1.3.1. The NumPy array object

1.3.1.1. What are NumPy and NumPy arrays?

NumPy arrays

Python objects:

• high-level number objects: integers, floating point

• containers: lists (costless insertion and append), dictionaries (fast lookup)

NumPy provides:

• extension package to Python for multi-dimensional arrays

• closer to hardware (efficiency)

• designed for scientific computation (convenience)

• Also known as array oriented computing

>>> import numpy as np
>>> a = np.array([0, 1, 2, 3])
>>> a
array([0, 1, 2, 3])

Tip

For example, An array containing:

• values of an experiment/simulation at discrete time steps

• signal recorded by a measurement device, e.g. sound wave

• pixels of an image, grey-level or colour

• 3-D data measured at different X-Y-Z positions, e.g. MRI scan

• …

Why it is useful: Memory-efficient container that provides fast numerical operations.

In [1]: L = range(1000)

In [2]: %timeit [i**2 for i in L]
50.6 us +- 725 ns per loop (mean +- std. dev. of 7 runs, 10,000 loops each)

In [3]: a = np.arange(1000)

In [4]: %timeit a**2
920 ns +- 7.16 ns per loop (mean +- std. dev. of 7 runs, 1,000,000 loops each)

NumPy Reference documentation

• On the web: https://numpy.org/doc/

• Interactive help:

  In [5]: np.array?
  Docstring:
  array(object, dtype=None, *, copy=True, order='K', subok=False, ndmin=0,
        like=None)
  
  Create an array.
  
  Parameters
  ----------
  object : array_like
      An array, any object exposing the array interface, an object whose
      ``__array__`` method returns an array, or any (nested) sequence.
      If object is a scalar, a 0-dimensional array containing object is
      returned.
  dtype : data-type, optional
      The desired data-type for the array. If not given, NumPy will try to use
      a default ``dtype`` that can represent the values (by applying promotion
      rules when necessary.)
  copy : bool, optional
      If ``True`` (default), then the array data is copied. If ``None``,
      a copy will only be made if ``__array__`` returns a copy, if obj is
      a nested sequence, or if a copy is needed to satisfy any of the other
      requirements (``dtype``, ``order``, etc.). Note that any copy of
      the data is shallow, i.e., for arrays with object dtype, the new
      array will point to the same objects. See Examples for `ndarray.copy`.
      For ``False`` it raises a ``ValueError`` if a copy cannot be avoided.
      Default: ``True``.
  order : {'K', 'A', 'C', 'F'}, optional
      Specify the memory layout of the array. If object is not an array, the
      newly created array will be in C order (row major) unless 'F' is
      specified, in which case it will be in Fortran order (column major).
      If object is an array the following holds.
  
      ===== ========= ===================================================
      order  no copy                     copy=True
      ===== ========= ===================================================
      'K'   unchanged F & C order preserved, otherwise most similar order
      'A'   unchanged F order if input is F and not C, otherwise C order
      'C'   C order   C order
      'F'   F order   F order
      ===== ========= ===================================================
  
      When ``copy=None`` and a copy is made for other reasons, the result is
      the same as if ``copy=True``, with some exceptions for 'A', see the
      Notes section. The default order is 'K'.
  subok : bool, optional
      If True, then sub-classes will be passed-through, otherwise
      the returned array will be forced to be a base-class array (default).
  ndmin : int, optional
      Specifies the minimum number of dimensions that the resulting
      array should have.  Ones will be prepended to the shape as
      needed to meet this requirement.
  like : array_like, optional
      Reference object to allow the creation of arrays which are not
      NumPy arrays. If an array-like passed in as ``like`` supports
      the ``__array_function__`` protocol, the result will be defined
      by it. In this case, it ensures the creation of an array object
      compatible with that passed in via this argument.
  
      .. versionadded:: 1.20.0
  
  Returns
  -------
  out : ndarray
      An array object satisfying the specified requirements.
  
  See Also
  --------
  empty_like : Return an empty array with shape and type of input.
  ones_like : Return an array of ones with shape and type of input.
  zeros_like : Return an array of zeros with shape and type of input.
  full_like : Return a new array with shape of input filled with value.
  empty : Return a new uninitialized array.
  ones : Return a new array setting values to one.
  zeros : Return a new array setting values to zero.
  full : Return a new array of given shape filled with value.
  copy: Return an array copy of the given object.
  
  
  Notes
  -----
  When order is 'A' and ``object`` is an array in neither 'C' nor 'F' order,
  and a copy is forced by a change in dtype, then the order of the result is
  not necessarily 'C' as expected. This is likely a bug.
  
  Examples
  --------
  >>> import numpy as np
  >>> np.array([1, 2, 3])
  array([1, 2, 3])
  
  Upcasting:
  
  >>> np.array([1, 2, 3.0])
  array([ 1.,  2.,  3.])
  
  More than one dimension:
  
  >>> np.array([[1, 2], [3, 4]])
  array([[1, 2],
         [3, 4]])
  
  Minimum dimensions 2:
  
  >>> np.array([1, 2, 3], ndmin=2)
  array([[1, 2, 3]])
  
  Type provided:
  
  >>> np.array([1, 2, 3], dtype=complex)
  array([ 1.+0.j,  2.+0.j,  3.+0.j])
  
  Data-type consisting of more than one element:
  
  >>> x = np.array([(1,2),(3,4)],dtype=[('a','<i4'),('b','<i4')])
  >>> x['a']
  array([1, 3])
  
  Creating an array from sub-classes:
  
  >>> np.array(np.asmatrix('1 2; 3 4'))
  array([[1, 2],
         [3, 4]])
  
  >>> np.array(np.asmatrix('1 2; 3 4'), subok=True)
  matrix([[1, 2],
          [3, 4]])
  Type:      builtin_function_or_method
  

  Tip

  >>> help(np.array)
  Help on built-in function array in module numpy:
  
  array(...)
      array(object, dtype=None, ...
  

• Looking for something:

  In [6]: np.con*?
  np.concat
  np.concatenate
  np.conj
  np.conjugate
  np.convolve
  

Import conventions

The recommended convention to import NumPy is:

>>> import numpy as np

1.3.1.2. Creating arrays

Manual construction of arrays

• 1-D:

  >>> a = np.array([0, 1, 2, 3])
  >>> a
  array([0, 1, 2, 3])
  >>> a.ndim
  1
  >>> a.shape
  (4,)
  >>> len(a)
  4
  

• 2-D, 3-D, …:

  >>> b = np.array([[0, 1, 2], [3, 4, 5]])    # 2 x 3 array
  >>> b
  array([[0, 1, 2],
         [3, 4, 5]])
  >>> b.ndim
  2
  >>> b.shape
  (2, 3)
  >>> len(b)     # returns the size of the first dimension
  2
  
  >>> c = np.array([[[1], [2]], [[3], [4]]])
  >>> c
  array([[[1],
          [2]],
  
         [[3],
          [4]]])
  >>> c.shape
  (2, 2, 1)
  

Exercise: Simple arrays

• Create a simple two dimensional array. First, redo the examples from above. And then create your own: how about odd numbers counting backwards on the first row, and even numbers on the second?

• Use the functions len(), numpy.shape() on these arrays. How do they relate to each other? And to the ndim attribute of the arrays?

Functions for creating arrays

Tip

In practice, we rarely enter items one by one…

• Evenly spaced:

  >>> a = np.arange(10) # 0 .. n-1  (!)
  >>> a
  array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
  >>> b = np.arange(1, 9, 2) # start, end (exclusive), step
  >>> b
  array([1, 3, 5, 7])
  

• or by number of points:

  >>> c = np.linspace(0, 1, 6)   # start, end, num-points
  >>> c
  array([0. ,  0.2,  0.4,  0.6,  0.8,  1. ])
  >>> d = np.linspace(0, 1, 5, endpoint=False)
  >>> d
  array([0. ,  0.2,  0.4,  0.6,  0.8])
  

• Common arrays:

  >>> a = np.ones((3, 3))  # reminder: (3, 3) is a tuple
  >>> a
  array([[1.,  1.,  1.],
         [1.,  1.,  1.],
         [1.,  1.,  1.]])
  >>> b = np.zeros((2, 2))
  >>> b
  array([[0.,  0.],
         [0.,  0.]])
  >>> c = np.eye(3)
  >>> c
  array([[1.,  0.,  0.],
         [0.,  1.,  0.],
         [0.,  0.,  1.]])
  >>> d = np.diag(np.array([1, 2, 3, 4]))
  >>> d
  array([[1, 0, 0, 0],
         [0, 2, 0, 0],
         [0, 0, 3, 0],
         [0, 0, 0, 4]])
  

• np.random: random numbers (Mersenne Twister PRNG):

  >>> rng = np.random.default_rng(27446968)
  >>> a = rng.random(4)       # uniform in [0, 1]
  >>> a
  array([0.64613018, 0.48984931, 0.50851229, 0.22563948])
  
  >>> b = rng.standard_normal(4)      # Gaussian
  >>> b
  array([-0.38250769, -0.61536465,  0.98131732,  0.59353096])
  

Exercise: Creating arrays using functions

• Experiment with arange, linspace, ones, zeros, eye and diag.

• Create different kinds of arrays with random numbers.

• Try setting the seed before creating an array with random values.

• Look at the function np.empty. What does it do? When might this be useful?

1.3.1.3. Basic data types

You may have noticed that, in some instances, array elements are displayed with a trailing dot (e.g. 2. vs 2). This is due to a difference in the data-type used:

>>> a = np.array([1, 2, 3])
>>> a.dtype
dtype('int64')

>>> b = np.array([1., 2., 3.])
>>> b.dtype
dtype('float64')

Tip

Different data-types allow us to store data more compactly in memory, but most of the time we simply work with floating point numbers. Note that, in the example above, NumPy auto-detects the data-type from the input.

---

You can explicitly specify which data-type you want:

>>> c = np.array([1, 2, 3], dtype=float)
>>> c.dtype
dtype('float64')

The default data type is floating point:

>>> a = np.ones((3, 3))
>>> a.dtype
dtype('float64')

There are also other types:

Complex:

>>> d = np.array([1+2j, 3+4j, 5+6*1j])
>>> d.dtype
dtype('complex128')

Bool:

>>> e = np.array([True, False, False, True])
>>> e.dtype
dtype('bool')

Strings:

>>> f = np.array(['Bonjour', 'Hello', 'Hallo'])
>>> f.dtype     # <--- strings containing max. 7 letters
dtype('<U7')

Much more:

• int32

• int64

• uint32

• uint64

1.3.1.4. Basic visualization

Now that we have our first data arrays, we are going to visualize them.

Start by launching IPython:

$ ipython # or ipython3 depending on your install

Or the notebook:

$ jupyter notebook

Once IPython has started, enable interactive plots:

>>> %matplotlib  

Or, from the notebook, enable plots in the notebook:

>>> %matplotlib inline 

The inline is important for the notebook, so that plots are displayed in the notebook and not in a new window.

Matplotlib is a 2D plotting package. We can import its functions as below:

>>> import matplotlib.pyplot as plt  # the tidy way

And then use (note that you have to use show explicitly if you have not enabled interactive plots with %matplotlib):

>>> plt.plot(x, y)       # line plot    
>>> plt.show()           # <-- shows the plot (not needed with interactive plots) 

Or, if you have enabled interactive plots with %matplotlib:

>>> plt.plot(x, y)       # line plot    

• 1D plotting:

>>> x = np.linspace(0, 3, 20)
>>> y = np.linspace(0, 9, 20)
>>> plt.plot(x, y)       # line plot
[<matplotlib.lines.Line2D object at ...>]
>>> plt.plot(x, y, 'o')  # dot plot
[<matplotlib.lines.Line2D object at ...>]

• 2D arrays (such as images):

>>> rng = np.random.default_rng(27446968)
>>> image = rng.random((30, 30))
>>> plt.imshow(image, cmap=plt.cm.hot)
<matplotlib.image.AxesImage object at ...>
>>> plt.colorbar()
<matplotlib.colorbar.Colorbar object at ...>

See also

More in the: matplotlib chapter

Exercise: Simple visualizations

• Plot some simple arrays: a cosine as a function of time and a 2D matrix.

• Try using the gray colormap on the 2D matrix.

1.3.1.5. Indexing and slicing

The items of an array can be accessed and assigned to the same way as other Python sequences (e.g. lists):

>>> a = np.arange(10)
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> a[0], a[2], a[-1]
(np.int64(0), np.int64(2), np.int64(9))

Warning

Indices begin at 0, like other Python sequences (and C/C++). In contrast, in Fortran or Matlab, indices begin at 1.

The usual python idiom for reversing a sequence is supported:

>>> a[::-1]
array([9, 8, 7, 6, 5, 4, 3, 2, 1, 0])

For multidimensional arrays, indices are tuples of integers:

>>> a = np.diag(np.arange(3))
>>> a
array([[0, 0, 0],
       [0, 1, 0],
       [0, 0, 2]])
>>> a[1, 1]
np.int64(1)
>>> a[2, 1] = 10 # third line, second column
>>> a
array([[ 0,  0,  0],
       [ 0,  1,  0],
       [ 0, 10,  2]])
>>> a[1]
array([0, 1, 0])

Note

• In 2D, the first dimension corresponds to rows, the second to columns.

• for multidimensional a, a[0] is interpreted by taking all elements in the unspecified dimensions.

Slicing: Arrays, like other Python sequences can also be sliced:

>>> a = np.arange(10)
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> a[2:9:3] # [start:end:step]
array([2, 5, 8])

Note that the last index is not included! :

>>> a[:4]
array([0, 1, 2, 3])

All three slice components are not required: by default, start is 0, end is the last and step is 1:

>>> a[1:3]
array([1, 2])
>>> a[::2]
array([0, 2, 4, 6, 8])
>>> a[3:]
array([3, 4, 5, 6, 7, 8, 9])

A small illustrated summary of NumPy indexing and slicing…

You can also combine assignment and slicing:

>>> a = np.arange(10)
>>> a[5:] = 10
>>> a
array([ 0,  1,  2,  3,  4, 10, 10, 10, 10, 10])
>>> b = np.arange(5)
>>> a[5:] = b[::-1]
>>> a
array([0, 1, 2, 3, 4, 4, 3, 2, 1, 0])

Exercise: Indexing and slicing

• Try the different flavours of slicing, using start, end and step: starting from a linspace, try to obtain odd numbers counting backwards, and even numbers counting forwards.

• Reproduce the slices in the diagram above. You may use the following expression to create the array:

  >>> np.arange(6) + np.arange(0, 51, 10)[:, np.newaxis]
  array([[ 0,  1,  2,  3,  4,  5],
         [10, 11, 12, 13, 14, 15],
         [20, 21, 22, 23, 24, 25],
         [30, 31, 32, 33, 34, 35],
         [40, 41, 42, 43, 44, 45],
         [50, 51, 52, 53, 54, 55]])
  

Exercise: Array creation

Create the following arrays (with correct data types):

[[1, 1, 1, 1],
 [1, 1, 1, 1],
 [1, 1, 1, 2],
 [1, 6, 1, 1]]

[[0., 0., 0., 0., 0.],
 [2., 0., 0., 0., 0.],
 [0., 3., 0., 0., 0.],
 [0., 0., 4., 0., 0.],
 [0., 0., 0., 5., 0.],
 [0., 0., 0., 0., 6.]]

Par on course: 3 statements for each

Hint: Individual array elements can be accessed similarly to a list, e.g. a[1] or a[1, 2].

Hint: Examine the docstring for diag.

Exercise: Tiling for array creation

Skim through the documentation for np.tile, and use this function to construct the array:

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

1.3.1.6. Copies and views

A slicing operation creates a view on the original array, which is just a way of accessing array data. Thus the original array is not copied in memory. You can use np.may_share_memory() to check if two arrays share the same memory block. Note however, that this uses heuristics and may give you false positives.

When modifying the view, the original array is modified as well:

>>> a = np.arange(10)
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> b = a[::2]
>>> b
array([0, 2, 4, 6, 8])
>>> np.may_share_memory(a, b)
True
>>> b[0] = 12
>>> b
array([12,  2,  4,  6,  8])
>>> a   # (!)
array([12,  1,  2,  3,  4,  5,  6,  7,  8,  9])

>>> a = np.arange(10)
>>> c = a[::2].copy()  # force a copy
>>> c[0] = 12
>>> a
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

>>> np.may_share_memory(a, c)
False

This behavior can be surprising at first sight… but it allows to save both memory and time.

Worked example: Prime number sieve

Compute prime numbers in 0–99, with a sieve

• Construct a shape (100,) boolean array is_prime, filled with True in the beginning:

>>> is_prime = np.ones((100,), dtype=bool)

• Cross out 0 and 1 which are not primes:

>>> is_prime[:2] = 0

• For each integer j starting from 2, cross out its higher multiples:

>>> N_max = int(np.sqrt(len(is_prime) - 1))
>>> for j in range(2, N_max + 1):
...     is_prime[2*j::j] = False

• Skim through help(np.nonzero), and print the prime numbers

• Follow-up:

  • Move the above code into a script file named prime_sieve.py

  • Run it to check it works

  • Use the optimization suggested in the sieve of Eratosthenes:

  1. Skip j which are already known to not be primes

  2. The first number to cross out is

1.3.1.7. Fancy indexing

Tip

NumPy arrays can be indexed with slices, but also with boolean or integer arrays (masks). This method is called fancy indexing. It creates copies not views.

Using boolean masks

>>> rng = np.random.default_rng(27446968)
>>> a = rng.integers(0, 21, 15)
>>> a
array([ 3, 13, 12, 10, 10, 10, 18,  4,  8,  5,  6, 11, 12, 17,  3])
>>> (a % 3 == 0)
array([ True, False,  True, False, False, False,  True, False, False,
       False,  True, False,  True, False,  True])
>>> mask = (a % 3 == 0)
>>> extract_from_a = a[mask] # or,  a[a%3==0]
>>> extract_from_a           # extract a sub-array with the mask
array([ 3, 12, 18,  6, 12,  3])

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

>>> a[a % 3 == 0] = -1
>>> a
array([-1, 13, -1, 10, 10, 10, -1,  4,  8,  5, -1, 11, -1, 17, -1])

Indexing with an array of integers

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

Indexing can be done with an array of integers, where the same index is repeated several time:

>>> a[[2, 3, 2, 4, 2]]  # note: [2, 3, 2, 4, 2] is a Python list
array([20, 30, 20, 40, 20])

New values can be assigned with this kind of indexing:

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

Tip

When a new array is created by indexing with an array of integers, the new array has the same shape as the array of integers:

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

---

The image below illustrates various fancy indexing applications

Exercise: Fancy indexing

• Again, reproduce the fancy indexing shown in the diagram above.

• Use fancy indexing on the left and array creation on the right to assign values into an array, for instance by setting parts of the array in the diagram above to zero.