanks in advance for any help!

Top answer
1 of 1
2
If you have the following: import numpy as np vec = np.array([1, 2, 3, 4]) row = np.array([[1, 2, 3, 4]]) col = np.array([[1], [2], [3], [4]]) mat = np.array([[1, 2, 3, 4], [5, 6, 7, 8]]) book = np.array([[[1.0, 2.0, 3.0, 4.0], [5.0, 6.0, 7.0, 8.0]], [[1.1, 2.1, 3.1, 4.1], [5.1, 6.1, 7.1, 8.1]], [[1.2, 2.2, 3.2, 4.2], [5.2, 6.2, 7.2, 8.2]]]) The shape tuples are: > vec.shape (4 columns, ) >row.shape (1 row, 4 columns) > col.shape (4 rows, 1 column) > mat.shape (2 rows, 4 columns) > book.shape (3 sheets, 2 rows, 4 columns) axis 0, is element at index 0 in the shape tuple. axis=0 is columns for a 1d array, rows for a 2d array, sheets for a 3d array and relates to the outer []. axis 1, is element at index 1 in the shape tuple. axis=1 is columns for a 2d array, rows for a 2d array. Instead of approaching the shape tuple from left to right, approach the shape tuple from right to left. The last element axis=-1 is always columns, axis=-2 is always rows, axis=-3 is always sheets and so on... If we look at mat and think of the axis parameter as the following arrows: axis=-1 โ†’ operates along column axis np.array([[1, 2, 3, 4], [5, 6, 7, 8]]) axis=-2 โ†“ np.array([[1, 2, 3, 4], [5, 6, 7, 8]]) An operation along axis=-1, operates along columns and therefore returns a column: > mat.sum(axis=-1, keepdims=True) array([[10], [26]]) An operation along axis=-2, operates along rows and therefore returns a row: mat.sum(axis=-2, keepdims=True) > array([[ 6, 8, 10, 12]]) I've put a bit more detail in this markdown tutorial here covering the axis parameter and dimensionality of ndarrays. GitHub: numpy library: axis, shape and negative index . I'm still working on this tutorial but the section that covers the axis parameter should be helpful if you need a bit more information.
๐ŸŒ
W3Schools
w3schools.com โ€บ python โ€บ numpy โ€บ numpy_ufunc_summations.asp
NumPy ufuncs - Summations
import numpy as np arr1 = np.array([1, 2, 3]) arr2 = np.array([1, 2, 3]) newarr = np.sum([arr1, arr2]) print(newarr) Try it Yourself ยป ยท Returns: 12 ยท If you specify axis=1, NumPy will sum the numbers in each array.
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Note.nkmk.me
note.nkmk.me โ€บ home โ€บ python โ€บ numpy
NumPy: Meaning of the axis parameter (0, 1, -1) | note.nkmk.me
January 18, 2024 - # print(np.sum(a, axis=2)) # AxisError: axis 2 is out of bounds for array of dimension 2
๐ŸŒ
Stack Overflow
stackoverflow.com โ€บ questions โ€บ 40200070 โ€บ what-does-axis-0-do-in-numpys-sum-function
python - What does axis = 0 do in Numpy's sum function? - Stack Overflow
Top answer
1 of 7
180

In case someone needs this visual description:

2 of 7
43

All that is going on is that numpy is summing across the first (0th) and only axis. Consider the following:

In [2]: a = np.array([1, 2, 3])

In [3]: a.shape
Out[3]: (3,)

In [4]: len(a.shape) # number of dimensions
Out[4]: 1

In [5]: a1 = a.reshape(3,1)

In [6]: a2 = a.reshape(1,3)

In [7]: a1
Out[7]: 
array([[1],
       [2],
       [3]])

In [8]: a2
Out[8]: array([[1, 2, 3]])

In [9]: a1.sum(axis=1)
Out[9]: array([1, 2, 3])

In [10]: a1.sum(axis=0)
Out[10]: array([6])

In [11]: a2.sum(axis=1)
Out[11]: array([6])

In [12]: a2.sum(axis=0)
Out[12]: array([1, 2, 3])

So, to be more explicit:

In [15]: a1.shape
Out[15]: (3, 1)

a1 is 2-dimensional, the "long" axis being the first.

In [16]: a1[:,0] # give me everything in the first axis, and the first part of the second
Out[16]: array([1, 2, 3])

Now, sum along the first axis:

In [17]: a1.sum(axis=0)
Out[17]: array([6])

Now, consider a less trivial two-dimensional case:

In [20]: b = np.array([[1,2,3],[4,5,6]])

In [21]: b
Out[21]: 
array([[1, 2, 3],
       [4, 5, 6]])

In [22]: b.shape
Out[22]: (2, 3)

The first axis is the "rows". Sum along the rows:

In [23]: b.sum(axis=0)
Out[23]: array([5, 7, 9])

The second axis are the "columns". Sum along the columns:

In [24]: b.sum(axis=1)
Out[24]: array([ 6, 15])
๐ŸŒ
SciPy
docs.scipy.org โ€บ doc โ€บ numpy-1.9.1 โ€บ reference โ€บ generated โ€บ numpy.sum.html
numpy.sum โ€” NumPy v1.10 Manual
October 18, 2015 - Sum of array elements over a given axis. ... Equivalent method. ... Cumulative sum of array elements. ... Integration of array values using the composite trapezoidal rule. ... Arithmetic is modular when using integer types, and no error is raised on overflow. The sum of an empty array is the neutral element 0: ... >>> np.sum([0.5, 1.5]) 2.0 >>> np.sum([0.5, 0.7, 0.2, 1.5], dtype=np.int32) 1 >>> np.sum([[0, 1], [0, 5]]) 6 >>> np.sum([[0, 1], [0, 5]], axis=0) array([0, 6]) >>> np.sum([[0, 1], [0, 5]], axis=1) array([1, 5])
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March 6, 2023 - Am I the only one who is wondering this? The way to understand what โ€œaxisโ€ means in numpy sum is that it collapses the specified axis. So when it collapses the axis 0 (the row), it becomes just one row (it sums column-wise).
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May 28, 2024 - axis=2: Operates along columns within each depth level. The code is available in this colab notebook. import numpy as np # 1D Array arr_1d = np.array([1, 2, 3, 4]) print("1D Array:\n", arr_1d) # Output: [1 2 3 4] print("Sum (Axis 0):", ...
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August 23, 2018 - Sum of array elements over a given axis. ... Equivalent method. ... Cumulative sum of array elements. ... Integration of array values using the composite trapezoidal rule. ... Arithmetic is modular when using integer types, and no error is raised on overflow. The sum of an empty array is the neutral element 0: ... >>> np.sum([0.5, 1.5]) 2.0 >>> np.sum([0.5, 0.7, 0.2, 1.5], dtype=np.int32) 1 >>> np.sum([[0, 1], [0, 5]]) 6 >>> np.sum([[0, 1], [0, 5]], axis=0) array([0, 6]) >>> np.sum([[0, 1], [0, 5]], axis=1) array([1, 5])
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Sum of array elements over a given axis. ... Equivalent method. ... Cumulative sum of array elements. ... Integration of array values using the composite trapezoidal rule. ... Arithmetic is modular when using integer types, and no error is raised on overflow. The sum of an empty array is the neutral element 0: ... >>> np.sum([0.5, 1.5]) 2.0 >>> np.sum([0.5, 0.7, 0.2, 1.5], dtype=np.int32) 1 >>> np.sum([[0, 1], [0, 5]]) 6 >>> np.sum([[0, 1], [0, 5]], axis=0) array([0, 6]) >>> np.sum([[0, 1], [0, 5]], axis=1) array([1, 5])
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numpy.org โ€บ doc โ€บ 2.1 โ€บ reference โ€บ generated โ€บ numpy.sum.html
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numpy.org โ€บ doc โ€บ 2.1 โ€บ reference โ€บ generated โ€บ numpy.matrix.sum.html
numpy.matrix.sum โ€” NumPy v2.1 Manual
Returns the sum of the matrix elements, along the given axis. Refer to numpy.sum for full documentation. ... This is the same as ndarray.sum, except that where an ndarray would be returned, a matrix object is returned instead. ... >>> x = np.matrix([[1, 2], [4, 3]]) >>> x.sum() 10 >>> x.sum(axis=1) ...
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numpy.org โ€บ doc โ€บ 2.2 โ€บ reference โ€บ generated โ€บ numpy.matrix.sum.html
numpy.matrix.sum โ€” NumPy v2.2 Manual
Returns the sum of the matrix elements, along the given axis. Refer to numpy.sum for full documentation. ... This is the same as ndarray.sum, except that where an ndarray would be returned, a matrix object is returned instead. ... >>> x = np.matrix([[1, 2], [4, 3]]) >>> x.sum() 10 >>> x.sum(axis=1) ...