Section contents
| Python objects: |
|
|---|---|
| Numpy provides: |
|
>>> import numpy as np
>>> a = np.array([0, 1, 2, 3])
>>> a
array([0, 1, 2, 3])
Tip
For example, An array containing:
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]
1000 loops, best of 3: 403 us per loop
In [3]: a = np.arange(1000)
In [4]: %timeit a**2
100000 loops, best of 3: 12.7 us per loop
On the web: http://docs.scipy.org/
Interactive help:
In [5]: np.array?
String Form:<built-in function array>
Docstring:
array(object, dtype=None, copy=True, order=None, subok=False, ndmin=0, ...
Looking for something:
>>> np.lookfor('create array')
Search results for 'create array'
---------------------------------
numpy.array
Create an array.
numpy.memmap
Create a memory-map to an array stored in a *binary* file on disk.
In [6]: np.con*?
np.concatenate
np.conj
np.conjugate
np.convolve
The recommended convention to import numpy is:
>>> import numpy as np
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
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):
>>> a = np.random.rand(4) # uniform in [0, 1]
>>> a
array([ 0.95799151, 0.14222247, 0.08777354, 0.51887998])
>>> b = np.random.randn(4) # Gaussian
>>> b
array([ 0.37544699, -0.11425369, -0.47616538, 1.79664113])
>>> np.random.seed(1234) # Setting the random seed
Exercise: Creating arrays using functions
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('S7')
|
| Much more: |
|
Now that we have our first data arrays, we are going to visualize them.
Start by launching IPython:
$ ipython
Or the notebook:
$ ipython 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:
>>> 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 ...>]
[source code, hires.png, pdf]
2D arrays (such as images):
>>> image = np.random.rand(30, 30)
>>> plt.imshow(image, cmap=plt.cm.hot)
>>> plt.colorbar()
<matplotlib.colorbar.Colorbar instance at ...>
[source code, hires.png, pdf]
See also
More in the: matplotlib chapter
Exercise: Simple visualizations
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]
(0, 2, 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, indexes are tuples of integers:
>>> a = np.diag(np.arange(3))
>>> a
array([[0, 0, 0],
[0, 1, 0],
[0, 0, 2]])
>>> a[1, 1]
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
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]]
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
>>> is_prime = np.ones((100,), dtype=bool)
>>> is_prime[:2] = 0
>>> N_max = int(np.sqrt(len(is_prime)))
>>> for j in range(2, N_max):
... is_prime[2*j::j] = False
Skim through help(np.nonzero), and print the prime numbers
Follow-up:
- Skip j which are already known to not be primes
- The first number to cross out is
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.
>>> np.random.seed(3)
>>> a = np.random.random_integers(0, 20, 15)
>>> a
array([10, 3, 8, 0, 19, 10, 11, 9, 10, 6, 0, 20, 12, 7, 14])
>>> (a % 3 == 0)
array([False, True, False, True, False, False, False, True, False,
True, True, False, True, False, False], dtype=bool)
>>> 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, 0, 9, 6, 0, 12])
Indexing with a mask can be very useful to assign a new value to a sub-array:
>>> a[a % 3 == 0] = -1
>>> a
array([10, -1, 8, -1, 19, 10, 11, -1, 10, -1, -1, 20, -1, 7, 14])
>>> 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 than 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