1.4.1.1. What are NumPy and NumPy arrays?¶
NumPy arrays¶
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>>> 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] 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
NumPy Reference documentation¶
On the web: //numpy.org/doc/
Interactive help:
In [5]: np.array? String Form: 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
Import conventions¶
The recommended convention to import numpy is:
1.4.1.2. 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]:
>>> 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
- 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?
1.4.1.3. 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:
>>> d = np.array[[1+2j, 3+4j, 5+6*1j]] >>> d.dtype dtype['complex128'] |
>>> e = np.array[[True, False, False, True]] >>> e.dtype dtype['bool'] |
>>> f = np.array[['Bonjour', 'Hello', 'Hallo']] >>> f.dtype # >> 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[] # >> plt.plot[x, y] # line plot
>>> x = np.linspace[0, 3, 20] >>> y = np.linspace[0, 9, 20] >>> plt.plot[x, y] # line plot [] >>> plt.plot[x, y, 'o'] # dot plot []
>>> image = np.random.rand[30, 30] >>> plt.imshow[image, cmap=plt.cm.hot] >>> plt.colorbar[] Exercise: Simple visualizations
1.4.1.5. 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] [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, 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] 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
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]] 1.4.1.6. 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
>>> is_prime = np.ones[[100,], dtype=bool]
>>> N_max = int[np.sqrt[len[is_prime] - 1]] >>> for j in range[2, N_max + 1]: ... is_prime[2*j::j] = False
1.4.1.7. 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¶>>> np.random.seed[3] >>> a = np.random.randint[0, 21, 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]] >>> 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]] 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
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