1.反变换法
设需产生分布函数为F(x)的连续随机数X。若已有[0,1]区间均匀分布随机数R,则产生X的反变换公式为：
F(x)=r, 即x=F-1(r)
反函数存在条件：如果函数y=f(x)是定义域D上的单调函数,那么f(x)一定有反函数存在,且反函数一定是单调的。分布函数F(x)为是一个单调递增函数，所以其反函数存在。从直观意义上理解，因为r一一对应着x,而在[0,1]均匀分布随机数R≤r的概率P(R≤r)=r。 因此,连续随机数X≤x的概率P(X≤x)=P(R≤r)=r=F(x)
即X的分布函数为F(x)。
例子:下面的代码使用反变换法在区间[0, 6]上生成随机数，其概率密度近似为P(x)=e-x
1 import numpy as np 2 import matplotlib.pyplot as plt 3 4 # probability distribution we're trying to calculate 5 p = lambda x: np.exp(-x) 6 7 # CDF of p 8 CDF = lambda x: 1-np.exp(-x) 9 10 # invert the CDF 11 invCDF = lambda x: -np.log(1-x) 12 13 # domain limits 14 xmin = 0 # the lower limit of our domain 15 xmax = 6 # the upper limit of our domain 16 17 # range limits 18 rmin = CDF(xmin) 19 rmax = CDF(xmax) 20 21 N = 10000 # the total of samples we wish to generate 22 23 # generate uniform samples in our range then invert the CDF 24 # to get samples of our target distribution 25 R = np.random.uniform(rmin, rmax, N) 26 X = invCDF(R) 27 28 # get the histogram info 29 hinfo = np.histogram(X,100) 30 31 # plot the histogram 32 plt.hist(X,bins=100, label=u'Samples'); 33 34 # plot our (normalized) function 35 xvals=np.linspace(xmin, xmax, 1000) 36 plt.plot(xvals, hinfo[0][0]*p(xvals), 'r', label=u'p(x)') 37 38 # turn on the legend 39 plt.legend() 40 plt.show()
一般来说，直方图的外廓曲线接近于总体X的概率密度曲线。
2.舍选抽样法(Rejection Methold)
用反变换法生成随机数时，如果求不出F-1(x)的解析形式或者F(x)就没有解析形式，则可以用F-1(x)的近似公式代替。但是由于反函数计算量较大，有时也是很不适宜的。另一种方法是由Von Neumann提出的舍选抽样法。下图中曲线w(x)为概率密度函数，按该密度函数产生随机数的方法如下：
基本的rejection methold步骤如下：
1. Draw x uniformly from [xmin xmax]
2. Draw x uniformly from [0, ymax]
3. if y < w(x)，accept the sample, otherwise reject it
4. repeat
即落在曲线w(x)和X轴所围成区域内的点接受，落在该区域外的点舍弃。
例子:下面的代码使用basic rejection sampling methold在区间[0, 10]上生成随机数，其概率密度近似为P(x)=e-x
1 # -*- coding: utf-8 -*- 2 ''' 3 The following code produces samples that follow the distribution P(x)=e^−x 4 for x=[0, 10] and generates a histogram of the sampled distribution. 5 ''' 6 import numpy as np 7 import matplotlib.pyplot as plt 8 9 10 P = lambda x: np.exp(-x) 11 12 # domain limits 13 xmin = 0 # the lower limit of our domain 14 xmax = 10 # the upper limit of our domain 15 16 # range limit (supremum) for y 17 ymax = 1 18 19 N = 10000 # the total of samples we wish to generate 20 accepted = 0 # the number of accepted samples 21 samples = np.zeros(N) 22 count = 0 # the total count of proposals 23 24 # generation loop 25 while (accepted < N): 26 27 # pick a uniform number on [xmin, xmax) (e.g. 0...10) 28 x = np.random.uniform(xmin, xmax) 29 30 # pick a uniform number on [0, ymax) 31 y = np.random.uniform(0,ymax) 32 33 # Do the accept/reject comparison 34 if y < P(x): 35 samples[accepted] = x 36 accepted += 1 37 38 count +=1 39 40 print count, accepted 41 42 # get the histogram info 43 # If bins is an int, it defines the number of equal-width bins in the given range 44 (n, bins)= np.histogram(samples, bins=30) # Returns: n-The values of the histogram,n是直方图中柱子的高度 45 46 # plot the histogram 47 plt.hist(samples,bins=30,label=u'Samples') # bins=30即直方图中有30根柱子 48 49 # plot our (normalized) function 50 xvals=np.linspace(xmin, xmax, 1000) 51 plt.plot(xvals, n[0]*P(xvals), 'r', label=u'P(x)') 52 53 # turn on the legend 54 plt.legend() 55 plt.show()
>>>
99552 10000
3.推广的舍取抽样法
从上图中可以看出，基本的rejection methold法抽样效率很低，因为随机数x和y是在区间[xmin xmax]和区间[0 ymax]上均匀分布的，产生的大部分点不会落在w(x)曲线之下(曲线e-x的形状一边高一边低，其曲线下的面积占矩形面积的比例很小，则舍选抽样效率很低)。为了改进简单舍选抽样法的效率，可以构造一个新的密度函数q(x)(called a proposal distribution from which we can readily draw samples)，使它的形状接近p(x)，并选择一个常数k使得kq(x)≥w(x)对于x定义域内的值都成立。对应下图，首先从分布q(z)中生成随机数z0，然后按均匀分布从区间[0 kq(z0)]生成一个随机数u0。 if u0 > p(z0) then the sample is rejected,otherwise u0 is retained. 即下图中灰色区域内的点都要舍弃。可见，由于随机点u0只出现在曲线kq(x)之下，且在q(x)较大处出现次数较多，从而大大提高了采样效率。显然q(x)形状越接近p(x)，则采样效率越高。
根据上述思想，也可以表达采样规则如下：
1. Draw x from your proposal distribution q(x)
2. Draw y uniformly from [0 1]
3. if y < p(x)/kq(x) , accept the sample, otherwise reject it
4. repeat
下面例子中选择函数p(x)=1/(x+1)作为proposal distribution，k=1。曲线1/(x+1)的形状与e-x相近。
1 import numpy as np 2 import matplotlib.pyplot as plt 3 4 p = lambda x: np.exp(-x) # our distribution 5 g = lambda x: 1/(x+1) # our proposal pdf (we're choosing k to be 1) 6 CDFg = lambda x: np.log(x +1) # generates our proposal using inverse sampling 7 8 # domain limits 9 xmin = 0 # the lower limit of our domain 10 xmax = 10 # the upper limit of our domain 11 12 # range limits for inverse sampling 13 umin = CDFg(xmin) 14 umax = CDFg(xmax) 15 16 N = 10000 # the total of samples we wish to generate 17 accepted = 0 # the number of accepted samples 18 samples = np.zeros(N) 19 count = 0 # the total count of proposals 20 21 # generation loop 22 while (accepted < N): 23 24 # Sample from g using inverse sampling 25 u = np.random.uniform(umin, umax) 26 xproposal = np.exp(u) - 1 27 28 # pick a uniform number on [0, 1) 29 y = np.random.uniform(0, 1) 30 31 # Do the accept/reject comparison 32 if y < p(xproposal)/g(xproposal): 33 samples[accepted] = xproposal 34 accepted += 1 35 36 count +=1 37 38 print count, accepted 39 40 # get the histogram info 41 hinfo = np.histogram(samples,50) 42 43 # plot the histogram 44 plt.hist(samples,bins=50, label=u'Samples'); 45 46 # plot our (normalized) function 47 xvals=np.linspace(xmin, xmax, 1000) 48 plt.plot(xvals, hinfo[0][0]*p(xvals), 'r', label=u'p(x)') 49 50 # turn on the legend 51 plt.legend() 52 plt.show()
>>>
24051 10000
可以对比基本的舍取法和改进的舍取法的结果，前者产生符合要求分布的10000个随机数运算了99552步，后者运算了24051步，可以看到效率明显提高。
参考：
http://iacs-courses.seas.harvard.edu/courses/am207/blog/lecture-3.html
http://blog.csdn.net/xianlingmao/article/details/7768833
http://blog.sina.com.cn/s/blog_60b44d6a0101l45z.html
http://www.ruanyifeng.com/blog/2015/07/monte-carlo-method.html
来源：https://www.cnblogs.com/21207-iHome/p/5266399.html