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reshape numpy array with a[:,None] [duplicate]

What is numpy.newaxis and when should I use it?
Using it on a 1-D array x produces:
>>> x
array([0, 1, 2, 3])
>>> x[np.newaxis, :]
array([[0, 1, 2, 3]])
>>> x[:, np.newaxis]
array([[0],
[1],
[2],
[3]])

Simply put, numpy.newaxis is used to increase the dimension of the existing array by one more dimension, when used once. Thus,
1D array will become 2D array
2D array will become 3D array
3D array will become 4D array
4D array will become 5D array
and so on..
Here is a visual illustration which depicts promotion of 1D array to 2D arrays.
Scenario-1: np.newaxis might come in handy when you want to explicitly convert a 1D array to either a row vector or a column vector, as depicted in the above picture.
Example:
# 1D array
In [7]: arr = np.arange(4)
In [8]: arr.shape
Out[8]: (4,)
# make it as row vector by inserting an axis along first dimension
In [9]: row_vec = arr[np.newaxis, :] # arr[None, :]
In [10]: row_vec.shape
Out[10]: (1, 4)
# make it as column vector by inserting an axis along second dimension
In [11]: col_vec = arr[:, np.newaxis] # arr[:, None]
In [12]: col_vec.shape
Out[12]: (4, 1)
Scenario-2: When we want to make use of numpy broadcasting as part of some operation, for instance while doing addition of some arrays.
Example:
Let's say you want to add the following two arrays:
x1 = np.array([1, 2, 3, 4, 5])
x2 = np.array([5, 4, 3])
If you try to add these just like that, NumPy will raise the following ValueError :
ValueError: operands could not be broadcast together with shapes (5,) (3,)
In this situation, you can use np.newaxis to increase the dimension of one of the arrays so that NumPy can broadcast.
In [2]: x1_new = x1[:, np.newaxis] # x1[:, None]
# now, the shape of x1_new is (5, 1)
# array([[1],
# [2],
# [3],
# [4],
# [5]])
Now, add:
In [3]: x1_new + x2
Out[3]:
array([[ 6, 5, 4],
[ 7, 6, 5],
[ 8, 7, 6],
[ 9, 8, 7],
[10, 9, 8]])
Alternatively, you can also add new axis to the array x2:
In [6]: x2_new = x2[:, np.newaxis] # x2[:, None]
In [7]: x2_new # shape is (3, 1)
Out[7]:
array([[5],
[4],
[3]])
Now, add:
In [8]: x1 + x2_new
Out[8]:
array([[ 6, 7, 8, 9, 10],
[ 5, 6, 7, 8, 9],
[ 4, 5, 6, 7, 8]])
Note: Observe that we get the same result in both cases (but one being the transpose of the other).
Scenario-3: This is similar to scenario-1. But, you can use np.newaxis more than once to promote the array to higher dimensions. Such an operation is sometimes needed for higher order arrays (i.e. Tensors).
Example:
In [124]: arr = np.arange(5*5).reshape(5,5)
In [125]: arr.shape
Out[125]: (5, 5)
# promoting 2D array to a 5D array
In [126]: arr_5D = arr[np.newaxis, ..., np.newaxis, np.newaxis] # arr[None, ..., None, None]
In [127]: arr_5D.shape
Out[127]: (1, 5, 5, 1, 1)
As an alternative, you can use numpy.expand_dims that has an intuitive axis kwarg.
# adding new axes at 1st, 4th, and last dimension of the resulting array
In [131]: newaxes = (0, 3, -1)
In [132]: arr_5D = np.expand_dims(arr, axis=newaxes)
In [133]: arr_5D.shape
Out[133]: (1, 5, 5, 1, 1)
More background on np.newaxis vs np.reshape
newaxis is also called as a pseudo-index that allows the temporary addition of an axis into a multiarray.
np.newaxis uses the slicing operator to recreate the array while numpy.reshape reshapes the array to the desired layout (assuming that the dimensions match; And this is must for a reshape to happen).
Example
In [13]: A = np.ones((3,4,5,6))
In [14]: B = np.ones((4,6))
In [15]: (A + B[:, np.newaxis, :]).shape # B[:, None, :]
Out[15]: (3, 4, 5, 6)
In the above example, we inserted a temporary axis between the first and second axes of B (to use broadcasting). A missing axis is filled-in here using np.newaxis to make the broadcasting operation work.
General Tip: You can also use None in place of np.newaxis; These are in fact the same objects.
In [13]: np.newaxis is None
Out[13]: True
P.S. Also see this great answer: newaxis vs reshape to add dimensions

What is np.newaxis?
The np.newaxis is just an alias for the Python constant None, which means that wherever you use np.newaxis you could also use None:
>>> np.newaxis is None
True
It's just more descriptive if you read code that uses np.newaxis instead of None.
How to use np.newaxis?
The np.newaxis is generally used with slicing. It indicates that you want to add an additional dimension to the array. The position of the np.newaxis represents where I want to add dimensions.
>>> import numpy as np
>>> a = np.arange(10)
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> a.shape
(10,)
In the first example I use all elements from the first dimension and add a second dimension:
>>> a[:, np.newaxis]
array([[0],
[1],
[2],
[3],
[4],
[5],
[6],
[7],
[8],
[9]])
>>> a[:, np.newaxis].shape
(10, 1)
The second example adds a dimension as first dimension and then uses all elements from the first dimension of the original array as elements in the second dimension of the result array:
>>> a[np.newaxis, :] # The output has 2 [] pairs!
array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]])
>>> a[np.newaxis, :].shape
(1, 10)
Similarly you can use multiple np.newaxis to add multiple dimensions:
>>> a[np.newaxis, :, np.newaxis] # note the 3 [] pairs in the output
array([[[0],
[1],
[2],
[3],
[4],
[5],
[6],
[7],
[8],
[9]]])
>>> a[np.newaxis, :, np.newaxis].shape
(1, 10, 1)
Are there alternatives to np.newaxis?
There is another very similar functionality in NumPy: np.expand_dims, which can also be used to insert one dimension:
>>> np.expand_dims(a, 1) # like a[:, np.newaxis]
>>> np.expand_dims(a, 0) # like a[np.newaxis, :]
But given that it just inserts 1s in the shape you could also reshape the array to add these dimensions:
>>> a.reshape(a.shape + (1,)) # like a[:, np.newaxis]
>>> a.reshape((1,) + a.shape) # like a[np.newaxis, :]
Most of the times np.newaxis is the easiest way to add dimensions, but it's good to know the alternatives.
When to use np.newaxis?
In several contexts is adding dimensions useful:
If the data should have a specified number of dimensions. For example if you want to use matplotlib.pyplot.imshow to display a 1D array.
If you want NumPy to broadcast arrays. By adding a dimension you could for example get the difference between all elements of one array: a - a[:, np.newaxis]. This works because NumPy operations broadcast starting with the last dimension 1.
To add a necessary dimension so that NumPy can broadcast arrays. This works because each length-1 dimension is simply broadcast to the length of the corresponding1 dimension of the other array.
1 If you want to read more about the broadcasting rules the NumPy documentation on that subject is very good. It also includes an example with np.newaxis:
>>> a = np.array([0.0, 10.0, 20.0, 30.0])
>>> b = np.array([1.0, 2.0, 3.0])
>>> a[:, np.newaxis] + b
array([[ 1., 2., 3.],
[ 11., 12., 13.],
[ 21., 22., 23.],
[ 31., 32., 33.]])

You started with a one-dimensional list of numbers. Once you used numpy.newaxis, you turned it into a two-dimensional matrix, consisting of four rows of one column each.
You could then use that matrix for matrix multiplication, or involve it in the construction of a larger 4 x n matrix.

newaxis object in the selection tuple serves to expand the dimensions of the resulting selection by one unit-length dimension.
It is not just conversion of row matrix to column matrix.
Consider the example below:
In [1]:x1 = np.arange(1,10).reshape(3,3)
print(x1)
Out[1]: array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
Now lets add new dimension to our data,
In [2]:x1_new = x1[:,np.newaxis]
print(x1_new)
Out[2]:array([[[1, 2, 3]],
[[4, 5, 6]],
[[7, 8, 9]]])
You can see that newaxis added the extra dimension here, x1 had dimension (3,3) and X1_new has dimension (3,1,3).
How our new dimension enables us to different operations:
In [3]:x2 = np.arange(11,20).reshape(3,3)
print(x2)
Out[3]:array([[11, 12, 13],
[14, 15, 16],
[17, 18, 19]])
Adding x1_new and x2, we get:
In [4]:x1_new+x2
Out[4]:array([[[12, 14, 16],
[15, 17, 19],
[18, 20, 22]],
[[15, 17, 19],
[18, 20, 22],
[21, 23, 25]],
[[18, 20, 22],
[21, 23, 25],
[24, 26, 28]]])
Thus, newaxis is not just conversion of row to column matrix. It increases the dimension of matrix, thus enabling us to do more operations on it.

What's the difference between shape(150,) and shape (150,1)?

What's the difference between shape(150,) and shape (150,1)?
I think they are the same, I mean they both represent a column vector.

Both have the same values, but one is a vector and the other one is a matrix of the vector. Here's an example:
import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([[1], [2], [3], [4], [5]])
print(x.shape)
print(y.shape)
And the output is:
(5,)
(5, 1)

Although they both occupy same space and positions in memory,
I think they are the same, I mean they both represent a column vector.
No they are not and certainly not according to NumPy (ndarrays).
The main difference is that the
shape (150,) => is a 1D array, whereas
shape (150,1) => is a 2D array

Questions like this see to come from two misconceptions.
not realizing that (5,) is a 1 element tuple.
expecting MATLAB like matrices
Make an array with the handy arange function:
In [424]: x = np.arange(5)
In [425]: x.shape
Out[425]: (5,) # 1 element tuple
In [426]: x.ndim
Out[426]: 1
numpy does not automatically make matrices, 2d arrays. It does not follow MATLAB in that regard.
We can reshape that array, adding a 2nd dimension. The result is a view (sooner or later you need to learn what that means):
In [427]: y = x.reshape(5,1)
In [428]: y.shape
Out[428]: (5, 1)
In [429]: y.ndim
Out[429]: 2
The display of these 2 arrays is very different. Same numbers, but the layout and number of brackets is very different, reflecting the respective shapes:
In [430]: x
Out[430]: array([0, 1, 2, 3, 4])
In [431]: y
Out[431]:
array([[0],
[1],
[2],
[3],
[4]])
The shape difference may seem academic - until you try to do math with the arrays:
In [432]: x+x
Out[432]: array([0, 2, 4, 6, 8]) # element wise sum
In [433]: x+y
Out[433]:
array([[0, 1, 2, 3, 4],
[1, 2, 3, 4, 5],
[2, 3, 4, 5, 6],
[3, 4, 5, 6, 7],
[4, 5, 6, 7, 8]])
How did that end up producing a (5,5) array? Broadcasting a (5,) array with a (5,1) array!

Numpy: Get matrices from tensor given list of indexes

I have a tensor with the shape (4, 3, 20). When I do X[:, 0, :].shape I get (4, 20). When I do X[:, [0,2,0,1], :].shape I get (4, 4, 20).
What I have is a list of indexes representing the second dimension of my tensor. I want to get a two-dimensional matrix like I get when I do X[:, 0, :] but I have different indexes for the second dimension instead of only one. How do I do that?

Your question is unclear, but I'll make a guess
In [58]: X=np.arange(24).reshape(4,3,2)
In [59]: X[range(4),[0,2,0,1],:]
Out[59]:
array([[ 0, 1],
[10, 11],
[12, 13],
[20, 21]])
This picks row 0 from the 1st plane; row 2 from the 2nd, etc. The result has the same shape as X[:,0,:], but values are pulled from different 1st dimension planes.
In [61]: X[:,0,:]
Out[61]:
array([[ 0, 1], # same
[ 6, 7],
[12, 13], # same
[18, 19]])

I think you are looking for np.squeeze. So, for cases when the indexing list, say L has just one element and upon indexing the input array with it would result in a 3D array with a singleton second dimension (dimension of length 1), would result in a 2D output with that squeez-ing. For L with more than one element, the indexing would result in a 3D array without any singleton dimension and thus, no change with that squeez-ing and hence the desired output. Thus, the solution with it would be -
np.squeeze(X[:,L,:])
Sample run to test out shapes on a random array -
In [25]: A = np.random.rand(4,3,20)
In [26]: L = [0]
In [27]: np.squeeze(A[:,L,:]).shape
Out[27]: (4, 20)
In [28]: L = [0,2,0,1]
In [29]: np.squeeze(A[:,L,:]).shape
Out[29]: (4, 4, 20)

Numpy 3D array transposed when indexed in single step vs two steps

import numpy as np
x = np.random.randn(2, 3, 4)
mask = np.array([1, 0, 1, 0], dtype=np.bool)
y = x[0, :, mask]
z = x[0, :, :][:, mask]
print(y)
print(z)
print(y.T)
Why does doing the above operation in two steps result in the transpose of doing it in one step?

Here's the same behavior with a list index:
In [87]: x=np.arange(2*3*4).reshape(2,3,4)
In [88]: x[0,:,[0,2]]
Out[88]:
array([[ 0, 4, 8],
[ 2, 6, 10]])
In [89]: x[0,:,:][:,[0,2]]
Out[89]:
array([[ 0, 2],
[ 4, 6],
[ 8, 10]])
In the 2nd case, x[0,:,:] returns a (3,4) array, and the next index picks 2 columns.
In the 1st case, it first selects on the first and last dimensions, and appends the slice (the middle dimension). The 0 and [0,2] produce a 2 dimension, and the 3 from the middle is appended, giving (2,3) shape.
This is a case of mixed basic and advanced indexing.
http://docs.scipy.org/doc/numpy/reference/arrays.indexing.html#combining-advanced-and-basic-indexing
In the first case, the dimensions resulting from the advanced indexing operation come first in the result array, and the subspace dimensions after that.
This is not an easy case to comprehend or explain. Basically there's some ambiguity as to what the final dimension should be. It tries to illustrate with an example x[:,ind_1,:,ind_2] where ind_1 and ind_2 are 3d (or together broadcast to that).
Earlier attempts to explain this are:
How does numpy order array slice indices?
Combining slicing and broadcasted indexing for multi-dimensional numpy arrays
===========================
A way around this problem is to replace the slice with an array - a column vector
In [221]: x[0,np.array([0,1,2])[:,None],[0,2]]
Out[221]:
array([[ 0, 2],
[ 4, 6],
[ 8, 10]])
In [222]: np.ix_([0],[0,1,2],[0,2])
Out[222]:
(array([[[0]]]), array([[[0],
[1],
[2]]]), array([[[0, 2]]]))
In [223]: x[np.ix_([0],[0,1,2],[0,2])]
Out[223]:
array([[[ 0, 2],
[ 4, 6],
[ 8, 10]]])
Though this last case is 3d, (1,3,2). ix_ didn't like the scalar 0. An alternate way of using ix_:
In [224]: i,j=np.ix_([0,1,2],[0,2])
In [225]: x[0,i,j]
Out[225]:
array([[ 0, 2],
[ 4, 6],
[ 8, 10]])
And here's a way of getting the same numbers, but in a (2,1,3) array:
In [232]: i,j=np.ix_([0,2],[0])
In [233]: x[j,:,i]
Out[233]:
array([[[ 0, 4, 8]],
[[ 2, 6, 10]]])

Numpy: add a vector to matrix column wise

a
Out[57]:
array([[1, 2],
[3, 4]])
b
Out[58]:
array([[5, 6],
[7, 8]])
In[63]: a[:,-1] + b
Out[63]:
array([[ 7, 10],
[ 9, 12]])
This is row wise addition. How do I add them column wise to get
In [65]: result
Out[65]:
array([[ 7, 8],
[11, 12]])
I don't want to transpose the whole array, add and then transpose back. Is there any other way?

Add a newaxis to the end of a[:,-1], so that it has shape (2,1). Addition with b would then broadcast along the column (the second axis) instead of the rows (which is the default).
In [47]: b + a[:,-1][:, np.newaxis]
Out[47]:
array([[ 7, 8],
[11, 12]])
a[:,-1] has shape (2,). b has shape (2,2). Broadcasting adds new axes on the left by default. So when NumPy computes a[:,-1] + b its broadcasting mechanism causes a[:,-1]'s shape to be changed to (1,2) and broadcasted to (2,2), with the values along its axis of length 1 (i.e. along its rows) to be broadcasted.
In contrast, a[:,-1][:, np.newaxis] has shape (2,1). So broadcasting changes its shape to (2,2) with the values along its axis of length 1 (i.e. along its columns) to be broadcasted.