Numpy概述
NumPy（Numerical Python的簡稱）是Python數(shù)值計算最重要的基礎(chǔ)包氨鹏。大多數(shù)提供科學(xué)計算的包都是用NumPy的數(shù)組作為構(gòu)建基礎(chǔ)挨下。
Why NumPy?
- 一個強大的N維數(shù)組對象ndarray窗宇，具有矢量算術(shù)運算和復(fù)雜廣播能力的快速且節(jié)省空間的多維數(shù)組
- 用于集成由C悲柱、C++叹阔、Fortran等語言類庫的C語言 API
- 線性代數(shù)截酷、隨機數(shù)生成以及傅里葉變換功能蔓同。
- 用于對整組數(shù)據(jù)進行快速運算的標(biāo)準(zhǔn)數(shù)學(xué)函數(shù)（無需編寫循環(huán)）,支持大量的數(shù)據(jù)運算
- 是眾多機器學(xué)習(xí)框架的基礎(chǔ)庫
Tips：Python的面向數(shù)組計算可以追溯到1995年饶辙，Jim Hugunin創(chuàng)建了Numeric庫蹲诀。接下來的10年斑粱，許多科學(xué)編程社區(qū)紛紛開始使用Python的數(shù)組編程，但是進入21世紀(jì)脯爪，庫的生態(tài)系統(tǒng)變得碎片化了则北。2005年，Travis Oliphant從Numeric和Numarray項目整了出了NumPy項目痕慢，進而所有社區(qū)都集合到了這個框架下尚揣。
NumPy之于數(shù)值計算特別重要的原因之一，是因為它可以高效處理大數(shù)組的數(shù)據(jù)掖举。這是因為：
- NumPy是在一個連續(xù)的內(nèi)存塊中存儲數(shù)據(jù)快骗，獨立于其他Python內(nèi)置對象。NumPy的C語言編寫的算法庫可以操作內(nèi)存塔次，而不必進行類型檢查或其它前期工作方篮。比起Python的內(nèi)置序列，NumPy數(shù)組使用的內(nèi)存更少励负。
- NumPy可以在整個數(shù)組上執(zhí)行復(fù)雜的計算藕溅，而不需要Python的for循環(huán)。
numpy.array 基礎(chǔ)
import numpy
numpy.__version__
'1.12.1'
import numpy as np
np.__version__
'1.12.1'
Python List的特點
L = [i for i in range(10)]
L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
L[5]
5
L[5] = 100
L
[0, 1, 2, 3, 4, 100, 6, 7, 8, 9]
L[5] = "Machine Learning"
L
[0, 1, 2, 3, 4, 'Machine Learning', 6, 7, 8, 9]
Python的List不要求存儲同樣的類型继榆，帶來效率問題巾表。
import array
arr = array.array('i', [i for i in range(10)])
arr
array('i', [0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
arr[5]
5
arr[5] = 100
arr
array('i', [0, 1, 2, 3, 4, 100, 6, 7, 8, 9])
arr[5] = "Machine Learning"
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-14-e74bffddd7b6> in <module>()
----> 1 arr[5] = "Machine Learning"
TypeError: an integer is required (got type str)
arr[5] = 5.0
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-15-f30bba6fbd5a> in <module>()
----> 1 arr[5] = 5.0
TypeError: integer argument expected, got float
array的缺點是沒有將數(shù)據(jù)當(dāng)做向量或者矩陣，不支持基本運算略吨。
numpy.array
nparr = np.array([i for i in range(10)])
nparr
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
nparr[5] = 100
nparr
array([ 0, 1, 2, 3, 4, 100, 6, 7, 8, 9])
nparr[5] = "Machine Learning"
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-21-df6cd071861b> in <module>()
----> 1 nparr[5] = "Machine Learning"
ValueError: invalid literal for int() with base 10: 'Machine Learning'
nparr.dtype
dtype('int64')
nparr[5] = 5.0
nparr
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
nparr.dtype
dtype('int64')
nparr[5] = 3.14
nparr
array([0, 1, 2, 3, 4, 3, 6, 7, 8, 9])
nparr2 = np.array([1, 2, 3.0])
nparr2.dtype
dtype('float64')
創(chuàng)建 numpy.array
import numpy as np
numpy.array
nparr = np.array([i for i in range(10)])
nparr
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
其他創(chuàng)建 numpy.array 的方法
zeros
np.zeros(10)
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
np.zeros(10, dtype=float)
array([ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
np.zeros((3, 5))
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.]])
np.zeros(shape=(3, 5), dtype=int)
array([[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]])
ones
np.ones(10)
array([ 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])
np.ones((3, 5))
array([[ 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1.],
[ 1., 1., 1., 1., 1.]])
full
np.full((3, 5), 666)
array([[666, 666, 666, 666, 666],
[666, 666, 666, 666, 666],
[666, 666, 666, 666, 666]])
np.full(fill_value=666, shape=(3, 5))
array([[666, 666, 666, 666, 666],
[666, 666, 666, 666, 666],
[666, 666, 666, 666, 666]])
arange
[i for i in range(0, 20, 2)]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
np.arange(0, 20, 2)
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
[i for i in range(0, 1, 0.2)]
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-43-d0579096bf02> in <module>()
----> 1 [i for i in range(0, 1, 0.2)]
TypeError: 'float' object cannot be interpreted as an integer
np.arange(0, 1, 0.2)
array([ 0. , 0.2, 0.4, 0.6, 0.8])
[i for i in range(0, 10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
np.arange(0, 10)
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
[i for i in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
np.arange(10)
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
linspace
np.linspace(0, 20, 10)
array([ 0. , 2.22222222, 4.44444444, 6.66666667,
8.88888889, 11.11111111, 13.33333333, 15.55555556,
17.77777778, 20. ])
np.linspace(0, 20, 11)
array([ 0., 2., 4., 6., 8., 10., 12., 14., 16., 18., 20.])
np.linspace(0, 1, 5)
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
random
randint
np.random.randint(0, 10) # [0, 10)之間的隨機數(shù)
5
np.random.randint(0, 10, 10)
array([2, 6, 1, 8, 1, 6, 8, 0, 1, 4])
np.random.randint(0, 1, 10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
np.random.randint(0, 10, size=10)
array([3, 4, 9, 9, 5, 2, 3, 3, 2, 1])
np.random.randint(0, 10, size=(3,5))
array([[1, 5, 3, 8, 5],
[2, 7, 9, 6, 0],
[0, 9, 9, 9, 7]])
np.random.randint(10, size=(3,5))
array([[4, 8, 3, 7, 2],
[9, 9, 2, 4, 4],
[1, 5, 1, 7, 7]])
seed
np.random.seed(666)
np.random.randint(0, 10, size=(3, 5))
array([[2, 6, 9, 4, 3],
[1, 0, 8, 7, 5],
[2, 5, 5, 4, 8]])
np.random.seed(666)
np.random.randint(0, 10, size=(3,5))
array([[2, 6, 9, 4, 3],
[1, 0, 8, 7, 5],
[2, 5, 5, 4, 8]])
random
np.random.random()
0.7315955468480113
np.random.random((3,5))
array([[ 0.8578588 , 0.76741234, 0.95323137, 0.29097383, 0.84778197],
[ 0.3497619 , 0.92389692, 0.29489453, 0.52438061, 0.94253896],
[ 0.07473949, 0.27646251, 0.4675855 , 0.31581532, 0.39016259]])
normal
np.random.normal()
0.9047266176428719
np.random.normal(10, 100)
-72.62832650185376
np.random.normal(0, 1, (3, 5))
array([[ 0.82101369, 0.36712592, 1.65399586, 0.13946473, -1.21715355],
[-0.99494737, -1.56448586, -1.62879004, 1.23174866, -0.91360034],
[-0.27084407, 1.42024914, -0.98226439, 0.80976498, 1.85205227]])
np.random.<TAB> 查看random中的更多方法
np.random?
np.random.normal?
help(np.random.normal)
Help on built-in function normal:
normal(...) method of mtrand.RandomState instance
normal(loc=0.0, scale=1.0, size=None)
Draw random samples from a normal (Gaussian) distribution.
The probability density function of the normal distribution, first
derived by De Moivre and 200 years later by both Gauss and Laplace
independently [2]_, is often called the bell curve because of
its characteristic shape (see the example below).
The normal distributions occurs often in nature. For example, it
describes the commonly occurring distribution of samples influenced
by a large number of tiny, random disturbances, each with its own
unique distribution [2]_.
Parameters
----------
loc : float or array_like of floats
Mean ("centre") of the distribution.
scale : float or array_like of floats
Standard deviation (spread or "width") of the distribution.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
See Also
--------
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gaussian distribution is
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
where :math:`\mu` is the mean and :math:`\sigma` the standard
deviation. The square of the standard deviation, :math:`\sigma^2`,
is called the variance.
The function has its peak at the mean, and its "spread" increases with
the standard deviation (the function reaches 0.607 times its maximum at
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
`numpy.random.normal` is more likely to return samples lying close to
the mean, rather than those far away.
References
----------
.. [1] Wikipedia, "Normal distribution",
http://en.wikipedia.org/wiki/Normal_distribution
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
Random Variables and Random Signal Principles", 4th ed., 2001,
pp. 51, 51, 125.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)
Verify the mean and the variance:
>>> abs(mu - np.mean(s)) < 0.01
True
>>> abs(sigma - np.std(s, ddof=1)) < 0.01
True
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
... linewidth=2, color='r')
>>> plt.show()
numpy.array 基本操作
import numpy as np
np.random.seed(0)
x = np.arange(10)
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
X = np.arange(15).reshape((3, 5))
X
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
numpy.array 的基本屬性
x.ndim
1
X.ndim
2
x.shape
(10,)
X.shape
(3, 5)
x.size
10
X.size
15
numpy.array 的數(shù)據(jù)訪問
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
x[0]
0
x[-1]
9
X
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
X[0][0] # 不建議集币！
0
X[0, 0]
0
X[0, -1]
4
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
x[0:5]
array([0, 1, 2, 3, 4])
x[:5]
array([0, 1, 2, 3, 4])
x[5:]
array([5, 6, 7, 8, 9])
x[4:7]
array([4, 5, 6])
x[::2]
array([0, 2, 4, 6, 8])
x[1::2]
array([1, 3, 5, 7, 9])
x[::-1]
array([9, 8, 7, 6, 5, 4, 3, 2, 1, 0])
X
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
X[:2, :3]
array([[0, 1, 2],
[5, 6, 7]])
X[:2][:3] # 結(jié)果不一樣，在numpy中使用","做多維索引
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
X[:2, ::2]
array([[0, 2, 4],
[5, 7, 9]])
X[::-1, ::-1]
array([[14, 13, 12, 11, 10],
[ 9, 8, 7, 6, 5],
[ 4, 3, 2, 1, 0]])
X[0, :]
array([0, 1, 2, 3, 4])
X[:, 0]
array([ 0, 5, 10])
Subarray of numpy.array
subX = X[:2, :3]
subX
array([[0, 1, 2],
[5, 6, 7]])
subX[0, 0] = 100
subX
array([[100, 1, 2],
[ 5, 6, 7]])
X
array([[100, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[ 10, 11, 12, 13, 14]])
X[0, 0] = 0
X
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
subX
array([[0, 1, 2],
[5, 6, 7]])
subX = X[:2, :3].copy()
subX[0, 0] = 100
subX
array([[100, 1, 2],
[ 5, 6, 7]])
X
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
Reshape
x.shape
(10,)
x.ndim
1
x.reshape(2, 5)
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
A = x.reshape(2, 5)
A
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
B = x.reshape(1, 10)
B
array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]])
B.ndim
2
B.shape
(1, 10)
x.reshape(-1, 10)
array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]])
x.reshape(10, -1)
array([[0],
[1],
[2],
[3],
[4],
[5],
[6],
[7],
[8],
[9]])
x.reshape(2, -1)
array([[0, 1, 2, 3, 4],
[5, 6, 7, 8, 9]])
x.reshape(3, -1)
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-53-12a588b09f7f> in <module>()
----> 1 x.reshape(3, -1)
ValueError: cannot reshape array of size 10 into shape (3,newaxis)
?
numpy.array 合并和分割
import numpy as np
numpy.array 的合并
x = np.array([1, 2, 3])
y = np.array([3, 2, 1])
np.concatenate([x, y])
array([1, 2, 3, 3, 2, 1])
z = np.array([666, 666, 666])
np.concatenate([x, y, z])
array([ 1, 2, 3, 3, 2, 1, 666, 666, 666])
A = np.array([[1, 2, 3],
[4, 5, 6]])
np.concatenate([A, A])
array([[1, 2, 3],
[4, 5, 6],
[1, 2, 3],
[4, 5, 6]])
np.concatenate([A, A], axis=1)
array([[1, 2, 3, 1, 2, 3],
[4, 5, 6, 4, 5, 6]])
np.concatenate([A, z])
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-7-148a822297cf> in <module>()
----> 1 np.concatenate([A, z])
ValueError: all the input arrays must have same number of dimensions
np.concatenate([A, z.reshape(1, -1)])
array([[ 1, 2, 3],
[ 4, 5, 6],
[666, 666, 666]])
np.vstack([A, z])
array([[ 1, 2, 3],
[ 4, 5, 6],
[666, 666, 666]])
B = np.full((2,2), 100)
np.hstack([A, B])
array([[ 1, 2, 3, 100, 100],
[ 4, 5, 6, 100, 100]])
np.hstack([A, z])
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-13-d5b9fc6fb0a8> in <module>()
----> 1 np.hstack([A, z])
/Users/yuanzhang/anaconda/lib/python3.6/site-packages/numpy/core/shape_base.py in hstack(tup)
286 return _nx.concatenate(arrs, 0)
287 else:
--> 288 return _nx.concatenate(arrs, 1)
289
290 def stack(arrays, axis=0):
ValueError: all the input arrays must have same number of dimensions
numpy.array 的分割
x = np.arange(10)
x
array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
x1, x2, x3 = np.split(x, [3, 7])
x1
array([0, 1, 2])
x2
array([3, 4, 5, 6])
x3
array([7, 8, 9])
x1, x2 = np.split(x, [5])
x1
array([0, 1, 2, 3, 4])
x2
array([5, 6, 7, 8, 9])
A = np.arange(16).reshape((4, 4))
A
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
A1, A2 = np.split(A, [2])
A1
array([[0, 1, 2, 3],
[4, 5, 6, 7]])
A2
array([[ 8, 9, 10, 11],
[12, 13, 14, 15]])
A1, A2 = np.split(A, [2], axis=1)
A1
array([[ 0, 1],
[ 4, 5],
[ 8, 9],
[12, 13]])
A2
array([[ 2, 3],
[ 6, 7],
[10, 11],
[14, 15]])
upper, lower = np.vsplit(A, [2])
upper
array([[0, 1, 2, 3],
[4, 5, 6, 7]])
lower
array([[ 8, 9, 10, 11],
[12, 13, 14, 15]])
left, right = np.hsplit(A, [2])
left
array([[ 0, 1],
[ 4, 5],
[ 8, 9],
[12, 13]])
right
array([[ 2, 3],
[ 6, 7],
[10, 11],
[14, 15]])
data = np.arange(16).reshape((4, 4))
data
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
X, y = np.hsplit(data, [-1])
X
array([[ 0, 1, 2],
[ 4, 5, 6],
[ 8, 9, 10],
[12, 13, 14]])
y
array([[ 3],
[ 7],
[11],
[15]])
y[:, 0]
array([ 3, 7, 11, 15])
numpy.array 中的運算
給定一個數(shù)組翠忠，讓數(shù)組中每一個數(shù)乘以2
n = 10
L = [i for i in range(n)]
2 * L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
A = []
for e in L:
A.append(2*e)
n = 1000000
L = [i for i in range(n)]
%%time
A = []
for e in L:
A.append(2*e)
CPU times: user 253 ms, sys: 30 ms, total: 283 ms
Wall time: 303 ms
%%time
A = [2*e for e in L]
CPU times: user 93.6 ms, sys: 25.8 ms, total: 119 ms
Wall time: 128 ms
import numpy as np
L = np.arange(n)
%%time
A = np.array(2*e for e in L)
CPU times: user 15.1 ms, sys: 8.97 ms, total: 24.1 ms
Wall time: 24.8 ms
%%time
A = 2 * L
CPU times: user 3.79 ms, sys: 4.36 ms, total: 8.14 ms
Wall time: 8.03 ms
n = 10
L = np.arange(n)
2 * L
array([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18])
NumPy’s UFuncs (Universal Functions)
X = np.arange(1, 16).reshape((3, 5))
X
array([[ 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10],
[11, 12, 13, 14, 15]])
X + 1
array([[ 2, 3, 4, 5, 6],
[ 7, 8, 9, 10, 11],
[12, 13, 14, 15, 16]])
X - 1
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14]])
X * 2
array([[ 2, 4, 6, 8, 10],
[12, 14, 16, 18, 20],
[22, 24, 26, 28, 30]])
X / 2
array([[ 0.5, 1. , 1.5, 2. , 2.5],
[ 3. , 3.5, 4. , 4.5, 5. ],
[ 5.5, 6. , 6.5, 7. , 7.5]])
X // 2
array([[0, 1, 1, 2, 2],
[3, 3, 4, 4, 5],
[5, 6, 6, 7, 7]])
X ** 2
array([[ 1, 4, 9, 16, 25],
[ 36, 49, 64, 81, 100],
[121, 144, 169, 196, 225]])
X % 2
array([[1, 0, 1, 0, 1],
[0, 1, 0, 1, 0],
[1, 0, 1, 0, 1]])
1 / X
array([[ 1. , 0.5 , 0.33333333, 0.25 , 0.2 ],
[ 0.16666667, 0.14285714, 0.125 , 0.11111111, 0.1 ],
[ 0.09090909, 0.08333333, 0.07692308, 0.07142857, 0.06666667]])
np.abs(X)
array([[ 1, 2, 3, 4, 5],
[ 6, 7, 8, 9, 10],
[11, 12, 13, 14, 15]])
np.sin(X)
array([[ 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]])
np.cos(X)
array([[ 0.54030231, -0.41614684, -0.9899925 , -0.65364362, 0.28366219],
[ 0.96017029, 0.75390225, -0.14550003, -0.91113026, -0.83907153],
[ 0.0044257 , 0.84385396, 0.90744678, 0.13673722, -0.75968791]])
np.tan(X)
array([[ 1.55740772e+00, -2.18503986e+00, -1.42546543e-01,
1.15782128e+00, -3.38051501e+00],
[ -2.91006191e-01, 8.71447983e-01, -6.79971146e+00,
-4.52315659e-01, 6.48360827e-01],
[ -2.25950846e+02, -6.35859929e-01, 4.63021133e-01,
7.24460662e+00, -8.55993401e-01]])
np.arctan(X)
array([[ 0.78539816, 1.10714872, 1.24904577, 1.32581766, 1.37340077],
[ 1.40564765, 1.42889927, 1.44644133, 1.46013911, 1.47112767],
[ 1.48013644, 1.48765509, 1.49402444, 1.49948886, 1.50422816]])
np.exp(X)
array([[ 2.71828183e+00, 7.38905610e+00, 2.00855369e+01,
5.45981500e+01, 1.48413159e+02],
[ 4.03428793e+02, 1.09663316e+03, 2.98095799e+03,
8.10308393e+03, 2.20264658e+04],
[ 5.98741417e+04, 1.62754791e+05, 4.42413392e+05,
1.20260428e+06, 3.26901737e+06]])
np.exp2(X)
array([[ 2.00000000e+00, 4.00000000e+00, 8.00000000e+00,
1.60000000e+01, 3.20000000e+01],
[ 6.40000000e+01, 1.28000000e+02, 2.56000000e+02,
5.12000000e+02, 1.02400000e+03],
[ 2.04800000e+03, 4.09600000e+03, 8.19200000e+03,
1.63840000e+04, 3.27680000e+04]])
np.power(3, X)
array([[ 3, 9, 27, 81, 243],
[ 729, 2187, 6561, 19683, 59049],
[ 177147, 531441, 1594323, 4782969, 14348907]])
np.log(X)
array([[ 0. , 0.69314718, 1.09861229, 1.38629436, 1.60943791],
[ 1.79175947, 1.94591015, 2.07944154, 2.19722458, 2.30258509],
[ 2.39789527, 2.48490665, 2.56494936, 2.63905733, 2.7080502 ]])
np.log2(X)
array([[ 0. , 1. , 1.5849625 , 2. , 2.32192809],
[ 2.5849625 , 2.80735492, 3. , 3.169925 , 3.32192809],
[ 3.45943162, 3.5849625 , 3.70043972, 3.80735492, 3.9068906 ]])
np.log10(X)
array([[ 0. , 0.30103 , 0.47712125, 0.60205999, 0.69897 ],
[ 0.77815125, 0.84509804, 0.90308999, 0.95424251, 1. ],
[ 1.04139269, 1.07918125, 1.11394335, 1.14612804, 1.17609126]])
矩陣運算
A = np.arange(4).reshape(2, 2)
A
array([[0, 1],
[2, 3]])
B = np.full((2, 2), 10)
B
array([[10, 10],
[10, 10]])
A + B
array([[10, 11],
[12, 13]])
A - B
array([[-10, -9],
[ -8, -7]])
A * B
array([[ 0, 10],
[20, 30]])
A.dot(B)
array([[10, 10],
[50, 50]])
A.T
array([[0, 2],
[1, 3]])
C = np.full((3, 3), 666)
A + C
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-145-cb7c4a36a7ba> in <module>()
----> 1 A + C
ValueError: operands could not be broadcast together with shapes (2,2) (3,3)
向量和矩陣的運算
加法
v = np.array([1, 2])
v + A
array([[1, 3],
[3, 5]])
v + A 是可以的鞠苟，但是在這個課程中，我們不研究其中的計算法則负间。有興趣的同學(xué)可以查詢資料自學(xué)numpy.array的broadcast
np.vstack([v] * A.shape[0])
array([[1, 2],
[1, 2]])
np.vstack([v] * A.shape[0]) + A
array([[1, 3],
[3, 5]])
np.tile(v, (2, 1))
array([[1, 2],
[1, 2]])
np.tile(v, (2, 1)) + A
array([[1, 3],
[3, 5]])
np.tile(v, (2, 2))
array([[1, 2, 1, 2],
[1, 2, 1, 2]])
乘法
v * A
array([[0, 2],
[2, 6]])
v.dot(A)
array([4, 7])
A.dot(v)
array([2, 8])
矩陣的逆
np.linalg.inv(A)
array([[-1.5, 0.5],
[ 1. , 0. ]])
invA = np.linalg.inv(A)
A.dot(invA)
array([[ 1., 0.],
[ 0., 1.]])
invA.dot(A)
array([[ 1., 0.],
[ 0., 1.]])
X = np.arange(16).reshape((2, 8))
invX = np.linalg.inv(X)
---------------------------------------------------------------------------
LinAlgError Traceback (most recent call last)
<ipython-input-207-60b1a25f4891> in <module>()
----> 1 invX = np.linalg.inv(X)
/Users/yuanzhang/anaconda/lib/python3.6/site-packages/numpy/linalg/linalg.py in inv(a)
515 a, wrap = _makearray(a)
516 _assertRankAtLeast2(a)
--> 517 _assertNdSquareness(a)
518 t, result_t = _commonType(a)
519
/Users/yuanzhang/anaconda/lib/python3.6/site-packages/numpy/linalg/linalg.py in _assertNdSquareness(*arrays)
210 for a in arrays:
211 if max(a.shape[-2:]) != min(a.shape[-2:]):
--> 212 raise LinAlgError('Last 2 dimensions of the array must be square')
213
214 def _assertFinite(*arrays):
LinAlgError: Last 2 dimensions of the array must be square
矩陣的偽逆
pinvX = np.linalg.pinv(X)
pinvX
array([[ -1.35416667e-01, 5.20833333e-02],
[ -1.01190476e-01, 4.16666667e-02],
[ -6.69642857e-02, 3.12500000e-02],
[ -3.27380952e-02, 2.08333333e-02],
[ 1.48809524e-03, 1.04166667e-02],
[ 3.57142857e-02, 8.67361738e-18],
[ 6.99404762e-02, -1.04166667e-02],
[ 1.04166667e-01, -2.08333333e-02]])
X.dot(pinvX)
array([[ 1.00000000e+00, -9.71445147e-17],
[ -1.33226763e-15, 1.00000000e+00]])
矩陣的偽逆又被稱為“廣義逆矩陣”偶妖，有興趣的同學(xué)可以翻看線性教材課本查看更多額廣義逆矩陣相關(guān)的性質(zhì)。中文wiki鏈接: https://zh.wikipedia.org/wiki/%E5%B9%BF%E4%B9%89%E9%80%86%E9%98%B5
Numpy 中的聚合操作
sum
import numpy as np
L = np.random.random(100)
sum(L)
52.675554310672098
np.sum(L)
52.675554310672105
big_array = np.random.rand(1000000)
%timeit sum(big_array)
%timeit np.sum(big_array)
10 loops, best of 3: 173 ms per loop
1000 loops, best of 3: 1.02 ms per loop
min, max
np.min(big_array)
2.2765289564574687e-07
np.max(big_array)
0.99999686126703025
big_array.min()
2.2765289564574687e-07
big_array.max()
0.99999686126703025
big_array.sum()
500454.89231729991
多維度聚合
X = np.arange(16).reshape(4,-1)
X
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
np.sum(X)
120
np.sum(X, axis=0)
array([24, 28, 32, 36])
np.sum(X, axis=1)
array([ 6, 22, 38, 54])
注意：axis描述的是將要被壓縮的維度政溃。
其他聚合操作
np.prod(X)
0
np.prod(X + 1)
20922789888000
np.mean(X)
7.5
np.median(X)
7.5
v = np.array([1, 1, 2, 2, 10])
np.mean(v)
3.2000000000000002
np.median(v)
2.0
np.percentile(big_array, q=50)
0.50056612640031206
np.median(big_array)
0.50056612640031206
np.percentile(big_array, q=100)
0.99999686126703025
np.max(big_array)
0.99999686126703025
for percent in [0, 25, 50, 75, 100]:
print(np.percentile(big_array, q=percent))
2.27652895646e-07
0.250501365819
0.5005661264
0.750543416185
0.999996861267
np.var(big_array)
0.083379660489048227
np.std(big_array)
0.28875536443336985
x = np.random.normal(0, 1, 1000000)
np.mean(x)
-0.00044876833100538597
np.std(x)
1.0000457010611321
Numpy 中arg運算
import numpy as np
x = np.random.normal(0, 1, 1000000)
索引
np.argmin(x)
886266
x[886266]
-4.8354963762015108
np.min(x)
-4.8354963762015108
np.argmax(x)
4851
x[4851]
4.5860138951376461
np.max(x)
4.5860138951376461
排序和使用索引
x = np.arange(16)
x
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
np.random.shuffle(x)
x
array([13, 2, 6, 7, 11, 10, 3, 4, 8, 0, 5, 1, 9, 14, 12, 15])
np.sort(x)
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
x
array([13, 2, 6, 7, 11, 10, 3, 4, 8, 0, 5, 1, 9, 14, 12, 15])
x.sort()
x
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
X = np.random.randint(10, size=(4,4))
X
array([[8, 8, 5, 8],
[1, 2, 2, 4],
[5, 5, 9, 9],
[3, 9, 3, 4]])
np.sort(X, axis=0)
array([[1, 2, 2, 4],
[3, 5, 3, 4],
[5, 8, 5, 8],
[8, 9, 9, 9]])
np.sort(X, axis=1)
array([[5, 8, 8, 8],
[1, 2, 2, 4],
[5, 5, 9, 9],
[3, 3, 4, 9]])
使用索引
x
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
np.random.shuffle(x)
x
array([14, 15, 8, 7, 10, 4, 9, 1, 6, 5, 3, 12, 2, 11, 0, 13])
np.argsort(x)
array([14, 7, 12, 10, 5, 9, 8, 3, 2, 6, 4, 13, 11, 15, 0, 1])
np.partition(x, 3)
array([ 1, 0, 2, 3, 4, 5, 7, 8, 6, 9, 10, 12, 11, 13, 15, 14])
np.argpartition(x, 3)
array([ 7, 14, 12, 10, 5, 9, 3, 2, 8, 6, 4, 11, 13, 15, 1, 0])
X
array([[8, 8, 5, 8],
[1, 2, 2, 4],
[5, 5, 9, 9],
[3, 9, 3, 4]])
np.argsort(X, axis=1)
array([[2, 0, 1, 3],
[0, 1, 2, 3],
[0, 1, 2, 3],
[0, 2, 3, 1]])
np.argpartition(X, 2, axis=1)
array([[2, 1, 0, 3],
[0, 1, 2, 3],
[0, 1, 2, 3],
[0, 2, 3, 1]])
Numpy 中的比較和Fancy Indexing
Fancy Indexing
import numpy as np
x = np.arange(16)
x
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
x[3]
3
x[3:9]
array([3, 4, 5, 6, 7, 8])
x[3:9:2]
array([3, 5, 7])
[x[3], x[5], x[7]]
[3, 5, 7]
ind = [3, 5, 7]
x[ind]
array([3, 5, 7])
ind = np.array([[0, 2], [1, 3]])
x[ind]
array([[0, 2],
[1, 3]])
Fancy Indexing 應(yīng)用在二維數(shù)組
X = x.reshape(4, -1)
X
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
row = np.array([0, 1, 2])
col = np.array([1, 2, 3])
X[row, col]
array([ 1, 6, 11])
X[0, col]
array([1, 2, 3])
X[:2, col]
array([[1, 2, 3],
[5, 6, 7]])
col = [True, False, True, True]
X[0, col]
array([0, 2, 3])
numpy.array 的比較
x
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15])
x < 3
array([ True, True, True, False, False, False, False, False, False,
False, False, False, False, False, False, False], dtype=bool)
x > 3
array([False, False, False, False, True, True, True, True, True,
True, True, True, True, True, True, True], dtype=bool)
x <= 3
array([ True, True, True, True, False, False, False, False, False,
False, False, False, False, False, False, False], dtype=bool)
x >= 3
array([False, False, False, True, True, True, True, True, True,
True, True, True, True, True, True, True], dtype=bool)
x == 3
array([False, False, False, True, False, False, False, False, False,
False, False, False, False, False, False, False], dtype=bool)
x != 3
array([ True, True, True, False, True, True, True, True, True,
True, True, True, True, True, True, True], dtype=bool)
2 * x == 24 - 4 * x
array([False, False, False, False, True, False, False, False, False,
False, False, False, False, False, False, False], dtype=bool)
X < 6
array([[ True, True, True, True],
[ True, True, False, False],
[False, False, False, False],
[False, False, False, False]], dtype=bool)
使用 numpy.array 的比較結(jié)果
np.count_nonzero( x <= 3)
4
np.sum(x <= 3)
4
np.sum(X % 2 == 0, axis=0)
array([4, 0, 4, 0])
np.sum(X % 2 == 0, axis=1)
array([2, 2, 2, 2])
np.any(x == 0)
True
np.any(x < 0)
False
np.all(x > 0)
False
np.all(x >= 0)
True
np.all(X > 0, axis=1)
array([False, True, True, True], dtype=bool)
np.sum((x > 3) & (x < 10))
6
np.sum((x > 3) && (x < 10))
File "<ipython-input-45-780ca9b7c144>", line 1
np.sum((x > 3) && (x < 10))
^
SyntaxError: invalid syntax
np.sum((x % 2 == 0) | (x > 10))
11
np.sum(~(x == 0))
15
比較結(jié)果和Fancy Indexing
x < 5
array([ True, True, True, True, True, False, False, False, False,
False, False, False, False, False, False, False], dtype=bool)
x[x < 5]
array([0, 1, 2, 3, 4])
x[x % 2 == 0]
array([ 0, 2, 4, 6, 8, 10, 12, 14])
X[X[:,3] % 3 == 0, :]
array([[ 0, 1, 2, 3],
[12, 13, 14, 15]])
