CHAPTER 4¶
NumPy basics: Arrays and vectorized computations¶
In [2]:
%matplotlib inline
/usr/local/lib/python3.5/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')
/usr/local/lib/python3.5/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')
In [3]:
from __future__ import division
from numpy.random import randn
import numpy as np
np.set_printoptions(precision = 4, suppress = True)
The NumPy ndarray: a multidimentional array object¶
In [4]:
data = randn(2, 3)
data
Out[4]:
array([[-1.1288, 0.1897, 0.7566],
[-0.4347, -0.5984, -0.6985]])
In [5]:
data * 10
Out[5]:
array([[-11.2878, 1.8974, 7.5657],
[ -4.3468, -5.9838, -6.9849]])
In [6]:
data + data
Out[6]:
array([[-2.2576, 0.3795, 1.5131],
[-0.8694, -1.1968, -1.397 ]])
In [7]:
data.shape
Out[7]:
(2, 3)
In [8]:
data.dtype
Out[8]:
dtype('float64')
Creating ndarrays¶
In [9]:
data1 = [6, 7.5, 8, 0, 1]
arr1 = np.array(data1)
arr1
Out[9]:
array([ 6. , 7.5, 8. , 0. , 1. ])
In [10]:
data2 = [[1, 2, 3, 4], [5, 6, 7, 8]]
arr2 = np.array(data2)
arr2
Out[10]:
array([[1, 2, 3, 4],
[5, 6, 7, 8]])
In [11]:
arr2.ndim
Out[11]:
2
In [12]:
arr2.shape
Out[12]:
(2, 4)
In [13]:
arr1.dtype
Out[13]:
dtype('float64')
In [14]:
arr2.dtype
Out[14]:
dtype('int64')
In [15]:
np.zeros(10)
Out[15]:
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
In [16]:
np.zeros((3, 6))
Out[16]:
array([[ 0., 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0., 0.]])
In [17]:
np.ones(10)
Out[17]:
array([ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])
In [18]:
np.ones((2, 3))
Out[18]:
array([[ 1., 1., 1.],
[ 1., 1., 1.]])
In [19]:
np.empty((2, 3, 2))
Out[19]:
array([[[ 3.1050e+231, 1.7306e-077],
[ 2.9644e-323, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000]],
[[ 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 8.3440e-309]]])
In [20]:
np.arange(15)
Out[20]:
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14])
Data type for ndarrays¶
In [21]:
arr3 = np.array([1, 2, 3], dtype = np.float64)
In [22]:
arr3.dtype
Out[22]:
dtype('float64')
In [23]:
arr4 = np.array([1, 2, 3], dtype = np.int64)
In [24]:
arr4.dtype
Out[24]:
dtype('int64')
In [25]:
arr5 = np.array([1, 2, 3, 4, 5])
arr5.dtype
Out[25]:
dtype('int64')
In [26]:
arr_float = arr5.astype(np.float64)
arr_float.dtype
Out[26]:
dtype('float64')
In [27]:
arr6 = np.array([3.7, -1.2, -2.6, 0.5, 12.9, 10.1])
arr6.dtype
Out[27]:
dtype('float64')
In [28]:
arr6.astype(np.int32)
Out[28]:
array([ 3, -1, -2, 0, 12, 10], dtype=int32)
In [29]:
numeric_strings = np.array(['1.25', '-9.6', '42'], dtype = np.string_)
numeric_strings.astype(float)
Out[29]:
array([ 1.25, -9.6 , 42. ])
In [30]:
int_array = np.arange(10)
calibers = np.array([0.22, 0.270, .357, .380, .44, .50], dtype = np.float64)
int_array.astype(calibers.dtype)
Out[30]:
array([ 0., 1., 2., 3., 4., 5., 6., 7., 8., 9.])
In [31]:
empty_unit32 = np.empty(8, dtype = 'u4')
empty_unit32
Out[31]:
array([ 0, 1879048192, 3537412978, 805308408, 6,
0, 0, 393216], dtype=uint32)
Operations between arrays and scalas¶
In [32]:
arr7 = np.array([[1, 2, 3], [4, 5, 6]])
arr7
Out[32]:
array([[1, 2, 3],
[4, 5, 6]])
In [33]:
arr7 + arr7
Out[33]:
array([[ 2, 4, 6],
[ 8, 10, 12]])
In [34]:
arr7 * arr7
Out[34]:
array([[ 1, 4, 9],
[16, 25, 36]])
In [35]:
1/arr7
Out[35]:
array([[ 1. , 0.5 , 0.3333],
[ 0.25 , 0.2 , 0.1667]])
In [36]:
arr7**0.5
Out[36]:
array([[ 1. , 1.4142, 1.7321],
[ 2. , 2.2361, 2.4495]])
Basic indexing and slicing¶
In [37]:
arr8 = np.arange(10)
arr8
Out[37]:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [38]:
arr8[5]
Out[38]:
5
In [39]:
arr8[5:8]
Out[39]:
array([5, 6, 7])
In [40]:
arr8[5:8] = 12
arr8
Out[40]:
array([ 0, 1, 2, 3, 4, 12, 12, 12, 8, 9])
In [41]:
array_slice = arr8[5:8]
array_slice
Out[41]:
array([12, 12, 12])
In [42]:
array_slice[1] = 12345
array_slice
Out[42]:
array([ 12, 12345, 12])
In [43]:
array_slice[:] = 64
array_slice
Out[43]:
array([64, 64, 64])
In [44]:
arr2d = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
arr2d[2]
Out[44]:
array([7, 8, 9])
In [45]:
arr2d[0][2]
Out[45]:
3
In [46]:
arr2d[0, 2]
Out[46]:
3
In [47]:
arr3d = np.array([[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]])
arr3d
Out[47]:
array([[[ 1, 2, 3],
[ 4, 5, 6]],
[[ 7, 8, 9],
[10, 11, 12]]])
In [48]:
arr3d[0]
Out[48]:
array([[1, 2, 3],
[4, 5, 6]])
In [49]:
arr3d[0, 1]
Out[49]:
array([4, 5, 6])
In [50]:
arr3d[0][1]
Out[50]:
array([4, 5, 6])
In [51]:
arr3d[0][1][2]
Out[51]:
6
In [52]:
old_values = arr3d[0].copy()
In [53]:
arr3d[0] = 42
arr3d
Out[53]:
array([[[42, 42, 42],
[42, 42, 42]],
[[ 7, 8, 9],
[10, 11, 12]]])
In [54]:
arr3d[0] = old_values
arr3d
Out[54]:
array([[[ 1, 2, 3],
[ 4, 5, 6]],
[[ 7, 8, 9],
[10, 11, 12]]])
In [55]:
arr3d[1, 0]
Out[55]:
array([7, 8, 9])
Indexing with slicing¶
In [56]:
arr8
Out[56]:
array([ 0, 1, 2, 3, 4, 64, 64, 64, 8, 9])
In [57]:
arr8[1:6]
Out[57]:
array([ 1, 2, 3, 4, 64])
In [58]:
arr2d
Out[58]:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
In [59]:
arr2d[:2]
Out[59]:
array([[1, 2, 3],
[4, 5, 6]])
In [60]:
arr2d[:2, 1:]
Out[60]:
array([[2, 3],
[5, 6]])
In [61]:
arr2d[1, :2]
Out[61]:
array([4, 5])
In [62]:
arr2d[2, :1]
Out[62]:
array([7])
In [63]:
arr2d[:, :1]
Out[63]:
array([[1],
[4],
[7]])
In [64]:
arr2d[:2, 1:] = 0
arr2d
Out[64]:
array([[1, 0, 0],
[4, 0, 0],
[7, 8, 9]])
Boolean indexing¶
In [65]:
names = np.array(['Bob', 'Joe', 'Will', 'Bob', 'Will', 'Joe', 'Joe'])
names
Out[65]:
array(['Bob', 'Joe', 'Will', 'Bob', 'Will', 'Joe', 'Joe'],
dtype='<U4')
In [66]:
data = randn(7, 4)
data
Out[66]:
array([[ 1.5667, 1.332 , -1.1835, -0.45 ],
[-0.1619, 1.067 , -0.691 , -0.6781],
[ 1.9966, 1.0252, 1.5692, -0.6976],
[ 1.931 , 0.801 , -0.4198, 0.2899],
[ 0.0727, -0.0716, -0.3509, 1.3726],
[-1.3476, -2.0312, 0.4368, -0.1413],
[-0.3111, -0.5803, -0.8319, -0.2031]])
In [67]:
names == 'Bob'
Out[67]:
array([ True, False, False, True, False, False, False], dtype=bool)
In [68]:
data[names == 'Bob']
Out[68]:
array([[ 1.5667, 1.332 , -1.1835, -0.45 ],
[ 1.931 , 0.801 , -0.4198, 0.2899]])
In [69]:
data[names == 'Bob', 3:]
Out[69]:
array([[-0.45 ],
[ 0.2899]])
In [70]:
data[names == 'Bob', 2]
Out[70]:
array([-1.1835, -0.4198])
In [71]:
names != 'Bob'
Out[71]:
array([False, True, True, False, True, True, True], dtype=bool)
In [72]:
data[names != 'Bob']
Out[72]:
array([[-0.1619, 1.067 , -0.691 , -0.6781],
[ 1.9966, 1.0252, 1.5692, -0.6976],
[ 0.0727, -0.0716, -0.3509, 1.3726],
[-1.3476, -2.0312, 0.4368, -0.1413],
[-0.3111, -0.5803, -0.8319, -0.2031]])
In [74]:
data[~(names == 'Bob')]
Out[74]:
array([[-0.1619, 1.067 , -0.691 , -0.6781],
[ 1.9966, 1.0252, 1.5692, -0.6976],
[ 0.0727, -0.0716, -0.3509, 1.3726],
[-1.3476, -2.0312, 0.4368, -0.1413],
[-0.3111, -0.5803, -0.8319, -0.2031]])
In [75]:
mask = (names == 'Bob') | (names == 'Will')
mask
Out[75]:
array([ True, False, True, True, True, False, False], dtype=bool)
In [76]:
data[mask]
Out[76]:
array([[ 1.5667, 1.332 , -1.1835, -0.45 ],
[ 1.9966, 1.0252, 1.5692, -0.6976],
[ 1.931 , 0.801 , -0.4198, 0.2899],
[ 0.0727, -0.0716, -0.3509, 1.3726]])
In [77]:
data[data < 0] = 0
data
Out[77]:
array([[ 1.5667, 1.332 , 0. , 0. ],
[ 0. , 1.067 , 0. , 0. ],
[ 1.9966, 1.0252, 1.5692, 0. ],
[ 1.931 , 0.801 , 0. , 0.2899],
[ 0.0727, 0. , 0. , 1.3726],
[ 0. , 0. , 0.4368, 0. ],
[ 0. , 0. , 0. , 0. ]])
In [78]:
data[names != 'Joe'] = 7
data
Out[78]:
array([[ 7. , 7. , 7. , 7. ],
[ 0. , 1.067 , 0. , 0. ],
[ 7. , 7. , 7. , 7. ],
[ 7. , 7. , 7. , 7. ],
[ 7. , 7. , 7. , 7. ],
[ 0. , 0. , 0.4368, 0. ],
[ 0. , 0. , 0. , 0. ]])
Fancing Indexing¶
In [79]:
arr9 = np.empty((8, 4))
arr9
Out[79]:
array([[ 3.1050e+231, 3.1050e+231, 2.2318e-314, 2.4703e-323],
[ 0.0000e+000, 0.0000e+000, 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000, 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000, 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000, 0.0000e+000, 0.0000e+000],
[ 0.0000e+000, 0.0000e+000, -3.9843e-048, 0.0000e+000],
[ 0.0000e+000, -7.5927e-076, 0.0000e+000, 1.9469e-308],
[ 0.0000e+000, 3.9525e-323, 2.2318e-314, 2.4703e-323]])
In [80]:
for i in range(8):
arr9[i] = i
arr9
Out[80]:
array([[ 0., 0., 0., 0.],
[ 1., 1., 1., 1.],
[ 2., 2., 2., 2.],
[ 3., 3., 3., 3.],
[ 4., 4., 4., 4.],
[ 5., 5., 5., 5.],
[ 6., 6., 6., 6.],
[ 7., 7., 7., 7.]])
In [81]:
arr9[[4, 3, 0, 6]]
Out[81]:
array([[ 4., 4., 4., 4.],
[ 3., 3., 3., 3.],
[ 0., 0., 0., 0.],
[ 6., 6., 6., 6.]])
In [82]:
arr9[[-3, -5, -7]]
Out[82]:
array([[ 5., 5., 5., 5.],
[ 3., 3., 3., 3.],
[ 1., 1., 1., 1.]])
In [83]:
arr10 = np.arange(32).reshape((8, 4))
arr10
Out[83]:
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, 30, 31]])
In [84]:
arr10[[1, 5, 7, 2], [0, 3, 1, 2]]
Out[84]:
array([ 4, 23, 29, 10])
In [85]:
arr10[[1, 5, 7, 3]][:, [0, 3, 1, 2]]
Out[85]:
array([[ 4, 7, 5, 6],
[20, 23, 21, 22],
[28, 31, 29, 30],
[12, 15, 13, 14]])
In [86]:
arr10[np.ix_([1, 5, 7, 2], [0, 3, 1, 2])]
Out[86]:
array([[ 4, 7, 5, 6],
[20, 23, 21, 22],
[28, 31, 29, 30],
[ 8, 11, 9, 10]])
Transposing arrays and swaping axes¶
In [87]:
arr11 = np.arange(15).reshape((3, 5))
arr11
Out[87]:
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
In [88]:
arr11.T
Out[88]:
array([[ 0, 5, 10],
[ 1, 6, 11],
[ 2, 7, 12],
[ 3, 8, 13],
[ 4, 9, 14]])
In [89]:
arr12 = np.random.randn(6, 3)
arr12
Out[89]:
array([[-1.7875, -0.1715, 0.0374],
[ 1.9915, -0.8422, 0.9459],
[ 0.0683, -1.3781, 0.7558],
[-1.8136, 0.3604, -0.7077],
[ 1.632 , -0.3238, -0.3254],
[ 3.1159, 1.0252, -1.7004]])
In [90]:
np.dot(arr12, arr12.T)
Out[90]:
array([[ 3.226 , -3.3801, 0.1426, 3.1535, -2.8738, -5.8091],
[ -3.3801, 5.5703, 2.0115, -4.5847, 3.2152, 3.7336],
[ 0.1426, 2.0115, 2.475 , -1.1553, 0.3117, -2.4853],
[ 3.1535, -4.5847, -1.1553, 3.9197, -2.8462, -4.0781],
[ -2.8738, 3.2152, 0.3117, -2.8462, 2.8741, 5.3063],
[ -5.8091, 3.7336, -2.4853, -4.0781, 5.3063, 13.6511]])
In [91]:
np.dot(arr12.T, arr12)
Out[91]:
array([[ 22.8272, 0.5475, -2.6773],
[ 0.5475, 3.9235, -3.7375],
[ -2.6773, -3.7375, 4.9654]])
In [92]:
arr13 = np.arange(16).reshape((2, 2, 4))
arr13
Out[92]:
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7]],
[[ 8, 9, 10, 11],
[12, 13, 14, 15]]])
In [93]:
arr13.transpose((1, 0, 2))
Out[93]:
array([[[ 0, 1, 2, 3],
[ 8, 9, 10, 11]],
[[ 4, 5, 6, 7],
[12, 13, 14, 15]]])
In [94]:
arr13.swapaxes(1, 2)
Out[94]:
array([[[ 0, 4],
[ 1, 5],
[ 2, 6],
[ 3, 7]],
[[ 8, 12],
[ 9, 13],
[10, 14],
[11, 15]]])
Universal Functions: Fast element-wise array functions¶
In [95]:
arr14 = np.arange(10)
arr14
Out[95]:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [96]:
np.sqrt(arr14)
Out[96]:
array([ 0. , 1. , 1.4142, 1.7321, 2. , 2.2361, 2.4495,
2.6458, 2.8284, 3. ])
In [97]:
np.exp(arr14)
Out[97]:
array([ 1. , 2.7183, 7.3891, 20.0855, 54.5982,
148.4132, 403.4288, 1096.6332, 2980.958 , 8103.0839])
In [98]:
x = randn(8)
x
Out[98]:
array([-1.1566, 0.1062, 0.1054, 0.0743, 0.2924, -0.4461, -1.7097,
-0.3379])
In [99]:
y = randn(8)
y
Out[99]:
array([ 0.3624, 0.257 , 0.2404, 0.5182, 1.3033, 1.1561, 0.0866,
0.6654])
In [100]:
np.maximum(x, y)
Out[100]:
array([ 0.3624, 0.257 , 0.2404, 0.5182, 1.3033, 1.1561, 0.0866,
0.6654])
In [101]:
arr15 = randn(7) * 5
arr15
Out[101]:
array([-0.0813, 1.8273, -1.6541, 0.5038, -0.6916, 4.2674, -2.447 ])
In [102]:
np.modf(arr15)
Out[102]:
(array([-0.0813, 0.8273, -0.6541, 0.5038, -0.6916, 0.2674, -0.447 ]), array([-0., 1., -1., 0., -0., 4., -2.]))
Data processing using arrays¶
In [103]:
points = np.arange(-5, 5, 0.01)
In [104]:
xs, ys = np.meshgrid(points, points)
xs
Out[104]:
array([[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99],
[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99],
[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99],
...,
[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99],
[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99],
[-5. , -4.99, -4.98, ..., 4.97, 4.98, 4.99]])
In [108]:
from matplotlib.pyplot import imshow, title
import matplotlib.pyplot as plt
In [109]:
z = np.sqrt(xs ** 2 + ys ** 2)
z
Out[109]:
array([[ 7.0711, 7.064 , 7.0569, ..., 7.0499, 7.0569, 7.064 ],
[ 7.064 , 7.0569, 7.0499, ..., 7.0428, 7.0499, 7.0569],
[ 7.0569, 7.0499, 7.0428, ..., 7.0357, 7.0428, 7.0499],
...,
[ 7.0499, 7.0428, 7.0357, ..., 7.0286, 7.0357, 7.0428],
[ 7.0569, 7.0499, 7.0428, ..., 7.0357, 7.0428, 7.0499],
[ 7.064 , 7.0569, 7.0499, ..., 7.0428, 7.0499, 7.0569]])
In [113]:
plt.imshow(z, cmap = plt.cm.gray)
plt.colorbar()
plt.title('Image plot of $\sqrt{x^2 + y^2}$ for a grid of values')
Out[113]:
<matplotlib.text.Text at 0x118a56630>
Expressing conditional logic as array operations¶
In [114]:
xarr = np.array([1.1, 1.2, 1.3, 1.4, 1.5])
yarr = np.array([2.1, 2.2, 2.3, 2.4, 2.5])
cond = np.array([True, False, True, True, False])
In [115]:
result = [(x if c else y)
for x, y, c in zip(xarr, yarr, cond)]
result
Out[115]:
[1.1000000000000001, 2.2000000000000002, 1.3, 1.3999999999999999, 2.5]
In [116]:
result = np.where(cond, xarr, yarr)
result
Out[116]:
array([ 1.1, 2.2, 1.3, 1.4, 2.5])
In [117]:
arr16 = randn(4, 4)
arr16
Out[117]:
array([[ 0.6851, -0.6378, -0.8524, 1.3891],
[ 0.1806, 1.2007, 0.5101, -0.2702],
[-0.717 , 0.783 , 0.1876, 0.8501],
[ 0.0737, -0.3917, 0.6096, 1.2823]])
In [118]:
np.where(arr16 > 0, 2, -2)
Out[118]:
array([[ 2, -2, -2, 2],
[ 2, 2, 2, -2],
[-2, 2, 2, 2],
[ 2, -2, 2, 2]])
In [119]:
np.where(arr16 > 0, 2, arr16)
Out[119]:
array([[ 2. , -0.6378, -0.8524, 2. ],
[ 2. , 2. , 2. , -0.2702],
[-0.717 , 2. , 2. , 2. ],
[ 2. , -0.3917, 2. , 2. ]])
Mathematical and statistical methods¶
In [120]:
arr17 = np.random.randn(4, 5)
arr17
Out[120]:
array([[ 1.6533, 0.1569, 1.1953, 0.5076, -0.6969],
[ 0.3001, 1.3571, 0.7924, 0.5225, 0.777 ],
[ 0.326 , -0.4121, -0.4198, -0.44 , 1.7303],
[ 1.4647, 0.1069, 0.0008, -0.1721, 0.0581]])
In [121]:
arr17.mean()
Out[121]:
0.44040416796585308
In [122]:
np.mean(arr17)
Out[122]:
0.44040416796585308
In [123]:
arr17.sum()
Out[123]:
8.8080833593170613
In [124]:
arr17.mean(axis = 1) # row
Out[124]:
array([ 0.5632, 0.7498, 0.1569, 0.2917])
In [125]:
arr17.mean(axis = 0) # column
Out[125]:
array([ 0.9361, 0.3022, 0.3922, 0.1045, 0.4671])
In [126]:
arr18 = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
arr18
Out[126]:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
In [127]:
arr18.cumsum(0) # sum of column
Out[127]:
array([[ 1, 2, 3],
[ 5, 7, 9],
[12, 15, 18]])
In [128]:
arr18.cumprod(1) # prod of row
Out[128]:
array([[ 1, 2, 6],
[ 4, 20, 120],
[ 7, 56, 504]])
Methods for boolean arrays¶
In [129]:
arr19 = randn(100)
In [130]:
(arr19 > 0).sum()
Out[130]:
50
In [131]:
bools = np.array([False, False, True, False])
bools.any()
Out[131]:
True
In [132]:
bools.all()
Out[132]:
False
sorting¶
In [133]:
arr20 = randn(8)
arr20
Out[133]:
array([-0.5483, 1.4955, 0.5092, -0.8928, -1.7595, 1.3327, 0.4815,
-1.1086])
In [134]:
arr20.sort()
arr20
Out[134]:
array([-1.7595, -1.1086, -0.8928, -0.5483, 0.4815, 0.5092, 1.3327,
1.4955])
In [135]:
arr21 = randn(5, 3)
arr21
Out[135]:
array([[ 0.1766, -2.2486, 1.2854],
[ 0.3819, 0.1207, 0.5651],
[ 0.9679, 0.9795, 0.8815],
[ 0.2147, -1.4361, 0.7716],
[-0.2916, 0.5463, -0.827 ]])
In [136]:
arr21.sort(1)
arr21
Out[136]:
array([[-2.2486, 0.1766, 1.2854],
[ 0.1207, 0.3819, 0.5651],
[ 0.8815, 0.9679, 0.9795],
[-1.4361, 0.2147, 0.7716],
[-0.827 , -0.2916, 0.5463]])
In [137]:
large_array = randn(1000)
large_array.sort()
large_array[int(0.05 * len(large_array))]
Out[137]:
-1.6186837940856837
Unique and Other Set Logic¶
In [138]:
names = np.array(['Bob', 'Joe', 'Will', 'Bob', 'Will', 'Joe', 'Joe'])
np.unique(names)
Out[138]:
array(['Bob', 'Joe', 'Will'],
dtype='<U4')
In [139]:
ints = np.array([3, 2, 3, 2, 2, 1, 1, 4, 4])
np.unique(ints)
Out[139]:
array([1, 2, 3, 4])
In [140]:
sorted(set(names))
Out[140]:
['Bob', 'Joe', 'Will']
In [141]:
sorted(set(ints))
Out[141]:
[1, 2, 3, 4]
In [142]:
values = np.array([6, 0, 0, 3, 2, 5, 6])
np.in1d(values, [2, 3, 6])
Out[142]:
array([ True, False, False, True, True, False, True], dtype=bool)
File input and output with arrays¶
Storing arrays on disk in binary format¶
In [143]:
arr22 = np.arange(10)
arr22
Out[143]:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [144]:
np.save('some_array', arr22)
In [145]:
np.load('some_array.npy')
Out[145]:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [146]:
np.savez('array_achieve.npz', a = arr22, b = arr22)
In [147]:
arch = np.load('array_achieve.npz')
arch['b']
Out[147]:
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
In [148]:
!rm some_array.npy
!rm array_achieve.npz
Saving and loading text files¶
Linear algebra¶
In [150]:
x = np.array([[1., 2., 3.], [4., 5., 6.]])
x
Out[150]:
array([[ 1., 2., 3.],
[ 4., 5., 6.]])
In [151]:
y = np.array([[6., 23.], [-1, 7], [8, 9]])
y
Out[151]:
array([[ 6., 23.],
[ -1., 7.],
[ 8., 9.]])
In [152]:
x.dot(y)
Out[152]:
array([[ 28., 64.],
[ 67., 181.]])
In [153]:
np.dot(x, np.ones(3))
Out[153]:
array([ 6., 15.])
In [154]:
np.random.seed(12345)
In [155]:
from numpy.linalg import inv, qr
In [156]:
X = randn(5, 5)
X
Out[156]:
array([[-0.2047, 0.4789, -0.5194, -0.5557, 1.9658],
[ 1.3934, 0.0929, 0.2817, 0.769 , 1.2464],
[ 1.0072, -1.2962, 0.275 , 0.2289, 1.3529],
[ 0.8864, -2.0016, -0.3718, 1.669 , -0.4386],
[-0.5397, 0.477 , 3.2489, -1.0212, -0.5771]])
In [157]:
mat = X.T.dot(X)
mat
Out[157]:
array([[ 4.075 , -3.3059, -1.3073, 3.4466, 2.6197],
[ -3.3059, 6.1523, 1.7149, -4.3193, -0.0938],
[ -1.3073, 1.7149, 11.1187, -3.3702, -2.0097],
[ 3.4466, -4.3193, -3.3702, 4.7812, 0.0331],
[ 2.6197, -0.0938, -2.0097, 0.0331, 7.7736]])
In [158]:
inv(mat)
Out[158]:
array([[ 3.0361, -0.1808, -0.6878, -2.8285, -1.1911],
[-0.1808, 0.5035, 0.1215, 0.6702, 0.0956],
[-0.6878, 0.1215, 0.2904, 0.8081, 0.3049],
[-2.8285, 0.6702, 0.8081, 3.4152, 1.1557],
[-1.1911, 0.0956, 0.3049, 1.1557, 0.6051]])
In [159]:
mat.dot(inv(mat))
Out[159]:
array([[ 1., -0., -0., 0., -0.],
[ 0., 1., 0., -0., 0.],
[-0., -0., 1., 0., -0.],
[ 0., -0., -0., 1., 0.],
[-0., -0., 0., 0., 1.]])
In [160]:
q, r = qr(mat)
q
Out[160]:
array([[-0.5883, -0.2619, -0.2013, -0.3233, -0.6635],
[ 0.4772, -0.6609, 0.1734, -0.5501, 0.0532],
[ 0.1887, -0.0806, -0.9635, -0.0266, 0.1698],
[-0.4975, 0.1618, 0.0179, -0.5581, 0.6438],
[-0.3782, -0.6797, 0.0276, 0.5298, 0.3371]])
In [161]:
r
Out[161]:
array([[ -6.9271, 7.389 , 6.1227, -7.1163, -4.9215],
[ 0. , -3.9735, -0.8671, 2.9747, -5.7402],
[ 0. , 0. , -10.2681, 1.8909, 1.6079],
[ 0. , 0. , 0. , -1.2996, 3.3577],
[ 0. , 0. , 0. , 0. , 0.5571]])
Random number generation¶
In [162]:
samples = np.random.normal(size = (4, 4))
samples
Out[162]:
array([[ 0.1241, 0.3026, 0.5238, 0.0009],
[ 1.3438, -0.7135, -0.8312, -2.3702],
[-1.8608, -0.8608, 0.5601, -1.2659],
[ 0.1198, -1.0635, 0.3329, -2.3594]])
In [163]:
from random import normalvariate
In [164]:
N = 1000000
In [166]:
%timeit samples = [normalvariate(0, 1) for _ in range(N)]
1 loop, best of 3: 876 ms per loop
In [167]:
%timeit np.random.normal(size = N)
10 loops, best of 3: 34.3 ms per loop
Example: Random Walks¶
In [168]:
import random
position = 0
walk = [position]
steps = 1000
for i in range(steps):
step = 1 if random.randint(0, 1) else -1
position += step
walk.append(position)
In [169]:
np.random.seed(123445)
In [170]:
nsteps = 1000
draws = np.random.randint(0, 2, size = nsteps)
steps = np.where(draws > 0, 1, -1)
walk = steps.cumsum()
In [171]:
walk.min()
Out[171]:
-25
In [172]:
walk.max()
Out[172]:
13
In [173]:
(np.abs(walk) >= 10).argmax()
Out[173]:
27
Simulating many random walks at once¶
In [174]:
nwalks = 5000
nsteps = 1000
draws = np.random.randint(0, 2, size = (nwalks, nsteps))
steps = np.where(draws > 0, 1, -1)
walks = steps.cumsum(1)
walks
Out[174]:
array([[ -1, -2, -1, ..., 38, 39, 38],
[ -1, -2, -1, ..., 50, 51, 50],
[ -1, 0, -1, ..., -8, -7, -8],
...,
[ -1, -2, -3, ..., -100, -99, -98],
[ -1, 0, -1, ..., -16, -17, -16],
[ 1, 0, -1, ..., -10, -11, -12]])
In [175]:
walks.max()
Out[175]:
119
In [176]:
walks.min()
Out[176]:
-118
In [177]:
hits30 = (np.abs(walks) >= 30).any(1)
hits30
Out[177]:
array([ True, True, False, ..., True, False, False], dtype=bool)
In [178]:
hits30.sum()
Out[178]:
3407
In [179]:
crossing_times = (np.abs(walks[hits30]) >= 30).argmax(1)
crossing_times.mean()
Out[179]:
503.94511300264162
In [180]:
steps = np.random.normal(loc=0, scale=0.25,
size=(nwalks, nsteps))
steps
Out[180]:
array([[ 0.0576, 0.4903, -0.0204, ..., 0.036 , 0.016 , 0.5973],
[-0.2707, -0.0213, 0.0457, ..., 0.0279, -0.3356, 0.2751],
[ 0.4423, -0.17 , 0.3218, ..., -0.1858, -0.2958, 0.0574],
...,
[-0.4958, 0.2782, 0.0975, ..., 0.2677, 0.1195, 0.1042],
[-0.1086, 0.2598, -0.0773, ..., 0.0493, 0.217 , -0.2588],
[ 0.0867, 0.1986, 0.2159, ..., 0.2175, -0.1233, 0.028 ]])