The Numpy array object¶

Section contents¶

What are Numpy and Numpy arrays?
Reference documentation
Import conventions
Creating arrays
Functions for creating arrays
Basic data types
Basic visualization
Indexing and slicing
Copies and views
Fancy indexing

What are Numpy and 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

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

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.

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L = range(1000)

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%timeit [i**2 for i in L]

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a = np.arange(1000)

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%timeit a**2

Reference documentation¶

On the web: http://docs.scipy.org/
Interactive help:

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np.array?

Looking for something:

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np.lookfor('create array') 

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np.con*?

Import conventions¶

The general convention to import numpy is:

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import numpy as np

Using this style of import is recommended.

Creating arrays¶

1-D:

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a = np.array([0, 1, 2, 3])
a

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a.ndim

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a.shape

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len(a)

2-D, 3-D, ...:

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b = np.array([[0, 1, 2], [3, 4, 5]])    # 2 x 3 array
b

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b.ndim

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b.shape

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len(b)     # returns the size of the first dimension

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c = np.array([[[1], [2]], [[3], [4]]])
c

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c.shape

Exercise: Simple arrays¶

Create simple one and two dimensional arrays. First, redo the examples

from above. And then create your own.

Use the functions len, shape and ndim on some of those arrays and

observe their output.

Functions for creating arrays¶

In practice, we rarely enter items one by one...

Evenly spaced:

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a = np.arange(10) # 0 .. n-1  (!)
a

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b = np.arange(1, 9, 2) # start, end (exclusive), step
b

or by number of points:

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c = np.linspace(0, 1, 6)   # start, end, num-points
c

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d = np.linspace(0, 1, 5, endpoint=False)
d

Common arrays:

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a = np.ones((3, 3))  # reminder: (3, 3) is a tuple
a

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b = np.zeros((2, 2))
b

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c = np.eye(3)
c

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d = np.diag(np.array([1, 2, 3, 4]))
d

np.random random numbers (Mersenne Twister PRNG):

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a = np.random.rand(4)       # uniform in [0, 1]
a  

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b = np.random.randn(4)      # Gaussian
b  

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np.random.seed(1234)        # Setting the random seed

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?

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:

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

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

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:

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

The default data type is floating point:

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a = np.ones((3, 3))
a.dtype

There are also other types:

Complex

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

Bool

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

Strings

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

Much more

int32
int64
unit32
unit64

Basic visualization¶

Tip¶

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

Start by launching IPython in pylab mode.

Or the notebook:

Alternatively, if IPython has already been started:

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%pylab

Or, from the notebook:

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%pylab 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:

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import matplotlib.pyplot as plt  # the tidy way

And then use (note that you have to use show explicitly):

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

Or, if you are using pylab:

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plot(x, y)       # line plot    

Using import matplotlib.pyplot as plt is recommended for use in scripts. Whereas pylab is recommended for interactive exploratory work.

1D plotting:

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x = np.linspace(0, 3, 20)
y = np.linspace(0, 9, 20)
plt.plot(x, y)       # line plot    

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plt.plot(x, y, 'o')  # dot plot    

2D arrays (such as images):

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image = np.random.rand(30, 30)
plt.imshow(image, cmap=plt.cm.hot)    
plt.colorbar()    

More in the Matplotlib tutorial this afternoon

Exercise: Simple visualizations¶

Plot some simple arrays.
Try to use both the IPython shell and the notebook, if possible.
Try using the gray colormap.

Indexing and slicing¶

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

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a = np.arange(10)
a

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a[0], a[2], a[-1]

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:

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a[::-1]

For multidimensional arrays, indexes are tuples of integers:

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a = np.diag(np.arange(3))
a

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a[1, 1]

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a[2, 1] = 10 # third line, second column
a

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a[1]

Note that:

In 2D, the first dimension corresponds to rows, the second to columns.
Let us repeat together: 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:

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a = np.arange(10)
a

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a[2:9:3] # [start:end:step]

Note that the last index is not included! :

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a[:4]

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

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a[1:3]

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a[::2]

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a[3:]

A small illustrated summary of Numpy indexing and slicing...

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from IPython.display import Image
Image(filename='images/numpy_indexing.png')

You can also combine assignment and slicing:

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a = np.arange(10)
a[5:] = 10
a

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b = np.arange(5)
a[5:] = b[::-1]
a

Exercise: Indexing and slicing¶

Try the different flavours of slicing, using start, end and step.
Verify that the slices in the diagram above are indeed correct. You may

use the following expression to create the array:

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np.arange(6) + np.arange(0, 51, 10)[:, np.newaxis]

Try assigning a smaller 2D array to a larger 2D array, like in the 1D

example above.

Use a different step, e.g. -2, in the reversal idiom above. What

effect does this have?

Exercise: Array creation¶

Create the following arrays (with correct data types):

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:

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:

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a = np.arange(10)
a

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b = a[::2]
b

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np.may_share_memory(a, b)

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b[0] = 12
b

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a   # (!)

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a = np.arange(10)
c = a[::2].copy()  # force a copy
c[0] = 12
a

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np.may_share_memory(a, c)

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

Worked example: Prime number sieve¶

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from IPython.display import Image
Image(filename='images/prime-sieve.png')

Compute prime numbers in 0--99, with a sieve

Construct a shape (100,) boolean array is_prime, filled with True in

the beginning:

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

Cross out 0 and 1 which are not primes:

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is_prime[:2] = 0

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

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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:
- 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](http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes):

  * Skip `j` which are already known to not be primes

  * The first number to cross out is $j^2$

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¶

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np.random.seed(3)
a = np.random.random_integers(0, 20, 15)
a

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(a % 3 == 0)

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mask = (a % 3 == 0)
extract_from_a = a[mask] # or,  a[a%3==0]
extract_from_a           # extract a sub-array with the mask

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

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a[a % 3 == 0] = -1
a

Indexing with an array of integers¶

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a = np.arange(0, 100, 10)
a

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

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a[[2, 3, 2, 4, 2]]  # note: [2, 3, 2, 4, 2] is a Python list

New values can be assigned with this kind of indexing:

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a[[9, 7]] = -100
a

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:

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

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a[idx]

The image below illustrates various fancy indexing applications

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from IPython.display import Image
Image(filename='images/numpy_fancy_indexing.png')

Exercise: Fancy indexing¶

Again, verify the fancy indexing shown in the diagram above.
Use fancy indexing on the left and array creation on the right to assign

values from a smaller array to a larger array.