02-indexing.md

layout	title	subtitle	minutes
page	Advanced NumPy	Indexing	30

Learning Objectives {.objectives}

After the lesson learner:

• Can get the value of any element of a N-dimensional array knowing its row, column etc.
• Can use slices to get and modify ranges of elements.
• Can explain the difference between a copy and a view. Knows which methods of indexing return a copy or view.
• Knows how to select elements from an array based on some criteria applied to their values.
• Can obtain a sub-array of non-contiguous of elements using fancy indexing.

Integer indexing and slicing

Individual items of an array can be accessed by the integer index of the element (starting with 0):

>>> a = np.arange(10)
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> np.array([a[0], a[2], a[-1]])
array([0, 2, 9])

For two- or more dimensional arrays multiple indices should be specified:

>>> b = np.arange(6).reshape(2,3)
>>> b
array([[0, 1, 2],
       [3, 4, 5]])
>>> b[1, 2]
5

Slicing allows to extract sub-arrays of multiple elements from an array. It's defined by three integers separated by a colon, i.e. start:end:increment. Any of the integers can be skipped in which case they are replaced by defaults (0 for start, end of array for end and 1 for increment):

>>> c = np.arange(9)
>>> c[1:3]
array([1, 2])
>>> c[:3]
array([0, 1, 2])
>>> c[1:]
array([1, 2, 3, 4, 5, 6, 7, 8])

You can also assign elements with slices and indexes:

>>> c
array([0, 1, 2, 3, 4, 5, 6, 7, 8])
>>> c[1:8:2]=1000
>>> c
array([   0, 1000,    2, 1000,    4, 1000,    6, 1000,    8])

View or copy {.challenge}

Create a 3x4 array of values from 0 to 11. Create another array as follows: y = x[2]. What happens when you modify y — does x also change? Now try y = x[:2] and modify it's first element. What happens now?

Checkerboard {.challenge}

Create an array of zeros and fill it with a checkerboard pattern with of size 8x8.

Boolean mask

Sometimes we may want to select array elements based on their values. For this case boolean mask is very useful. The mask is an array of the same length as the indexed array containg only False or True values:

>>> a = np.arange(4)
>>> a
array([0, 1, 2, 3])
>>> mask = np.array([False, True, True, False])
>>> a[mask]
array([1, 2])

In most cases the mask is constructed from the values of the array itself. For example, to select only odd numbers we could use the following mask:

>>> odd = (a % 2) == 1
>>> odd
array([False,  True, False,  True], dtype=bool)
>>> a[odd]
array([1, 3])

This could be also done in a single step:

>>> a[(a % 2) == 1]
array([1, 3])

Indexing with a mask can be also useful to assign a new value to a sub-array:

>>> a[(a % 2) == 1] = -1
>>> a
array([ 0, -1,  2, -1])

View or copy? {.challenge}

What are the final values of a and b at the end of the following program? Explain why.

a = np.arange(5)
b = a[a < 3]
b[::2] = 0

a) a = [0, 1, 2, 3, 4], b = [0, 1, 2] b) a = [0, 1, 0, 3, 4], b = [0, 1, 0] c) a = [0, 0, 2, 3, 4], b = [0, 0, 2] d) a = [0, 1, 2, 3, 4], b = [0, 1, 0] e) a = [0, 1, 2, 3, 4], b = [0, 1, 0, 3, 0]

Rectification {.challenge}

Rectify an array (replace negative elements with zeros) of random numbers from normal distribution (generated with np.random.randn) using boolean indexing.

Fancy indexing

Indexing can be done with an array of integers. In this case the same index can be also repeated several times:

>>> a = np.arange(0, 100, 10)
>>> a
array([ 0, 10, 20, 30, 40, 50, 60, 70, 80, 90])
>>> a[[2, 3, 2, 4, 2]] 
array([20, 30, 20, 40, 20])

New values can be also assigned with this kind of indexing:

>>> a[[9, 7]] = -100
>>> a
array([   0,   10,   20,   30,   40,   50,   60, -100,   80, -100])

When a new array is created by indexing with an array of integers, the new array has the same shape than the array of integers. Note that fancing indexing returns a copy and not a view.

>>> a = np.arange(10)
>>> idx = np.array([[3, 4], [9, 7]])
>>> idx.shape
(2, 2)
>>> a[idx]
array([[3, 4],
       [9, 7]])

Fancy indexing is often used to re-order or sort data. You can easily obtain the indices required to sort data using np.argsort:

>>> a = np.random.randint(10, size=5)
>>> a
array([4, 0, 6, 1, 2])
>>> i = np.argsort(a)
>>> a[i]
array([0, 1, 2, 4, 6])

Sub-arrays {.challenge}

Let x = np.array([1, 5, 10]).

Which of the following will show [1, 10]:

a) x[::2]

b) x[[1, 3]]

c) x[[0, 2]]

d) x[0, 2]

e) x[[1, -1]]

f) x[[False, True, False]]

For each statement predict whether it returns a copy or a view.

Random elements {.challenge}

Using fancy indexing select randomly with repetition 10 elements from a random array of 100 elements (Hint: you can use np.random.randint(max_int, size=n) to generate n random numbers from 0 to max_int)

Drawing random integers without repetition {.challenge}

Generate a random sequence of 10 integers from 1 to 100 without repetition (Hint: you may want to use np.random.rand and np.argsort).