蒙特卡洛方法的数学基础-1

• 蒙特卡洛方法的数学基础-1

概率论

Bayes 公式

常用分布

Binominal Distribution

Poisson Distribution

Gaussian Distribution 

Exponential Distribution

Uniform Distribution

大数定理

• 均匀概率分布随机地取N个数xi ，函数值之和的算术平均收敛于函数的期望值
• 算术平均收敛于真值

中心极限定理

• n个相互独立分布各异的随机变量的 总和服从正态分布
• 置信水平下的统计误差

Monte Carlo方法简单应用

案例：Buffon 投针实验

• 略

案例：投点法求定积分

• 略

任意分布的伪随机变量的抽样

反函数直接抽样法

变换抽样法

dim=1

dim=2

改进的Maraglia方法

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(0)

Num = 10000
x = np.zeros(Num)
y = np.zeros(Num)
z = np.zeros(Num)

Times = 0
while Times < Num:
    u,v = np.random.rand(2)
    w = (2*u-1)**2 + (2*v-1)**2
    if w<=1:
        z[Times] = (-2*np.log(w)/w)**0.5        
        x[Times] = u * z[Times]
        y[Times] = v * z[Times]
        Times += 1
    else:
        continue

fig = plt.figure(figsize=(12, 6))
ax1 = fig.add_subplot(1, 2, 1)
ax2 = fig.add_subplot(1, 2, 2)

ax1.hist(x, np.arange(0, max(x), 1))
ax1.set_title("distribution x")

ax2.hist(y, np.arange(0, max(y), 1))
ax2.set_title("distribution y")

plt.savefig("3.jpg")

舍选抽样法

...

复合抽样法

---

• 依据概率密度函数抽样
• 依然条件概率密度函数抽样

---

加分布与减分布抽样法

• 加分布抽样

其中

• 取\zeta \sim U[0,1]，解对n的不等式

• 对，对应的hn(x) 抽样得 \zeta = \zeta_{h_n}

• 减分布抽样

乘加与乘减分布抽样法

...

反函数近似抽样法

近似替代

最小二乘拟合 

极限近似法

...

蒙特卡洛方法处理定积分

近似积分公式

• 近似积分公式在某种意义上是对区间内某些函数值的加权平均
  • 蒙卡方法的目的是优化样本点的选择（位置和权重）
• 矩形积分方法(The Leftpoint Rule)
  • 1阶近似
• 矩形积分方法(The Rightpoint Rule)
  • 1阶近似
• 矩形积分方法(The Midpoint Rule)
  • 2阶近似
• 梯形积分方法(The Trapezoidal Rule)
  • 2阶近似
• Simpson's Rule 
  • 4阶近似

蒙特卡洛积分思路

• iid 
  • independent and identically distributed

---

• example 

import matplotlib.pyplot as plt
import numpy as np
import scipy.integrate as integrate

np.random.seed(0)
a = 0
b = 5
N = 100000

f = lambda x:x**2 + np.exp(x)
result1 = integrate.quad(f, a, b)

points = np.random.rand(N)*(b-a) + a
ybar = sum(f(points))/N
Sbar = (sum((points-ybar)**2)/N)**0.5 
result2 = ((b-a)*ybar, (b-a)*Sbar/N**0.5)

X = np.linspace(a, b, N+1)
result3 = f(X[0]) + f(X[N])
flag = 1
for i in range(1, N-1):
    if flag == 1:
        result3 += 4*f(X[i])
        flag *= -1
        
    elif flag == -1:
        result3 += 2*f(X[i])
        flag *= -1

    else:
        print("ERROR!")
        
result3 *= (b-a)/N/3

print("result1:", result1)
print("result2:", result2)
print("result3:", result3)

result1: (189.07982576924329, 2.099207760611269e-12)
result2: (188.98624801068067, 0.5586055984291508)
result3: 189.06826542000084
>>> 

---

• 如何评价哩，只能说，蒙卡方法至少能积出正确的答案，关键是速度慢啊
• 好吧，有人说是代码写得差，嗯 合理
• 改一下

import matplotlib.pyplot as plt
import numpy as np
import scipy.integrate as integrate

np.random.seed(0)
a = 0
b = 5
N = 100000

f = lambda x:x**2 + np.exp(x)
result1 = integrate.quad(f, a, b)

points = np.random.rand(N)*(b-a) + a
ybar = sum(f(points))/N
Sbar = (sum((points-ybar)**2)/N)**0.5 
result2 = ((b-a)*ybar, (b-a)*Sbar/N**0.5)

X = np.linspace(a, b, N+1)
result3 = f(X)
coeff = np.ones(N+1)
coeff[1::2] = 4
coeff[2::2] = 2
result3 *= coeff
result3 = sum(result3)

        
result3 *= (b-a)/N/3

print("result1:", result1)
print("result2:", result2)
print("result3:", result3)

• 测一下用时吧，时间复杂度都是同阶的（大误，这是Python的numpy 特性）

import matplotlib.pyplot as plt
import numpy as np
import scipy.integrate as integrate
import time

np.random.seed(0)
a = 0
b = 5
N = 100000

f = lambda x:x**2 + np.exp(x)
s1 = time.time()
for i in range(100):
    result1 = integrate.quad(f, a, b)
e1 = time.time()

s2 = time.time()
for i in range(100):
    points = np.random.rand(N)*(b-a) + a
    ybar = sum(f(points))/N
    Sbar = (sum((points-ybar)**2)/N)**0.5 
    result2 = ((b-a)*ybar, (b-a)*Sbar/N**0.5)
e2 = time.time()

X = np.linspace(a, b, N+1)
coeff = np.ones(N+1)
coeff[1::2] = 4
coeff[2::2] = 2
s3 = time.time()
for i in range(100):
    result3 = f(X)
    result3 *= coeff
    result3 = sum(result3)
    result3 *= (b-a)/N/3
e3 = time.time()

print("result1:", result1, "used time:", round(e1 - s1, 2))
print("result2:", result2, "used time:", round(e2 - s2, 2))
print("result3:", result3, "used time:", round(e3 - s3, 2))

---

result1: (189.07982576924329, 2.099207760611269e-12) used time: 0.0
result2: (188.83626394308098, 0.5580722224407848) used time: 1.8
result3: 189.0827159885614 used time: 0.9
>>> 

• 蒙卡在这里，精度低，速度慢......