python：numpy.random模塊生成隨機數

簡介

所謂生成隨機數，即按照某種概率分佈，從給定的區間內隨機選取一個數。常用的分佈有：均勻分佈（uniform distribution），正態分佈（normal distribution），泊松分佈（poisson distribution）等。

python中的numpy.random模塊提供了常用的隨機數生成方法，下面簡要總結。

按均勻分佈生成隨機數

rand

功能
按照均勻分佈，在[0,1)內生成隨機數。

接口

Docstring:
rand(d0, d1, ..., dn)

Random values in a given shape.

Create an array of the given shape and populate it with
random samples from a uniform distribution
over ``[0, 1)``.

實例

In [3]: np.random.rand()
Out[3]: 0.08621180209775481

In [4]: np.random.rand(5)
Out[4]: array([0.85086433, 0.46726857, 0.48144885, 0.36369815, 0.9181539 ])

In [5]: np.random.rand(2,5)
Out[5]: 
array([[0.70784333, 0.08966827, 0.50090152, 0.22517888, 0.86735194],
       [0.92224362, 0.98468415, 0.19134583, 0.06757754, 0.19330571]])

驗證下是按均勻分佈生成的隨機數
方法：利用直方圖可驗證其概率密度函數是否與均勻分佈一致。

x = np.random.rand(10000)
plt.hist(x,bins=20,density=True)

結果：符合均勻分佈的概率密度函數，即P(x)=1, x=[0,1]

uniform

功能
按照均勻分佈，在[low,high)內生成隨機數。

接口

Docstring:
uniform(low=0.0, high=1.0, size=None)

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval
``[low, high)`` (includes low, but excludes high).  In other words,
any value within the given interval is equally likely to be drawn
by `uniform`.

實例

In [8]: np.random.uniform(-1,1)
Out[8]: -0.34092928027362945

In [9]: np.random.uniform(-1,1,5)
Out[9]: array([ 0.84319637, -0.00449745, -0.21958133, -0.16755161, -0.94383758])

In [10]: np.random.uniform(-1,1,(2,5))
Out[10]: 
array([[-0.66176972,  0.71703011,  0.61658043,  0.0373402 ,  0.95824383],
       [ 0.31743693,  0.48948276,  0.0393428 ,  0.64896449,  0.3659116 ]])

驗證下是按均勻分佈生成的隨機數

x = np.random.uniform(-1,1,10000)
plt.hist(x,bins=20,density=True)

結果：與均勻分佈一致。

randint

功能
按照離散均勻分佈，在[low,high)內生成隨機整數。

接口

Docstring:
randint(low, high=None, size=None, dtype='l')

Return random integers from `low` (inclusive) to `high` (exclusive).

Return random integers from the "discrete uniform" distribution of
the specified dtype in the "half-open" interval [`low`, `high`). If
`high` is None (the default), then results are from [0, `low`).

實例

In [12]: np.random.randint(0,101)
Out[12]: 27

In [13]: np.random.randint(0,101,5)
Out[13]: array([71, 31, 22, 85, 64])

In [14]: np.random.randint(0,101,(2,5))
Out[14]: 
array([[94, 29, 80, 40,  6],
       [17, 19, 85, 11, 48]])

驗證下是按均勻分佈生成的隨機數

x = np.random.randint(0,100,10000)
plt.hist(x,bins=20,density=True)

結果：與均勻分佈一致。

註釋：當區間爲[0,101),即x = np.random.randint(0,100,10000)時，最後一個區間的概率密度總是會較大些。猜測跟隨機數生成機制有關，暫不深入，後續有時間研究下。

按正態分佈生成隨機數

randn

功能
按照正態分佈($\mu=0,\sigma=1$)，生成隨機數

接口

Docstring:
randn(d0, d1, ..., dn)

Return a sample (or samples) from the "standard normal" distribution.

If positive, int_like or int-convertible arguments are provided,
`randn` generates an array of shape ``(d0, d1, ..., dn)``, filled
with random floats sampled from a univariate "normal" (Gaussian)
distribution of mean 0 and variance 1 (if any of the :math:`d_i` are
floats, they are first converted to integers by truncation). A single
float randomly sampled from the distribution is returned if no
argument is provided.

This is a convenience function.  If you want an interface that takes a
tuple as the first argument, use `numpy.random.standard_normal` instead.

實例

In [15]: np.random.randn()
Out[15]: 0.2594770528010187

In [16]: np.random.randn(5)
Out[16]: array([ 0.51858431, -0.56406054,  0.39934797,  0.87223161,  0.56642685])

In [17]: np.random.randn(2,5)
Out[17]: 
array([[-0.09649912, -0.51282169,  0.10047756,  1.03372611, -1.54014928],
       [-1.29894642,  0.46580577,  0.77321341,  2.16154836, -0.99208604]])

驗證下是按正態分佈生成的隨機數

x = np.random.randn(10000)
n, bins, patches =plt.hist(x,bins=50,density=True)
y1 = scipy.stats.norm.pdf(bins, loc=0, scale=1)
plt.plot(bins,y1,'r--')

結果：符合正態分佈。

normal

功能
按照正態分佈($\mu,\sigma$)，生成隨機數

接口

Docstring:
normal(loc=0.0, scale=1.0, size=None)

Draw random samples from a normal (Gaussian) distribution.

實例

In [22]: np.random.normal(170,10)
Out[22]: 169.24973434084185

In [23]: np.random.normal(170,10,5)
Out[23]:
array([167.0171718 , 169.47192942, 169.61260805, 151.53654937,
       172.61541579])

In [24]: np.random.normal(170,10,(2,5))
Out[24]:
array([[187.28916343, 170.30553783, 175.69009811, 171.87050032,
        179.26404146],
       [171.17078449, 173.25610155, 169.60757285, 178.79427632,
        163.25814631]])

驗證下是按正態分佈生成的隨機數

mu,sigma = 170,10
x = np.random.normal(mu,sigma,10000)
n, bins, patches = plt.hist(x,bins=50,density=True)
y1 = scipy.stats.norm.pdf(bins, loc=mu, scale=sigma)
plt.plot(bins,y1,'r--')

結果：符合正態分佈

給定一個數組，隨機排列

permutation

功能
給定一個數組(或list，隨機排列

接口

Randomly permute a sequence, or return a permuted range.

If `x` is a multi-dimensional array, it is only shuffled along its
first index.

Parameters
----------
x : int or array_like
    If `x` is an integer, randomly permute ``np.arange(x)``.
    If `x` is an array, make a copy and shuffle the elements
    randomly.

Returns
-------
out : ndarray
    Permuted sequence or array range.

實例

In [28]: np.random.permutation(6)
Out[28]: array([2, 1, 3, 4, 0, 5])

In [29]: np.random.permutation(np.arange(1,7))
Out[29]: array([5, 3, 4, 2, 1, 6])

shuffle

功能
給定一個數組或list，打亂順序（同permutation）

接口

Docstring:
shuffle(x)

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a
multi-dimensional array. The order of sub-arrays is changed but
their contents remains the same.

Parameters
----------
x : array_like
    The array or list to be shuffled.

Returns
-------
None

實例

In [32]: x = np.arange(1,7)

In [33]: x
Out[33]: array([1, 2, 3, 4, 5, 6])

In [34]: np.random.shuffle(x)

In [35]: x
Out[35]: array([4, 1, 6, 3, 5, 2])

自定義概率分佈抽樣

choice

功能
給定一個數組或list，從中抽樣。可自定義概率分佈，即自定義各元素被抽到的概率（默認爲均勻分佈）

接口

Docstring:
choice(a, size=None, replace=True, p=None)

Generates a random sample from a given 1-D array

注意：replace爲True，則抽樣出的n個元素可以重複，即每次抽樣出一個元素後，還會將其放回樣本池，默認可重複；反之。

實例
1、元素爲均勻分佈

In [107]: x = np.arange(1,7)

In [108]: np.random.choice(x,5)
Out[108]: array([3, 4, 5, 6, 3])

In [109]: np.random.choice(x,5,replace=False)
Out[109]: array([2, 1, 5, 6, 4])

2、自定義元素的概率分佈

In [110]: x = ['石頭','剪刀','布']

In [111]: np.random.choice(x,5,p=[0.2,0.3,0.5])
Out[111]: array(['布', '布', '布', '布', '剪刀'], dtype='<U2')

同理，可驗證是否按照自定義的概率分佈進行抽樣。

x = ['石頭','剪刀','布']
res = np.random.choice(x,5000,p=[0.2,0.3,0.5])

_, ax = plt.subplots()
n,bins,patches = ax.hist(res,bins=3,density=1)

p = n*(bins[1]-bins[0])
s = [ax.get_xticklabels()[i].get_text() for i in range(3)]
print(list(zip(s,p)))

結果：

[('石頭', 0.19440000000000002), ('布', 0.5024), ('剪刀', 0.3031999999999999)]

總結

• 用uniform(low, high, size)生成[low, high)內均勻分佈的隨機數(float)
  low,high默認爲[0,1)，即等價於rand()
• 用randint(low, high, size)生成[low, high)內均勻分佈的隨機整數(int)
• 用normal(loc, scale,size)隨機生成正態分佈($\mu,\sigma$)的隨機數(float)
  loc,scale默認爲0,1，即等價於randn()
• 用choice生成自定義概率分佈的隨機數（或其他類型元素）