In a recent session of Python Foundations for Scientists & Engineers, a question came up about indexing a NumPy ndarray. Beyond getting and setting single values, NumPy enables some powerful efficiencies through slicing, which produces views of an array’s data without copying, and fancy indexing, which allows use of more-complex expressions to extract portions of arrays. We have written on the efficiency of array operations, and the details of slicing are pretty well covered, from the NumPy docs on slicing, to this chapter of “Beautiful Code” by the original author of NumPy, Travis Oliphant.
Slicing is pretty cool because it allows fast efficient computations of things like finite difference, for say, computing numerical derivatives. Recall that the derivative of a function describes the change in one variable with respect to another:
\frac{dy}{dx}And in numerical computations, we can use a discrete approximation:
\frac{dy}{dx} \approx \frac{\Delta x}{\Delta y}And to find the derivative at any particular location i, you compute the ratio of differences:
\frac{\Delta x}{\Delta y}\big|_i = \frac{x_{i+1} - x_{i}}{y_{i+1} - y{i}}NumPy allows you to use slicing to avoid setting up costly-for-Python for: loops by specifying start, stop, and step values in the array indices. This lets you subtracting all of the i indices from the i+1 indices at the same time by specifying one slice that starts at element 1 and goes to the end (the i+1 indices), and another that starts at 0 and goes up to but not including the last element. No copies are made during the slicing operations. I use examples like this to show how you can get 2 and sometimes 3 or more orders of magnitude speedups of the same operation with for loops.
>>> import numpy as np
>>> x = np.linspace(-np.pi, np.pi, 101)
>>> y = np.sin(x)
>>> dy_dx = (
... (y[1:] - y[:-1]) /
... (x[1:] - x[:-1])
... )
>>> np.sum(dy_dx - np.cos(x[:-1] + (x[1]-x[0]) / 2)) # compare to cos(x)
np.float64(-6.245004513516506e-16) # This is pretty close to 0Fancy indexing is also well documented (but the NumPy docs now use the more staid term “Advanced Integer Indexing“, but I wanted to draw attention to a “Gotcha” that has bitten me a couple of times. With fancy indexing, you can either make a mask of Boolean values, typically using some kind of boolean operator:
>>> a = np.arange(10)
>>> evens_mask = a % 2 == 0
>>> odds_mask = a % 2 == 1
>>> print(a[evens_mask])
[0 2 4 6 8]
>>> print(a[odds_mask])
[1 3 5 7 9]Or you can specify the indices you want, and this is the Gotcha, with tuples or lists, but the behavior is different either way. Let’s construct an example like one we use in class. We’ll make a 2-D array b and construct at positional fancy index that specifies elements in a diagonal. Notice that it’s a tuple, as shown by the (,) and each element is a list of coordinates in the array.
>>> b = np.arange(25).reshape(5, 5)
>>> print(b)
[[ 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]]
>>> upper_diagonal = (
... [0, 1, 2, 3], # row indices
... [1, 2, 3, 4], # column indices
... )
>>> print(b[upper_diagonal])
[ 1 7 13 19]In this case, the tuple has as many elements as there are dimensions, and each element is a list (or tuple, or array) of the indices to that dimension. So in the example above, the first element comes from b[0, 1], the second from b[1, 2] so on pair-wise through the lists of indices. The result is substantially different if you try to construct a fancy index from a list instead of a tuple:
>>> upper_diagonal_list = [
[0, 1, 2, 3],
[1, 2, 3, 4]
]
>>> b_with_a_list = b[upper_diagonal_list]
>>> print(b_with_a_list)
[[[ 0 1 2 3 4]
[ 5 6 7 8 9]
[10 11 12 13 14]
[15 16 17 18 19]]
[[ 5 6 7 8 9]
[10 11 12 13 14]
[15 16 17 18 19]
[20 21 22 23 24]]]What just happened?? In many places, lists and tuples have similar behaviors, but not here. What’s happening with the list version is different. This is in fact a form of broadcasting, where we’re repeating rows. Look at the shape of b_with_a_list:
>>> print(b_with_a_list.shape)
(2, 4, 5)Notice that its dimension 0 has 2 elements, which is the same as the number of items in upper_diagoal_list. Notice the dimension 1 has 4 elements, corresponding to the size of each element in upper_diagoal_list. Then notice that dimension 2 matches the size of the rows of b, and hopefully it will be clear what’s happening. In upper_diagoal_list we’re constructing a new array by specifying the rows to use, so the first element of b_with_a_list (seen as the first block above) consist of rows 0, 1, 2, and 3 from b, and the second element is the rows from the second element of upper_diagonal_list. Let’s print it again with comments:
>>> print(b_with_a_list)
[[[ 0 1 2 3 4] # b[0] \
[ 5 6 7 8 9] # b[1] | indices are first element of
[10 11 12 13 14] # b[2] | upper_diagonal_list
[15 16 17 18 19]] # b[3] /
[[ 5 6 7 8 9] # b[1] \
[10 11 12 13 14] # b[2] | indices are second element of
[15 16 17 18 19] # b[3] | upper_diagonal_list
[20 21 22 23 24]]] # b[4] /Forgetting this convention has bitten me more than once, so I hope this explanation helps you resolve some confusion if you should ever run into it.
