Monte Carlo algorithm, in my opinion, is a very magical algorithm that can simulate a lot of scenes and simulated data, may be almost the same with real data, but far less than the cost of acquiring simulated real data .
Today, I use a Monte Carlo algorithm, to do two simple simulation. A π value is calculated, a further quadrature.
A, π value
How the value of π is an irrational number, not an infinite loop, around AD 480, the mathematician Zu rushed into the Northern and Southern Dynasties of the outcome of seven decimal place, today, we use a computer to simulate my simulation results .
First, to illustrate the idea of simulation, as shown below, is derived by proportional relationship of the inscribed circle and a square area, as long as the area ratio is calculated Talia, we can obtain π.
Then how, without the use of π to calculate the area of the inscribed circle. We can use the dot manner, a square region of n random dot, if there are x within the inner region of the inscribed circle, the area ratio is Talia n / x. If this large n, the result is very accurate.
Specific code as follows
list_p=[]
list_p_x=[]
max_count = 100000000
first_count =10
rate = 2
count_s = 0
j=0
while first_count < max_count:
print(first_count)
while j < first_count:
x = random.uniform(-1, 1)
y = random.uniform(0, 1)
if x**2 + y**2 < 1:
count_s = count_s + 1
j = j + 1
list_p_x.append(first_count)
list_p.append(count_s/first_count*4)
j=0
count_s=0
first_count =first_count * rate
plt.xlim(0,first_count/2)
plt.ylim (3,3.3)
plt.plot(list_p_x,list_p)
plt.hlines(y=np.pi,xmin= first_count,xmax=list_p_x, colors = "c", linestyles = "dashed")
模拟出来的结果如下，在模拟超过1w次后，结果已经趋于稳定，基本等于3.14，这已经基本我们大部分使用场景。
二、积分
积分实际也可以理解是计算面试，比如下图，是y = -x^2+1 的函数图形，现在用蒙特卡罗求一下该函数的积分。
思路和求π得方法一致，也是通过随机打点的方式，根据在积分区域的散点数与矩形区域内散点数之比，乘以矩形面积，就是该积分区域面积。
分析模拟结果如下图，可以看到模拟3w到多次时，准确率很高了，与1.33不断接近，在9w次之后，基本保持重叠。
通过蒙特卡罗模拟，生成一系列符合预期要求的随机数，就可以模拟出一个十分接近实际值得近似值，十分适应于对数值计算精度要求不是很高的场景，比如，我们在计算圆面积的是，通常都会取3.14，而不会取3.1415926.....等。
注：
公众号：数据志（原：跟着菜鸟一起学R语言）
原文链接：https://www.cnblogs.com/wheng/articles/11832476.html