Numpy python руководство

Prerequisites#

You’ll need to know a bit of Python. For a refresher, see the Python
tutorial.

To work the examples, you’ll need matplotlib installed
in addition to NumPy.

Learner profile

This is a quick overview of arrays in NumPy. It demonstrates how n-dimensional
((n>=2)) arrays are represented and can be manipulated. In particular, if
you don’t know how to apply common functions to n-dimensional arrays (without
using for-loops), or if you want to understand axis and shape properties for
n-dimensional arrays, this article might be of help.

Learning Objectives

After reading, you should be able to:

• Understand the difference between one-, two- and n-dimensional arrays in
  NumPy;

• Understand how to apply some linear algebra operations to n-dimensional
  arrays without using for-loops;

• Understand axis and shape properties for n-dimensional arrays.

The Basics#

NumPy’s main object is the homogeneous multidimensional array. It is a
table of elements (usually numbers), all of the same type, indexed by a
tuple of non-negative integers. In NumPy dimensions are called axes.

For example, the array for the coordinates of a point in 3D space,
[1, 2, 1], has one axis. That axis has 3 elements in it, so we say
it has a length of 3. In the example pictured below, the array has 2
axes. The first axis has a length of 2, the second axis has a length of
3.

[[1., 0., 0.],
 [0., 1., 2.]]

NumPy’s array class is called ndarray. It is also known by the alias
array. Note that numpy.array is not the same as the Standard
Python Library class array.array, which only handles one-dimensional
arrays and offers less functionality. The more important attributes of
an ndarray object are:

ndarray.ndim

the number of axes (dimensions) of the array.

ndarray.shape

the dimensions of the array. This is a tuple of integers indicating
the size of the array in each dimension. For a matrix with n rows
and m columns, shape will be (n,m). The length of the
shape tuple is therefore the number of axes, ndim.

ndarray.size

the total number of elements of the array. This is equal to the
product of the elements of shape.

ndarray.dtype

an object describing the type of the elements in the array. One can
create or specify dtype’s using standard Python types. Additionally
NumPy provides types of its own. numpy.int32, numpy.int16, and
numpy.float64 are some examples.

ndarray.itemsize

the size in bytes of each element of the array. For example, an
array of elements of type float64 has itemsize 8 (=64/8),
while one of type complex32 has itemsize 4 (=32/8). It is
equivalent to ndarray.dtype.itemsize.

ndarray.data

the buffer containing the actual elements of the array. Normally, we
won’t need to use this attribute because we will access the elements
in an array using indexing facilities.

>>> import numpy as np
>>> a = np.arange(15).reshape(3, 5)
>>> a
array([[ 0,  1,  2,  3,  4],
       [ 5,  6,  7,  8,  9],
       [10, 11, 12, 13, 14]])
>>> a.shape
(3, 5)
>>> a.ndim
2
>>> a.dtype.name
'int64'
>>> a.itemsize
8
>>> a.size
15
>>> type(a)
<class 'numpy.ndarray'>
>>> b = np.array([6, 7, 8])
>>> b
array([6, 7, 8])
>>> type(b)
<class 'numpy.ndarray'>

>>> import numpy as np
>>> a = np.array([2, 3, 4])
>>> a
array([2, 3, 4])
>>> a.dtype
dtype('int64')
>>> b = np.array([1.2, 3.5, 5.1])
>>> b.dtype
dtype('float64')

>>> a = np.array(1, 2, 3, 4)    # WRONG
Traceback (most recent call last):
  ...
TypeError: array() takes from 1 to 2 positional arguments but 4 were given
>>> a = np.array([1, 2, 3, 4])  # RIGHT

>>> b = np.array([(1.5, 2, 3), (4, 5, 6)])
>>> b
array([[1.5, 2. , 3. ],
       [4. , 5. , 6. ]])

>>> c = np.array([[1, 2], [3, 4]], dtype=complex)
>>> c
array([[1.+0.j, 2.+0.j],
       [3.+0.j, 4.+0.j]])

>>> np.zeros((3, 4))
array([[0., 0., 0., 0.],
       [0., 0., 0., 0.],
       [0., 0., 0., 0.]])
>>> np.ones((2, 3, 4), dtype=np.int16)
array([[[1, 1, 1, 1],
        [1, 1, 1, 1],
        [1, 1, 1, 1]],

       [[1, 1, 1, 1],
        [1, 1, 1, 1],
        [1, 1, 1, 1]]], dtype=int16)
>>> np.empty((2, 3)) 
array([[3.73603959e-262, 6.02658058e-154, 6.55490914e-260],  # may vary
       [5.30498948e-313, 3.14673309e-307, 1.00000000e+000]])

>>> np.arange(10, 30, 5)
array([10, 15, 20, 25])
>>> np.arange(0, 2, 0.3)  # it accepts float arguments
array([0. , 0.3, 0.6, 0.9, 1.2, 1.5, 1.8])

>>> from numpy import pi
>>> np.linspace(0, 2, 9)                   # 9 numbers from 0 to 2
array([0.  , 0.25, 0.5 , 0.75, 1.  , 1.25, 1.5 , 1.75, 2.  ])
>>> x = np.linspace(0, 2 * pi, 100)        # useful to evaluate function at lots of points
>>> f = np.sin(x)

>>> a = np.arange(6)                    # 1d array
>>> print(a)
[0 1 2 3 4 5]
>>>
>>> b = np.arange(12).reshape(4, 3)     # 2d array
>>> print(b)
[[ 0  1  2]
 [ 3  4  5]
 [ 6  7  8]
 [ 9 10 11]]
>>>
>>> c = np.arange(24).reshape(2, 3, 4)  # 3d array
>>> print(c)
[[[ 0  1  2  3]
  [ 4  5  6  7]
  [ 8  9 10 11]]

 [[12 13 14 15]
  [16 17 18 19]
  [20 21 22 23]]]

>>> print(np.arange(10000))
[   0    1    2 ... 9997 9998 9999]
>>>
>>> print(np.arange(10000).reshape(100, 100))
[[   0    1    2 ...   97   98   99]
 [ 100  101  102 ...  197  198  199]
 [ 200  201  202 ...  297  298  299]
 ...
 [9700 9701 9702 ... 9797 9798 9799]
 [9800 9801 9802 ... 9897 9898 9899]
 [9900 9901 9902 ... 9997 9998 9999]]

>>> np.set_printoptions(threshold=sys.maxsize)  # sys module should be imported

>>> a = np.array([20, 30, 40, 50])
>>> b = np.arange(4)
>>> b
array([0, 1, 2, 3])
>>> c = a - b
>>> c
array([20, 29, 38, 47])
>>> b**2
array([0, 1, 4, 9])
>>> 10 * np.sin(a)
array([ 9.12945251, -9.88031624,  7.4511316 , -2.62374854])
>>> a < 35
array([ True,  True, False, False])

>>> A = np.array([[1, 1],
...               [0, 1]])
>>> B = np.array([[2, 0],
...               [3, 4]])
>>> A * B     # elementwise product
array([[2, 0],
       [0, 4]])
>>> A @ B     # matrix product
array([[5, 4],
       [3, 4]])
>>> A.dot(B)  # another matrix product
array([[5, 4],
       [3, 4]])

>>> rg = np.random.default_rng(1)  # create instance of default random number generator
>>> a = np.ones((2, 3), dtype=int)
>>> b = rg.random((2, 3))
>>> a *= 3
>>> a
array([[3, 3, 3],
       [3, 3, 3]])
>>> b += a
>>> b
array([[3.51182162, 3.9504637 , 3.14415961],
       [3.94864945, 3.31183145, 3.42332645]])
>>> a += b  # b is not automatically converted to integer type
Traceback (most recent call last):
    ...
numpy.core._exceptions._UFuncOutputCastingError: Cannot cast ufunc 'add' output from dtype('float64') to dtype('int64') with casting rule 'same_kind'

>>> a = np.ones(3, dtype=np.int32)
>>> b = np.linspace(0, pi, 3)
>>> b.dtype.name
'float64'
>>> c = a + b
>>> c
array([1.        , 2.57079633, 4.14159265])
>>> c.dtype.name
'float64'
>>> d = np.exp(c * 1j)
>>> d
array([ 0.54030231+0.84147098j, -0.84147098+0.54030231j,
       -0.54030231-0.84147098j])
>>> d.dtype.name
'complex128'

>>> a = rg.random((2, 3))
>>> a
array([[0.82770259, 0.40919914, 0.54959369],
       [0.02755911, 0.75351311, 0.53814331]])
>>> a.sum()
3.1057109529998157
>>> a.min()
0.027559113243068367
>>> a.max()
0.8277025938204418

>>> b = np.arange(12).reshape(3, 4)
>>> b
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])
>>>
>>> b.sum(axis=0)     # sum of each column
array([12, 15, 18, 21])
>>>
>>> b.min(axis=1)     # min of each row
array([0, 4, 8])
>>>
>>> b.cumsum(axis=1)  # cumulative sum along each row
array([[ 0,  1,  3,  6],
       [ 4,  9, 15, 22],
       [ 8, 17, 27, 38]])

>>> B = np.arange(3)
>>> B
array([0, 1, 2])
>>> np.exp(B)
array([1.        , 2.71828183, 7.3890561 ])
>>> np.sqrt(B)
array([0.        , 1.        , 1.41421356])
>>> C = np.array([2., -1., 4.])
>>> np.add(B, C)
array([2., 0., 6.])

>>> a = np.arange(10)**3
>>> a
array([  0,   1,   8,  27,  64, 125, 216, 343, 512, 729])
>>> a[2]
8
>>> a[2:5]
array([ 8, 27, 64])
>>> # equivalent to a[0:6:2] = 1000;
>>> # from start to position 6, exclusive, set every 2nd element to 1000
>>> a[:6:2] = 1000
>>> a
array([1000,    1, 1000,   27, 1000,  125,  216,  343,  512,  729])
>>> a[::-1]  # reversed a
array([ 729,  512,  343,  216,  125, 1000,   27, 1000,    1, 1000])
>>> for i in a:
...     print(i**(1 / 3.))
...
9.999999999999998
1.0
9.999999999999998
3.0
9.999999999999998
4.999999999999999
5.999999999999999
6.999999999999999
7.999999999999999
8.999999999999998

>>> def f(x, y):
...     return 10 * x + y
...
>>> b = np.fromfunction(f, (5, 4), dtype=int)
>>> b
array([[ 0,  1,  2,  3],
       [10, 11, 12, 13],
       [20, 21, 22, 23],
       [30, 31, 32, 33],
       [40, 41, 42, 43]])
>>> b[2, 3]
23
>>> b[0:5, 1]  # each row in the second column of b
array([ 1, 11, 21, 31, 41])
>>> b[:, 1]    # equivalent to the previous example
array([ 1, 11, 21, 31, 41])
>>> b[1:3, :]  # each column in the second and third row of b
array([[10, 11, 12, 13],
       [20, 21, 22, 23]])

>>> b[-1]   # the last row. Equivalent to b[-1, :]
array([40, 41, 42, 43])

>>> c = np.array([[[  0,  1,  2],  # a 3D array (two stacked 2D arrays)
...                [ 10, 12, 13]],
...               [[100, 101, 102],
...                [110, 112, 113]]])
>>> c.shape
(2, 2, 3)
>>> c[1, ...]  # same as c[1, :, :] or c[1]
array([[100, 101, 102],
       [110, 112, 113]])
>>> c[..., 2]  # same as c[:, :, 2]
array([[  2,  13],
       [102, 113]])

>>> for row in b:
...     print(row)
...
[0 1 2 3]
[10 11 12 13]
[20 21 22 23]
[30 31 32 33]
[40 41 42 43]

>>> for element in b.flat:
...     print(element)
...
0
1
2
3
10
11
12
13
20
21
22
23
30
31
32
33
40
41
42
43

Shape Manipulation#

>>> a = np.floor(10 * rg.random((3, 4)))
>>> a
array([[3., 7., 3., 4.],
       [1., 4., 2., 2.],
       [7., 2., 4., 9.]])
>>> a.shape
(3, 4)

>>> a.ravel()  # returns the array, flattened
array([3., 7., 3., 4., 1., 4., 2., 2., 7., 2., 4., 9.])
>>> a.reshape(6, 2)  # returns the array with a modified shape
array([[3., 7.],
       [3., 4.],
       [1., 4.],
       [2., 2.],
       [7., 2.],
       [4., 9.]])
>>> a.T  # returns the array, transposed
array([[3., 1., 7.],
       [7., 4., 2.],
       [3., 2., 4.],
       [4., 2., 9.]])
>>> a.T.shape
(4, 3)
>>> a.shape
(3, 4)

>>> a
array([[3., 7., 3., 4.],
       [1., 4., 2., 2.],
       [7., 2., 4., 9.]])
>>> a.resize((2, 6))
>>> a
array([[3., 7., 3., 4., 1., 4.],
       [2., 2., 7., 2., 4., 9.]])

>>> a.reshape(3, -1)
array([[3., 7., 3., 4.],
       [1., 4., 2., 2.],
       [7., 2., 4., 9.]])

>>> a = np.floor(10 * rg.random((2, 2)))
>>> a
array([[9., 7.],
       [5., 2.]])
>>> b = np.floor(10 * rg.random((2, 2)))
>>> b
array([[1., 9.],
       [5., 1.]])
>>> np.vstack((a, b))
array([[9., 7.],
       [5., 2.],
       [1., 9.],
       [5., 1.]])
>>> np.hstack((a, b))
array([[9., 7., 1., 9.],
       [5., 2., 5., 1.]])

>>> from numpy import newaxis
>>> np.column_stack((a, b))  # with 2D arrays
array([[9., 7., 1., 9.],
       [5., 2., 5., 1.]])
>>> a = np.array([4., 2.])
>>> b = np.array([3., 8.])
>>> np.column_stack((a, b))  # returns a 2D array
array([[4., 3.],
       [2., 8.]])
>>> np.hstack((a, b))        # the result is different
array([4., 2., 3., 8.])
>>> a[:, newaxis]  # view `a` as a 2D column vector
array([[4.],
       [2.]])
>>> np.column_stack((a[:, newaxis], b[:, newaxis]))
array([[4., 3.],
       [2., 8.]])
>>> np.hstack((a[:, newaxis], b[:, newaxis]))  # the result is the same
array([[4., 3.],
       [2., 8.]])

>>> np.column_stack is np.hstack
False
>>> np.row_stack is np.vstack
True

>>> np.r_[1:4, 0, 4]
array([1, 2, 3, 0, 4])

>>> a = np.floor(10 * rg.random((2, 12)))
>>> a
array([[6., 7., 6., 9., 0., 5., 4., 0., 6., 8., 5., 2.],
       [8., 5., 5., 7., 1., 8., 6., 7., 1., 8., 1., 0.]])
>>> # Split `a` into 3
>>> np.hsplit(a, 3)
[array([[6., 7., 6., 9.],
       [8., 5., 5., 7.]]), array([[0., 5., 4., 0.],
       [1., 8., 6., 7.]]), array([[6., 8., 5., 2.],
       [1., 8., 1., 0.]])]
>>> # Split `a` after the third and the fourth column
>>> np.hsplit(a, (3, 4))
[array([[6., 7., 6.],
       [8., 5., 5.]]), array([[9.],
       [7.]]), array([[0., 5., 4., 0., 6., 8., 5., 2.],
       [1., 8., 6., 7., 1., 8., 1., 0.]])]

Copies and Views#

When operating and manipulating arrays, their data is sometimes copied
into a new array and sometimes not. This is often a source of confusion
for beginners. There are three cases:

>>> a = np.array([[ 0,  1,  2,  3],
...               [ 4,  5,  6,  7],
...               [ 8,  9, 10, 11]])
>>> b = a            # no new object is created
>>> b is a           # a and b are two names for the same ndarray object
True

>>> def f(x):
...     print(id(x))
...
>>> id(a)  # id is a unique identifier of an object 
148293216  # may vary
>>> f(a)   
148293216  # may vary

>>> c = a.view()
>>> c is a
False
>>> c.base is a            # c is a view of the data owned by a
True
>>> c.flags.owndata
False
>>>
>>> c = c.reshape((2, 6))  # a's shape doesn't change
>>> a.shape
(3, 4)
>>> c[0, 4] = 1234         # a's data changes
>>> a
array([[   0,    1,    2,    3],
       [1234,    5,    6,    7],
       [   8,    9,   10,   11]])

>>> s = a[:, 1:3]
>>> s[:] = 10  # s[:] is a view of s. Note the difference between s = 10 and s[:] = 10
>>> a
array([[   0,   10,   10,    3],
       [1234,   10,   10,    7],
       [   8,   10,   10,   11]])

>>> d = a.copy()  # a new array object with new data is created
>>> d is a
False
>>> d.base is a  # d doesn't share anything with a
False
>>> d[0, 0] = 9999
>>> a
array([[   0,   10,   10,    3],
       [1234,   10,   10,    7],
       [   8,   10,   10,   11]])

>>> a = np.arange(int(1e8))
>>> b = a[:100].copy()
>>> del a  # the memory of ``a`` can be released.

Less Basic#

Advanced indexing and index tricks#

NumPy offers more indexing facilities than regular Python sequences. In
addition to indexing by integers and slices, as we saw before, arrays
can be indexed by arrays of integers and arrays of booleans.

>>> a = np.arange(12)**2  # the first 12 square numbers
>>> i = np.array([1, 1, 3, 8, 5])  # an array of indices
>>> a[i]  # the elements of `a` at the positions `i`
array([ 1,  1,  9, 64, 25])
>>>
>>> j = np.array([[3, 4], [9, 7]])  # a bidimensional array of indices
>>> a[j]  # the same shape as `j`
array([[ 9, 16],
       [81, 49]])

>>> palette = np.array([[0, 0, 0],         # black
...                     [255, 0, 0],       # red
...                     [0, 255, 0],       # green
...                     [0, 0, 255],       # blue
...                     [255, 255, 255]])  # white
>>> image = np.array([[0, 1, 2, 0],  # each value corresponds to a color in the palette
...                   [0, 3, 4, 0]])
>>> palette[image]  # the (2, 4, 3) color image
array([[[  0,   0,   0],
        [255,   0,   0],
        [  0, 255,   0],
        [  0,   0,   0]],

       [[  0,   0,   0],
        [  0,   0, 255],
        [255, 255, 255],
        [  0,   0,   0]]])

>>> a = np.arange(12).reshape(3, 4)
>>> a
array([[ 0,  1,  2,  3],
       [ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])
>>> i = np.array([[0, 1],  # indices for the first dim of `a`
...               [1, 2]])
>>> j = np.array([[2, 1],  # indices for the second dim
...               [3, 3]])
>>>
>>> a[i, j]  # i and j must have equal shape
array([[ 2,  5],
       [ 7, 11]])
>>>
>>> a[i, 2]
array([[ 2,  6],
       [ 6, 10]])
>>>
>>> a[:, j]
array([[[ 2,  1],
        [ 3,  3]],

       [[ 6,  5],
        [ 7,  7]],

       [[10,  9],
        [11, 11]]])

>>> l = (i, j)
>>> # equivalent to a[i, j]
>>> a[l]
array([[ 2,  5],
       [ 7, 11]])

>>> s = np.array([i, j])
>>> # not what we want
>>> a[s]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
IndexError: index 3 is out of bounds for axis 0 with size 3
>>> # same as `a[i, j]`
>>> a[tuple(s)]
array([[ 2,  5],
       [ 7, 11]])

>>> time = np.linspace(20, 145, 5)  # time scale
>>> data = np.sin(np.arange(20)).reshape(5, 4)  # 4 time-dependent series
>>> time
array([ 20.  ,  51.25,  82.5 , 113.75, 145.  ])
>>> data
array([[ 0.        ,  0.84147098,  0.90929743,  0.14112001],
       [-0.7568025 , -0.95892427, -0.2794155 ,  0.6569866 ],
       [ 0.98935825,  0.41211849, -0.54402111, -0.99999021],
       [-0.53657292,  0.42016704,  0.99060736,  0.65028784],
       [-0.28790332, -0.96139749, -0.75098725,  0.14987721]])
>>> # index of the maxima for each series
>>> ind = data.argmax(axis=0)
>>> ind
array([2, 0, 3, 1])
>>> # times corresponding to the maxima
>>> time_max = time[ind]
>>>
>>> data_max = data[ind, range(data.shape[1])]  # => data[ind[0], 0], data[ind[1], 1]...
>>> time_max
array([ 82.5 ,  20.  , 113.75,  51.25])
>>> data_max
array([0.98935825, 0.84147098, 0.99060736, 0.6569866 ])
>>> np.all(data_max == data.max(axis=0))
True

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

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

>>> a = np.arange(5)
>>> a[[0, 0, 2]] += 1
>>> a
array([1, 1, 3, 3, 4])

>>> a = np.arange(12).reshape(3, 4)
>>> b = a > 4
>>> b  # `b` is a boolean with `a`'s shape
array([[False, False, False, False],
       [False,  True,  True,  True],
       [ True,  True,  True,  True]])
>>> a[b]  # 1d array with the selected elements
array([ 5,  6,  7,  8,  9, 10, 11])

>>> a[b] = 0  # All elements of `a` higher than 4 become 0
>>> a
array([[0, 1, 2, 3],
       [4, 0, 0, 0],
       [0, 0, 0, 0]])

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> def mandelbrot(h, w, maxit=20, r=2):
...     """Returns an image of the Mandelbrot fractal of size (h,w)."""
...     x = np.linspace(-2.5, 1.5, 4*h+1)
...     y = np.linspace(-1.5, 1.5, 3*w+1)
...     A, B = np.meshgrid(x, y)
...     C = A + B*1j
...     z = np.zeros_like(C)
...     divtime = maxit + np.zeros(z.shape, dtype=int)
...
...     for i in range(maxit):
...         z = z**2 + C
...         diverge = abs(z) > r                    # who is diverging
...         div_now = diverge & (divtime == maxit)  # who is diverging now
...         divtime[div_now] = i                    # note when
...         z[diverge] = r                          # avoid diverging too much
...
...     return divtime
>>> plt.clf()
>>> plt.imshow(mandelbrot(400, 400))

>>> a = np.arange(12).reshape(3, 4)
>>> b1 = np.array([False, True, True])         # first dim selection
>>> b2 = np.array([True, False, True, False])  # second dim selection
>>>
>>> a[b1, :]                                   # selecting rows
array([[ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])
>>>
>>> a[b1]                                      # same thing
array([[ 4,  5,  6,  7],
       [ 8,  9, 10, 11]])
>>>
>>> a[:, b2]                                   # selecting columns
array([[ 0,  2],
       [ 4,  6],
       [ 8, 10]])
>>>
>>> a[b1, b2]                                  # a weird thing to do
array([ 4, 10])

>>> a = np.array([2, 3, 4, 5])
>>> b = np.array([8, 5, 4])
>>> c = np.array([5, 4, 6, 8, 3])
>>> ax, bx, cx = np.ix_(a, b, c)
>>> ax
array([[[2]],

       [[3]],

       [[4]],

       [[5]]])
>>> bx
array([[[8],
        [5],
        [4]]])
>>> cx
array([[[5, 4, 6, 8, 3]]])
>>> ax.shape, bx.shape, cx.shape
((4, 1, 1), (1, 3, 1), (1, 1, 5))
>>> result = ax + bx * cx
>>> result
array([[[42, 34, 50, 66, 26],
        [27, 22, 32, 42, 17],
        [22, 18, 26, 34, 14]],

       [[43, 35, 51, 67, 27],
        [28, 23, 33, 43, 18],
        [23, 19, 27, 35, 15]],

       [[44, 36, 52, 68, 28],
        [29, 24, 34, 44, 19],
        [24, 20, 28, 36, 16]],

       [[45, 37, 53, 69, 29],
        [30, 25, 35, 45, 20],
        [25, 21, 29, 37, 17]]])
>>> result[3, 2, 4]
17
>>> a[3] + b[2] * c[4]
17

>>> def ufunc_reduce(ufct, *vectors):
...    vs = np.ix_(*vectors)
...    r = ufct.identity
...    for v in vs:
...        r = ufct(r, v)
...    return r

>>> ufunc_reduce(np.add, a, b, c)
array([[[15, 14, 16, 18, 13],
        [12, 11, 13, 15, 10],
        [11, 10, 12, 14,  9]],

       [[16, 15, 17, 19, 14],
        [13, 12, 14, 16, 11],
        [12, 11, 13, 15, 10]],

       [[17, 16, 18, 20, 15],
        [14, 13, 15, 17, 12],
        [13, 12, 14, 16, 11]],

       [[18, 17, 19, 21, 16],
        [15, 14, 16, 18, 13],
        [14, 13, 15, 17, 12]]])

Tricks and Tips#

Here we give a list of short and useful tips.

>>> a = np.arange(30)
>>> b = a.reshape((2, -1, 3))  # -1 means "whatever is needed"
>>> b.shape
(2, 5, 3)
>>> b
array([[[ 0,  1,  2],
        [ 3,  4,  5],
        [ 6,  7,  8],
        [ 9, 10, 11],
        [12, 13, 14]],

       [[15, 16, 17],
        [18, 19, 20],
        [21, 22, 23],
        [24, 25, 26],
        [27, 28, 29]]])

>>> x = np.arange(0, 10, 2)
>>> y = np.arange(5)
>>> m = np.vstack([x, y])
>>> m
array([[0, 2, 4, 6, 8],
       [0, 1, 2, 3, 4]])
>>> xy = np.hstack([x, y])
>>> xy
array([0, 2, 4, 6, 8, 0, 1, 2, 3, 4])

>>> import numpy as np
>>> rg = np.random.default_rng(1)
>>> import matplotlib.pyplot as plt
>>> # Build a vector of 10000 normal deviates with variance 0.5^2 and mean 2
>>> mu, sigma = 2, 0.5
>>> v = rg.normal(mu, sigma, 10000)
>>> # Plot a normalized histogram with 50 bins
>>> plt.hist(v, bins=50, density=True)       # matplotlib version (plot)
(array...)
>>> # Compute the histogram with numpy and then plot it
>>> (n, bins) = np.histogram(v, bins=50, density=True)  # NumPy version (no plot)
>>> plt.plot(.5 * (bins[1:] + bins[:-1]), n) 

Further reading#

• The Python tutorial

• NumPy Reference

• SciPy Tutorial

• SciPy Lecture Notes

• A matlab, R, IDL, NumPy/SciPy dictionary

• tutorial-svd

Welcome to NumPy!#

NumPy (Numerical Python) is an open source Python library that’s used in
almost every field of science and engineering. It’s the universal standard for
working with numerical data in Python, and it’s at the core of the scientific
Python and PyData ecosystems. NumPy users include everyone from beginning coders
to experienced researchers doing state-of-the-art scientific and industrial
research and development. The NumPy API is used extensively in Pandas, SciPy,
Matplotlib, scikit-learn, scikit-image and most other data science and
scientific Python packages.

The NumPy library contains multidimensional array and matrix data structures
(you’ll find more information about this in later sections). It provides
ndarray, a homogeneous n-dimensional array object, with methods to
efficiently operate on it. NumPy can be used to perform a wide variety of
mathematical operations on arrays. It adds powerful data structures to Python
that guarantee efficient calculations with arrays and matrices and it supplies
an enormous library of high-level mathematical functions that operate on these
arrays and matrices.

Learn more about NumPy here!

Installing NumPy#

To install NumPy, we strongly recommend using a scientific Python distribution.
If you’re looking for the full instructions for installing NumPy on your
operating system, see Installing NumPy.

If you already have Python, you can install NumPy with:

or

If you don’t have Python yet, you might want to consider using Anaconda. It’s the easiest way to get started. The good
thing about getting this distribution is the fact that you don’t need to worry
too much about separately installing NumPy or any of the major packages that
you’ll be using for your data analyses, like pandas, Scikit-Learn, etc.

How to import NumPy#

To access NumPy and its functions import it in your Python code like this:

We shorten the imported name to np for better readability of code using
NumPy. This is a widely adopted convention that you should follow so that
anyone working with your code can easily understand it.

Reading the example code#

If you aren’t already comfortable with reading tutorials that contain a lot of code,
you might not know how to interpret a code block that looks
like this:

>>> a = np.arange(6)
>>> a2 = a[np.newaxis, :]
>>> a2.shape
(1, 6)

If you aren’t familiar with this style, it’s very easy to understand.
If you see >>>, you’re looking at input, or the code that
you would enter. Everything that doesn’t have >>> in front of it
is output, or the results of running your code. This is the style
you see when you run python on the command line, but if you’re using
IPython, you might see a different style. Note that it is not part of the
code and will cause an error if typed or pasted into the Python
shell. It can be safely typed or pasted into the IPython shell; the >>>
is ignored.

What’s the difference between a Python list and a NumPy array?#

NumPy gives you an enormous range of fast and efficient ways of creating arrays
and manipulating numerical data inside them. While a Python list can contain
different data types within a single list, all of the elements in a NumPy array
should be homogeneous. The mathematical operations that are meant to be performed
on arrays would be extremely inefficient if the arrays weren’t homogeneous.

Why use NumPy?

NumPy arrays are faster and more compact than Python lists. An array consumes
less memory and is convenient to use. NumPy uses much less memory to store data
and it provides a mechanism of specifying the data types. This allows the code
to be optimized even further.

What is an array?#

An array is a central data structure of the NumPy library. An array is a grid of
values and it contains information about the raw data, how to locate an element,
and how to interpret an element. It has a grid of elements that can be indexed
in various ways.
The elements are all of the same type, referred to as the array dtype.

An array can be indexed by a tuple of nonnegative integers, by booleans, by
another array, or by integers. The rank of the array is the number of
dimensions. The shape of the array is a tuple of integers giving the size of
the array along each dimension.

One way we can initialize NumPy arrays is from Python lists, using nested lists
for two- or higher-dimensional data.

For example:

>>> a = np.array([1, 2, 3, 4, 5, 6])

or:

>>> a = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

We can access the elements in the array using square brackets. When you’re
accessing elements, remember that indexing in NumPy starts at 0. That means that
if you want to access the first element in your array, you’ll be accessing
element “0”.

>>> print(a[0])
[1 2 3 4]

More information about arrays#

This section covers 1D array, 2D array, ndarray, vector, matrix

You might occasionally hear an array referred to as a “ndarray,” which is
shorthand for “N-dimensional array.” An N-dimensional array is simply an array
with any number of dimensions. You might also hear 1-D, or one-dimensional
array, 2-D, or two-dimensional array, and so on. The NumPy ndarray class
is used to represent both matrices and vectors. A vector is an array with a
single dimension (there’s no difference
between row and column vectors), while a matrix refers to an
array with two dimensions. For 3-D or higher dimensional arrays, the term
tensor is also commonly used.

What are the attributes of an array?

An array is usually a fixed-size container of items of the same type and size.
The number of dimensions and items in an array is defined by its shape. The
shape of an array is a tuple of non-negative integers that specify the sizes of
each dimension.

In NumPy, dimensions are called axes. This means that if you have a 2D array
that looks like this:

[[0., 0., 0.],
 [1., 1., 1.]]

Your array has 2 axes. The first axis has a length of 2 and the second axis has
a length of 3.

Just like in other Python container objects, the contents of an array can be
accessed and modified by indexing or slicing the array. Unlike the typical container
objects, different arrays can share the same data, so changes made on one array might
be visible in another.

Array attributes reflect information intrinsic to the array itself. If you
need to get, or even set, properties of an array without creating a new array,
you can often access an array through its attributes.

Read more about array attributes here and learn about
array objects here.

How to create a basic array#

This section covers np.array(), np.zeros(), np.ones(),
np.empty(), np.arange(), np.linspace(), dtype

To create a NumPy array, you can use the function np.array().

All you need to do to create a simple array is pass a list to it. If you choose
to, you can also specify the type of data in your list.
You can find more information about data types here.

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

You can visualize your array this way:

Be aware that these visualizations are meant to simplify ideas and give you a basic understanding of NumPy concepts and mechanics. Arrays and array operations are much more complicated than are captured here!

Besides creating an array from a sequence of elements, you can easily create an
array filled with 0’s:

>>> np.zeros(2)
array([0., 0.])

Or an array filled with 1’s:

>>> np.ones(2)
array([1., 1.])

Or even an empty array! The function empty creates an array whose initial
content is random and depends on the state of the memory. The reason to use
empty over zeros (or something similar) is speed — just make sure to
fill every element afterwards!

>>> # Create an empty array with 2 elements
>>> np.empty(2) 
array([3.14, 42.  ])  # may vary

You can create an array with a range of elements:

>>> np.arange(4)
array([0, 1, 2, 3])

And even an array that contains a range of evenly spaced intervals. To do this,
you will specify the first number, last number, and the step size.

>>> np.arange(2, 9, 2)
array([2, 4, 6, 8])

You can also use np.linspace() to create an array with values that are
spaced linearly in a specified interval:

>>> np.linspace(0, 10, num=5)
array([ 0. ,  2.5,  5. ,  7.5, 10. ])

Specifying your data type

While the default data type is floating point (np.float64), you can explicitly
specify which data type you want using the dtype keyword.

>>> x = np.ones(2, dtype=np.int64)
>>> x
array([1, 1])

Learn more about creating arrays here

Adding, removing, and sorting elements#

This section covers np.sort(), np.concatenate()

Sorting an element is simple with np.sort(). You can specify the axis, kind,
and order when you call the function.

If you start with this array:

>>> arr = np.array([2, 1, 5, 3, 7, 4, 6, 8])

You can quickly sort the numbers in ascending order with:

>>> np.sort(arr)
array([1, 2, 3, 4, 5, 6, 7, 8])

In addition to sort, which returns a sorted copy of an array, you can use:

• argsort, which is an indirect sort along a specified axis,

• lexsort, which is an indirect stable sort on multiple keys,

• searchsorted, which will find elements in a sorted array, and

• partition, which is a partial sort.

To read more about sorting an array, see: sort.

If you start with these arrays:

>>> a = np.array([1, 2, 3, 4])
>>> b = np.array([5, 6, 7, 8])

You can concatenate them with np.concatenate().

>>> np.concatenate((a, b))
array([1, 2, 3, 4, 5, 6, 7, 8])

Or, if you start with these arrays:

>>> x = np.array([[1, 2], [3, 4]])
>>> y = np.array([[5, 6]])

You can concatenate them with:

>>> np.concatenate((x, y), axis=0)
array([[1, 2],
       [3, 4],
       [5, 6]])

In order to remove elements from an array, it’s simple to use indexing to select
the elements that you want to keep.

To read more about concatenate, see: concatenate.

How do you know the shape and size of an array?#

This section covers ndarray.ndim, ndarray.size, ndarray.shape

ndarray.ndim will tell you the number of axes, or dimensions, of the array.

ndarray.size will tell you the total number of elements of the array. This
is the product of the elements of the array’s shape.

ndarray.shape will display a tuple of integers that indicate the number of
elements stored along each dimension of the array. If, for example, you have a
2-D array with 2 rows and 3 columns, the shape of your array is (2, 3).

For example, if you create this array:

>>> array_example = np.array([[[0, 1, 2, 3],
...                            [4, 5, 6, 7]],
...
...                           [[0, 1, 2, 3],
...                            [4, 5, 6, 7]],
...
...                           [[0 ,1 ,2, 3],
...                            [4, 5, 6, 7]]])

To find the number of dimensions of the array, run:

To find the total number of elements in the array, run:

>>> array_example.size
24

And to find the shape of your array, run:

>>> array_example.shape
(3, 2, 4)

Can you reshape an array?#

This section covers arr.reshape()

Yes!

Using arr.reshape() will give a new shape to an array without changing the
data. Just remember that when you use the reshape method, the array you want to
produce needs to have the same number of elements as the original array. If you
start with an array with 12 elements, you’ll need to make sure that your new
array also has a total of 12 elements.

If you start with this array:

>>> a = np.arange(6)
>>> print(a)
[0 1 2 3 4 5]

You can use reshape() to reshape your array. For example, you can reshape
this array to an array with three rows and two columns:

>>> b = a.reshape(3, 2)
>>> print(b)
[[0 1]
 [2 3]
 [4 5]]

With np.reshape, you can specify a few optional parameters:

>>> np.reshape(a, newshape=(1, 6), order='C')
array([[0, 1, 2, 3, 4, 5]])

a is the array to be reshaped.

newshape is the new shape you want. You can specify an integer or a tuple of
integers. If you specify an integer, the result will be an array of that length.
The shape should be compatible with the original shape.

order: C means to read/write the elements using C-like index order,
F means to read/write the elements using Fortran-like index order, A
means to read/write the elements in Fortran-like index order if a is Fortran
contiguous in memory, C-like order otherwise. (This is an optional parameter and
doesn’t need to be specified.)

If you want to learn more about C and Fortran order, you can
read more about the internal organization of NumPy arrays here.
Essentially, C and Fortran orders have to do with how indices correspond
to the order the array is stored in memory. In Fortran, when moving through
the elements of a two-dimensional array as it is stored in memory, the first
index is the most rapidly varying index. As the first index moves to the next
row as it changes, the matrix is stored one column at a time.
This is why Fortran is thought of as a Column-major language.
In C on the other hand, the last index changes
the most rapidly. The matrix is stored by rows, making it a Row-major
language. What you do for C or Fortran depends on whether it’s more important
to preserve the indexing convention or not reorder the data.

Learn more about shape manipulation here.

How to convert a 1D array into a 2D array (how to add a new axis to an array)#

This section covers np.newaxis, np.expand_dims

You can use np.newaxis and np.expand_dims to increase the dimensions of
your existing array.

Using np.newaxis will increase the dimensions of your array by one dimension
when used once. This means that a 1D array will become a 2D array, a
2D array will become a 3D array, and so on.

For example, if you start with this array:

>>> a = np.array([1, 2, 3, 4, 5, 6])
>>> a.shape
(6,)

You can use np.newaxis to add a new axis:

>>> a2 = a[np.newaxis, :]
>>> a2.shape
(1, 6)

You can explicitly convert a 1D array with either a row vector or a column
vector using np.newaxis. For example, you can convert a 1D array to a row
vector by inserting an axis along the first dimension:

>>> row_vector = a[np.newaxis, :]
>>> row_vector.shape
(1, 6)

Or, for a column vector, you can insert an axis along the second dimension:

>>> col_vector = a[:, np.newaxis]
>>> col_vector.shape
(6, 1)

You can also expand an array by inserting a new axis at a specified position
with np.expand_dims.

For example, if you start with this array:

>>> a = np.array([1, 2, 3, 4, 5, 6])
>>> a.shape
(6,)

You can use np.expand_dims to add an axis at index position 1 with:

>>> b = np.expand_dims(a, axis=1)
>>> b.shape
(6, 1)

You can add an axis at index position 0 with:

>>> c = np.expand_dims(a, axis=0)
>>> c.shape
(1, 6)

Find more information about newaxis here and
expand_dims at expand_dims.

Indexing and slicing#

You can index and slice NumPy arrays in the same ways you can slice Python
lists.

>>> data = np.array([1, 2, 3])

>>> data[1]
2
>>> data[0:2]
array([1, 2])
>>> data[1:]
array([2, 3])
>>> data[-2:]
array([2, 3])

You can visualize it this way:

You may want to take a section of your array or specific array elements to use
in further analysis or additional operations. To do that, you’ll need to subset,
slice, and/or index your arrays.

If you want to select values from your array that fulfill certain conditions,
it’s straightforward with NumPy.

For example, if you start with this array:

>>> a = np.array([[1 , 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

You can easily print all of the values in the array that are less than 5.

>>> print(a[a < 5])
[1 2 3 4]

You can also select, for example, numbers that are equal to or greater than 5,
and use that condition to index an array.

>>> five_up = (a >= 5)
>>> print(a[five_up])
[ 5  6  7  8  9 10 11 12]

You can select elements that are divisible by 2:

>>> divisible_by_2 = a[a%2==0]
>>> print(divisible_by_2)
[ 2  4  6  8 10 12]

Or you can select elements that satisfy two conditions using the & and |
operators:

>>> c = a[(a > 2) & (a < 11)]
>>> print(c)
[ 3  4  5  6  7  8  9 10]

You can also make use of the logical operators & and | in order to
return boolean values that specify whether or not the values in an array fulfill
a certain condition. This can be useful with arrays that contain names or other
categorical values.

>>> five_up = (a > 5) | (a == 5)
>>> print(five_up)
[[False False False False]
 [ True  True  True  True]
 [ True  True  True True]]

You can also use np.nonzero() to select elements or indices from an array.

Starting with this array:

>>> a = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

You can use np.nonzero() to print the indices of elements that are, for
example, less than 5:

>>> b = np.nonzero(a < 5)
>>> print(b)
(array([0, 0, 0, 0]), array([0, 1, 2, 3]))

In this example, a tuple of arrays was returned: one for each dimension. The
first array represents the row indices where these values are found, and the
second array represents the column indices where the values are found.

If you want to generate a list of coordinates where the elements exist, you can
zip the arrays, iterate over the list of coordinates, and print them. For
example:

>>> list_of_coordinates= list(zip(b[0], b[1]))

>>> for coord in list_of_coordinates:
...     print(coord)
(0, 0)
(0, 1)
(0, 2)
(0, 3)

You can also use np.nonzero() to print the elements in an array that are less
than 5 with:

>>> print(a[b])
[1 2 3 4]

If the element you’re looking for doesn’t exist in the array, then the returned
array of indices will be empty. For example:

>>> not_there = np.nonzero(a == 42)
>>> print(not_there)
(array([], dtype=int64), array([], dtype=int64))

Learn more about indexing and slicing here
and here.

Read more about using the nonzero function at: nonzero.

How to create an array from existing data#

This section covers slicing and indexing, np.vstack(), np.hstack(),
np.hsplit(), .view(), copy()

You can easily create a new array from a section of an existing array.

Let’s say you have this array:

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

You can create a new array from a section of your array any time by specifying
where you want to slice your array.

>>> arr1 = a[3:8]
>>> arr1
array([4, 5, 6, 7, 8])

Here, you grabbed a section of your array from index position 3 through index
position 8.

You can also stack two existing arrays, both vertically and horizontally. Let’s
say you have two arrays, a1 and a2:

>>> a1 = np.array([[1, 1],
...                [2, 2]])

>>> a2 = np.array([[3, 3],
...                [4, 4]])

You can stack them vertically with vstack:

>>> np.vstack((a1, a2))
array([[1, 1],
       [2, 2],
       [3, 3],
       [4, 4]])

Or stack them horizontally with hstack:

>>> np.hstack((a1, a2))
array([[1, 1, 3, 3],
       [2, 2, 4, 4]])

You can split an array into several smaller arrays using hsplit. You can
specify either the number of equally shaped arrays to return or the columns
after which the division should occur.

Let’s say you have this array:

>>> x = np.arange(1, 25).reshape(2, 12)
>>> x
array([[ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12],
       [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]])

If you wanted to split this array into three equally shaped arrays, you would
run:

>>> np.hsplit(x, 3)
  [array([[ 1,  2,  3,  4],
         [13, 14, 15, 16]]), array([[ 5,  6,  7,  8],
         [17, 18, 19, 20]]), array([[ 9, 10, 11, 12],
         [21, 22, 23, 24]])]

If you wanted to split your array after the third and fourth column, you’d run:

>>> np.hsplit(x, (3, 4))
  [array([[ 1,  2,  3],
         [13, 14, 15]]), array([[ 4],
         [16]]), array([[ 5,  6,  7,  8,  9, 10, 11, 12],
         [17, 18, 19, 20, 21, 22, 23, 24]])]

Learn more about stacking and splitting arrays here.

You can use the view method to create a new array object that looks at the
same data as the original array (a shallow copy).

Views are an important NumPy concept! NumPy functions, as well as operations
like indexing and slicing, will return views whenever possible. This saves
memory and is faster (no copy of the data has to be made). However it’s
important to be aware of this — modifying data in a view also modifies the
original array!

Let’s say you create this array:

>>> a = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

Now we create an array b1 by slicing a and modify the first element of
b1. This will modify the corresponding element in a as well!

>>> b1 = a[0, :]
>>> b1
array([1, 2, 3, 4])
>>> b1[0] = 99
>>> b1
array([99,  2,  3,  4])
>>> a
array([[99,  2,  3,  4],
       [ 5,  6,  7,  8],
       [ 9, 10, 11, 12]])

Using the copy method will make a complete copy of the array and its data (a
deep copy). To use this on your array, you could run:

Learn more about copies and views here.

Basic array operations#

This section covers addition, subtraction, multiplication, division, and more

Once you’ve created your arrays, you can start to work with them. Let’s say,
for example, that you’ve created two arrays, one called “data” and one called
“ones”

You can add the arrays together with the plus sign.

>>> data = np.array([1, 2])
>>> ones = np.ones(2, dtype=int)
>>> data + ones
array([2, 3])

You can, of course, do more than just addition!

>>> data - ones
array([0, 1])
>>> data * data
array([1, 4])
>>> data / data
array([1., 1.])

Basic operations are simple with NumPy. If you want to find the sum of the
elements in an array, you’d use sum(). This works for 1D arrays, 2D arrays,
and arrays in higher dimensions.

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

>>> a.sum()
10

To add the rows or the columns in a 2D array, you would specify the axis.

If you start with this array:

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

You can sum over the axis of rows with:

>>> b.sum(axis=0)
array([3, 3])

You can sum over the axis of columns with:

>>> b.sum(axis=1)
array([2, 4])

Learn more about basic operations here.

Broadcasting#

There are times when you might want to carry out an operation between an array
and a single number (also called an operation between a vector and a scalar)
or between arrays of two different sizes. For example, your array (we’ll call it
“data”) might contain information about distance in miles but you want to
convert the information to kilometers. You can perform this operation with:

>>> data = np.array([1.0, 2.0])
>>> data * 1.6
array([1.6, 3.2])

NumPy understands that the multiplication should happen with each cell. That
concept is called broadcasting. Broadcasting is a mechanism that allows
NumPy to perform operations on arrays of different shapes. The dimensions of
your array must be compatible, for example, when the dimensions of both arrays
are equal or when one of them is 1. If the dimensions are not compatible, you
will get a ValueError.

Learn more about broadcasting here.

More useful array operations#

This section covers maximum, minimum, sum, mean, product, standard deviation, and more

NumPy also performs aggregation functions. In addition to min, max, and
sum, you can easily run mean to get the average, prod to get the
result of multiplying the elements together, std to get the standard
deviation, and more.

>>> data.max()
2.0
>>> data.min()
1.0
>>> data.sum()
3.0

Let’s start with this array, called “a”

>>> a = np.array([[0.45053314, 0.17296777, 0.34376245, 0.5510652],
...               [0.54627315, 0.05093587, 0.40067661, 0.55645993],
...               [0.12697628, 0.82485143, 0.26590556, 0.56917101]])

It’s very common to want to aggregate along a row or column. By default, every
NumPy aggregation function will return the aggregate of the entire array. To
find the sum or the minimum of the elements in your array, run:

Or:

You can specify on which axis you want the aggregation function to be computed.
For example, you can find the minimum value within each column by specifying
axis=0.

>>> a.min(axis=0)
array([0.12697628, 0.05093587, 0.26590556, 0.5510652 ])

The four values listed above correspond to the number of columns in your array.
With a four-column array, you will get four values as your result.

Read more about array methods here.

Creating matrices#

You can pass Python lists of lists to create a 2-D array (or “matrix”) to
represent them in NumPy.

>>> data = np.array([[1, 2], [3, 4], [5, 6]])
>>> data
array([[1, 2],
       [3, 4],
       [5, 6]])

Indexing and slicing operations are useful when you’re manipulating matrices:

>>> data[0, 1]
2
>>> data[1:3]
array([[3, 4],
       [5, 6]])
>>> data[0:2, 0]
array([1, 3])

You can aggregate matrices the same way you aggregated vectors:

>>> data.max()
6
>>> data.min()
1
>>> data.sum()
21

You can aggregate all the values in a matrix and you can aggregate them across
columns or rows using the axis parameter. To illustrate this point, let’s
look at a slightly modified dataset:

>>> data = np.array([[1, 2], [5, 3], [4, 6]])
>>> data
array([[1, 2],
       [5, 3],
       [4, 6]])
>>> data.max(axis=0)
array([5, 6])
>>> data.max(axis=1)
array([2, 5, 6])

Once you’ve created your matrices, you can add and multiply them using
arithmetic operators if you have two matrices that are the same size.

>>> data = np.array([[1, 2], [3, 4]])
>>> ones = np.array([[1, 1], [1, 1]])
>>> data + ones
array([[2, 3],
       [4, 5]])

You can do these arithmetic operations on matrices of different sizes, but only
if one matrix has only one column or one row. In this case, NumPy will use its
broadcast rules for the operation.

>>> data = np.array([[1, 2], [3, 4], [5, 6]])
>>> ones_row = np.array([[1, 1]])
>>> data + ones_row
array([[2, 3],
       [4, 5],
       [6, 7]])

Be aware that when NumPy prints N-dimensional arrays, the last axis is looped
over the fastest while the first axis is the slowest. For instance:

>>> np.ones((4, 3, 2))
array([[[1., 1.],
        [1., 1.],
        [1., 1.]],

       [[1., 1.],
        [1., 1.],
        [1., 1.]],

       [[1., 1.],
        [1., 1.],
        [1., 1.]],

       [[1., 1.],
        [1., 1.],
        [1., 1.]]])

There are often instances where we want NumPy to initialize the values of an
array. NumPy offers functions like ones() and zeros(), and the
random.Generator class for random number generation for that.
All you need to do is pass in the number of elements you want it to generate:

>>> np.ones(3)
array([1., 1., 1.])
>>> np.zeros(3)
array([0., 0., 0.])
>>> rng = np.random.default_rng()  # the simplest way to generate random numbers
>>> rng.random(3) 
array([0.63696169, 0.26978671, 0.04097352])

You can also use ones(), zeros(), and random() to create
a 2D array if you give them a tuple describing the dimensions of the matrix:

>>> np.ones((3, 2))
array([[1., 1.],
       [1., 1.],
       [1., 1.]])
>>> np.zeros((3, 2))
array([[0., 0.],
       [0., 0.],
       [0., 0.]])
>>> rng.random((3, 2)) 
array([[0.01652764, 0.81327024],
       [0.91275558, 0.60663578],
       [0.72949656, 0.54362499]])  # may vary

Read more about creating arrays, filled with 0’s, 1’s, other values or
uninitialized, at array creation routines.

Generating random numbers#

The use of random number generation is an important part of the configuration
and evaluation of many numerical and machine learning algorithms. Whether you
need to randomly initialize weights in an artificial neural network, split data
into random sets, or randomly shuffle your dataset, being able to generate
random numbers (actually, repeatable pseudo-random numbers) is essential.

With Generator.integers, you can generate random integers from low (remember
that this is inclusive with NumPy) to high (exclusive). You can set
endpoint=True to make the high number inclusive.

You can generate a 2 x 4 array of random integers between 0 and 4 with:

>>> rng.integers(5, size=(2, 4)) 
array([[2, 1, 1, 0],
       [0, 0, 0, 4]])  # may vary

Read more about random number generation here.

How to get unique items and counts#

This section covers np.unique()

You can find the unique elements in an array easily with np.unique.

For example, if you start with this array:

>>> a = np.array([11, 11, 12, 13, 14, 15, 16, 17, 12, 13, 11, 14, 18, 19, 20])

you can use np.unique to print the unique values in your array:

>>> unique_values = np.unique(a)
>>> print(unique_values)
[11 12 13 14 15 16 17 18 19 20]

To get the indices of unique values in a NumPy array (an array of first index
positions of unique values in the array), just pass the return_index
argument in np.unique() as well as your array.

>>> unique_values, indices_list = np.unique(a, return_index=True)
>>> print(indices_list)
[ 0  2  3  4  5  6  7 12 13 14]

You can pass the return_counts argument in np.unique() along with your
array to get the frequency count of unique values in a NumPy array.

>>> unique_values, occurrence_count = np.unique(a, return_counts=True)
>>> print(occurrence_count)
[3 2 2 2 1 1 1 1 1 1]

This also works with 2D arrays!
If you start with this array:

>>> a_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [1, 2, 3, 4]])

You can find unique values with:

>>> unique_values = np.unique(a_2d)
>>> print(unique_values)
[ 1  2  3  4  5  6  7  8  9 10 11 12]

If the axis argument isn’t passed, your 2D array will be flattened.

If you want to get the unique rows or columns, make sure to pass the axis
argument. To find the unique rows, specify axis=0 and for columns, specify
axis=1.

>>> unique_rows = np.unique(a_2d, axis=0)
>>> print(unique_rows)
[[ 1  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]

To get the unique rows, index position, and occurrence count, you can use:

>>> unique_rows, indices, occurrence_count = np.unique(
...      a_2d, axis=0, return_counts=True, return_index=True)
>>> print(unique_rows)
[[ 1  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]
>>> print(indices)
[0 1 2]
>>> print(occurrence_count)
[2 1 1]

To learn more about finding the unique elements in an array, see unique.

Transposing and reshaping a matrix#

This section covers arr.reshape(), arr.transpose(), arr.T

It’s common to need to transpose your matrices. NumPy arrays have the property
T that allows you to transpose a matrix.

You may also need to switch the dimensions of a matrix. This can happen when,
for example, you have a model that expects a certain input shape that is
different from your dataset. This is where the reshape method can be useful.
You simply need to pass in the new dimensions that you want for the matrix.

>>> data.reshape(2, 3)
array([[1, 2, 3],
       [4, 5, 6]])
>>> data.reshape(3, 2)
array([[1, 2],
       [3, 4],
       [5, 6]])

You can also use .transpose() to reverse or change the axes of an array
according to the values you specify.

If you start with this array:

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

You can transpose your array with arr.transpose().

>>> arr.transpose()
array([[0, 3],
       [1, 4],
       [2, 5]])

You can also use arr.T:

>>> arr.T
array([[0, 3],
       [1, 4],
       [2, 5]])

To learn more about transposing and reshaping arrays, see transpose and
reshape.

How to reverse an array#

This section covers np.flip()

NumPy’s np.flip() function allows you to flip, or reverse, the contents of
an array along an axis. When using np.flip(), specify the array you would like
to reverse and the axis. If you don’t specify the axis, NumPy will reverse the
contents along all of the axes of your input array.

Reversing a 1D array

If you begin with a 1D array like this one:

>>> arr = np.array([1, 2, 3, 4, 5, 6, 7, 8])

You can reverse it with:

>>> reversed_arr = np.flip(arr)

If you want to print your reversed array, you can run:

>>> print('Reversed Array: ', reversed_arr)
Reversed Array:  [8 7 6 5 4 3 2 1]

Reversing a 2D array

A 2D array works much the same way.

If you start with this array:

>>> arr_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

You can reverse the content in all of the rows and all of the columns with:

>>> reversed_arr = np.flip(arr_2d)
>>> print(reversed_arr)
[[12 11 10  9]
 [ 8  7  6  5]
 [ 4  3  2  1]]

You can easily reverse only the rows with:

>>> reversed_arr_rows = np.flip(arr_2d, axis=0)
>>> print(reversed_arr_rows)
[[ 9 10 11 12]
 [ 5  6  7  8]
 [ 1  2  3  4]]

Or reverse only the columns with:

>>> reversed_arr_columns = np.flip(arr_2d, axis=1)
>>> print(reversed_arr_columns)
[[ 4  3  2  1]
 [ 8  7  6  5]
 [12 11 10  9]]

You can also reverse the contents of only one column or row. For example, you
can reverse the contents of the row at index position 1 (the second row):

>>> arr_2d[1] = np.flip(arr_2d[1])
>>> print(arr_2d)
[[ 1  2  3  4]
 [ 8  7  6  5]
 [ 9 10 11 12]]

You can also reverse the column at index position 1 (the second column):

>>> arr_2d[:,1] = np.flip(arr_2d[:,1])
>>> print(arr_2d)
[[ 1 10  3  4]
 [ 8  7  6  5]
 [ 9  2 11 12]]

Read more about reversing arrays at flip.

Reshaping and flattening multidimensional arrays#

This section covers .flatten(), ravel()

There are two popular ways to flatten an array: .flatten() and .ravel().
The primary difference between the two is that the new array created using
ravel() is actually a reference to the parent array (i.e., a “view”). This
means that any changes to the new array will affect the parent array as well.
Since ravel does not create a copy, it’s memory efficient.

If you start with this array:

>>> x = np.array([[1 , 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])

You can use flatten to flatten your array into a 1D array.

>>> x.flatten()
array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12])

When you use flatten, changes to your new array won’t change the parent
array.

For example:

>>> a1 = x.flatten()
>>> a1[0] = 99
>>> print(x)  # Original array
[[ 1  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]
>>> print(a1)  # New array
[99  2  3  4  5  6  7  8  9 10 11 12]

But when you use ravel, the changes you make to the new array will affect
the parent array.

For example:

>>> a2 = x.ravel()
>>> a2[0] = 98
>>> print(x)  # Original array
[[98  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]
>>> print(a2)  # New array
[98  2  3  4  5  6  7  8  9 10 11 12]

Read more about flatten at ndarray.flatten and ravel at ravel.

How to access the docstring for more information#

This section covers help(), ?, ??

When it comes to the data science ecosystem, Python and NumPy are built with the
user in mind. One of the best examples of this is the built-in access to
documentation. Every object contains the reference to a string, which is known
as the docstring. In most cases, this docstring contains a quick and concise
summary of the object and how to use it. Python has a built-in help()
function that can help you access this information. This means that nearly any
time you need more information, you can use help() to quickly find the
information that you need.

For example:

>>> help(max)
Help on built-in function max in module builtins:

max(...)
    max(iterable, *[, default=obj, key=func]) -> value
    max(arg1, arg2, *args, *[, key=func]) -> value

    With a single iterable argument, return its biggest item. The
    default keyword-only argument specifies an object to return if
    the provided iterable is empty.
    With two or more arguments, return the largest argument.

Because access to additional information is so useful, IPython uses the ?
character as a shorthand for accessing this documentation along with other
relevant information. IPython is a command shell for interactive computing in
multiple languages.
You can find more information about IPython here.

For example:

In [0]: max?
max(iterable, *[, default=obj, key=func]) -> value
max(arg1, arg2, *args, *[, key=func]) -> value

With a single iterable argument, return its biggest item. The
default keyword-only argument specifies an object to return if
the provided iterable is empty.
With two or more arguments, return the largest argument.
Type:      builtin_function_or_method

You can even use this notation for object methods and objects themselves.

Let’s say you create this array:

>>> a = np.array([1, 2, 3, 4, 5, 6])

Then you can obtain a lot of useful information (first details about a itself,
followed by the docstring of ndarray of which a is an instance):

In [1]: a?
Type:            ndarray
String form:     [1 2 3 4 5 6]
Length:          6
File:            ~/anaconda3/lib/python3.9/site-packages/numpy/__init__.py
Docstring:       <no docstring>
Class docstring:
ndarray(shape, dtype=float, buffer=None, offset=0,
        strides=None, order=None)

An array object represents a multidimensional, homogeneous array
of fixed-size items.  An associated data-type object describes the
format of each element in the array (its byte-order, how many bytes it
occupies in memory, whether it is an integer, a floating point number,
or something else, etc.)

Arrays should be constructed using `array`, `zeros` or `empty` (refer
to the See Also section below).  The parameters given here refer to
a low-level method (`ndarray(...)`) for instantiating an array.

For more information, refer to the `numpy` module and examine the
methods and attributes of an array.

Parameters
----------
(for the __new__ method; see Notes below)

shape : tuple of ints
        Shape of created array.
...

This also works for functions and other objects that you create. Just
remember to include a docstring with your function using a string literal
(""" """ or ''' ''' around your documentation).

For example, if you create this function:

>>> def double(a):
...   '''Return a * 2'''
...   return a * 2

You can obtain information about the function:

In [2]: double?
Signature: double(a)
Docstring: Return a * 2
File:      ~/Desktop/<ipython-input-23-b5adf20be596>
Type:      function

You can reach another level of information by reading the source code of the
object you’re interested in. Using a double question mark (??) allows you to
access the source code.

For example:

In [3]: double??
Signature: double(a)
Source:
def double(a):
    '''Return a * 2'''
    return a * 2
File:      ~/Desktop/<ipython-input-23-b5adf20be596>
Type:      function

If the object in question is compiled in a language other than Python, using
?? will return the same information as ?. You’ll find this with a lot of
built-in objects and types, for example:

In [4]: len?
Signature: len(obj, /)
Docstring: Return the number of items in a container.
Type:      builtin_function_or_method

and :

In [5]: len??
Signature: len(obj, /)
Docstring: Return the number of items in a container.
Type:      builtin_function_or_method

have the same output because they were compiled in a programming language other
than Python.

Working with mathematical formulas#

The ease of implementing mathematical formulas that work on arrays is one of
the things that make NumPy so widely used in the scientific Python community.

For example, this is the mean square error formula (a central formula used in
supervised machine learning models that deal with regression):

Implementing this formula is simple and straightforward in NumPy:

What makes this work so well is that predictions and labels can contain
one or a thousand values. They only need to be the same size.

You can visualize it this way:

In this example, both the predictions and labels vectors contain three values,
meaning n has a value of three. After we carry out subtractions the values
in the vector are squared. Then NumPy sums the values, and your result is the
error value for that prediction and a score for the quality of the model.

How to save and load NumPy objects#

This section covers np.save, np.savez, np.savetxt,
np.load, np.loadtxt

You will, at some point, want to save your arrays to disk and load them back
without having to re-run the code. Fortunately, there are several ways to save
and load objects with NumPy. The ndarray objects can be saved to and loaded from
the disk files with loadtxt and savetxt functions that handle normal
text files, load and save functions that handle NumPy binary files with
a .npy file extension, and a savez function that handles NumPy files
with a .npz file extension.

The .npy and .npz files store data, shape, dtype, and other information
required to reconstruct the ndarray in a way that allows the array to be
correctly retrieved, even when the file is on another machine with different
architecture.

If you want to store a single ndarray object, store it as a .npy file using
np.save. If you want to store more than one ndarray object in a single file,
save it as a .npz file using np.savez. You can also save several arrays
into a single file in compressed npz format with savez_compressed.

It’s easy to save and load and array with np.save(). Just make sure to
specify the array you want to save and a file name. For example, if you create
this array:

>>> a = np.array([1, 2, 3, 4, 5, 6])

You can save it as “filename.npy” with:

>>> np.save('filename', a)

You can use np.load() to reconstruct your array.

>>> b = np.load('filename.npy')

If you want to check your array, you can run:

>>> print(b)
[1 2 3 4 5 6]

You can save a NumPy array as a plain text file like a .csv or .txt file
with np.savetxt.

For example, if you create this array:

>>> csv_arr = np.array([1, 2, 3, 4, 5, 6, 7, 8])

You can easily save it as a .csv file with the name “new_file.csv” like this:

>>> np.savetxt('new_file.csv', csv_arr)

You can quickly and easily load your saved text file using loadtxt():

>>> np.loadtxt('new_file.csv')
array([1., 2., 3., 4., 5., 6., 7., 8.])

The savetxt() and loadtxt() functions accept additional optional
parameters such as header, footer, and delimiter. While text files can be easier
for sharing, .npy and .npz files are smaller and faster to read. If you need more
sophisticated handling of your text file (for example, if you need to work with
lines that contain missing values), you will want to use the genfromtxt
function.

With savetxt, you can specify headers, footers, comments, and more.

Learn more about input and output routines here.

Importing and exporting a CSV#

It’s simple to read in a CSV that contains existing information. The best and
easiest way to do this is to use
Pandas.

>>> import pandas as pd

>>> # If all of your columns are the same type:
>>> x = pd.read_csv('music.csv', header=0).values
>>> print(x)
[['Billie Holiday' 'Jazz' 1300000 27000000]
 ['Jimmie Hendrix' 'Rock' 2700000 70000000]
 ['Miles Davis' 'Jazz' 1500000 48000000]
 ['SIA' 'Pop' 2000000 74000000]]

>>> # You can also simply select the columns you need:
>>> x = pd.read_csv('music.csv', usecols=['Artist', 'Plays']).values
>>> print(x)
[['Billie Holiday' 27000000]
 ['Jimmie Hendrix' 70000000]
 ['Miles Davis' 48000000]
 ['SIA' 74000000]]