蒙特卡洛方法的简单应用
- 蒙特卡洛方法的简单应用
圆周率估算
eastimate pi
python version 3.11
RNG:np.random.random
import os
figure_save_path = "file_fig"
import warnings
warnings.filterwarnings("error")
import numpy as np
np.random.seed(0)
import matplotlib.pyplot as plt
from PIL import Image
def main(times=100):
print("RNG:", "Python numpy.random.random")
print("开始估算")
Nlist = [1000000*i for i in range(1, times)]
Pilist = []
Slist = []
gif_frames = []
for i in range(1, times):
N = Nlist[i-1]
random_points = np.random.random((2, N))
radius = sum(random_points[:]**2)
count = np.sum(radius<=1)
ave = count/N
S2 = ((N-count)*ave**2+count*(1-ave)**2)/(N-1)
S = S2**0.5
Pilist.append(4*ave)
Slist.append(4*S/N**0.5)
plt.xlabel("test_times")
plt.ylabel("estimate_pi_value", rotation=90)
plt.title("Estimate Pi")
plt.errorbar(Nlist[:i], Pilist[:i], Slist[:i], label="estimate pi", c="green")
plt.axhline(np.pi,label="exact pi", c="red")
plt.legend()
if not os.path.exists(figure_save_path):
os.makedirs(figure_save_path)
else:
pass
plt.savefig(os.path.join(figure_save_path, str(i) + ".jpg"))
plt.cla()
img = Image.open(os.path.join(figure_save_path, str(i) + ".jpg"))
gif_frames.append(img)
print("估算完成")
print("开始绘制动画")
gif_frames[0].save("estimate-pi.gif",
save_all=True, append_images=gif_frames[1:], duration=200, loop=0)
print("动画绘制完成")
main()
input("press any key to exit")
- 动画
二维积分估算
- 2dintegral
- RNG:numpy.random.random
- Python version:3.11
- 允许积分类型 \int\int f(x,y)dxdy
- 若实现手动输入方程功能会增加程序解析负担,大约是十倍左右,故不予实现
- 若需要实现,可以仿照如下例子:
def main(X_start=0, X_end=1, Y_start=0, Y_end=1):
print("RNG:numpy.random.random")
print("")
funstr = input("输入方程")
#funstr = "x**2*y**2"
f = lambda x,y: eval(funstr)
#f = lambda x,y:x**2*y**2
print("运算开始")
N = 10000
start = time.time()
random_xpoints = np.random.uniform(X_start, X_end, N)
random_ypoints = np.random.uniform(Y_start, Y_end, N)
X = np.arange(X_start, X_end, 100)
Y = np.arange(Y_start, Y_end, 100)
guess = np.array(list(map(f, random_xpoints, random_ypoints)))
ave = sum(guess)/N
I = ave*(X_end-X_start)*(Y_end-Y_start)
S2 = sum((guess-ave)**2)/(N-1)
S = S2**0.5
end = time.time()
print(I, "\pm", S/N**0.5)
print("运算结束,用时:", round(end-start, 4))
拒绝抽样方法
import os
figure_save_path = "file_fig"
import warnings
warnings.filterwarnings("error")
import numpy as np
np.random.seed(0)
import matplotlib.pyplot as plt
from PIL import Image
from copy import deepcopy
plt.rcParams['font.sans-serif'] = ['FangSong']
plt.rcParams['axes.unicode_minus'] = False
######
#待抽样分布x~f(x)
#已有分布g(x)
#建议分布c*g(x)
#在概率空间有c*g(x)恒大于f(x)
#step 1 按f(x)抽样 得x,x_i~f(x) x.size
#step 2 生成u,u_i~U[0, max(c*g(x))] u.size = x.size
#step 3 从x中选取满足u_i= int(1e4):
N = int(num_wanted /10)
else:
pass
while has_get_num < num_wanted:
itertimes += 1
if itertimes >= maxiter:
print("无法生成足够多的随机数")
break
else:
pass
#拒绝方法：cg(x)<=f(x)
sample_try = np.random.random(N)
sample_try = (x_end - x_start) * sample_try + x_start
sample_test= f1(sample_try)
c = 1 + max(f1(x))
sample_test *= c
sample_help = np.random.random(N) * (max(sample_test)-0) + 0
sample = sample_try[sample_help num_wanted:
result[has_get_num:num_wanted-1] = sample[:num_wanted-has_get_num-1]
else:
result[has_get_num:has_get_num+sample.size] = sample
has_get_num += sample.size
plt.title("随机数生成分布")
plt.xlabel("随机数")
plt.ylabel("比例")
plt.plot(x, f1(x), color='red', label="待抽样分布", linewidth=2)
plt.hist(result, bins=200, label="随机数柱状分布图", color='green', density=True)
plt.legend(prop={'size': 10})
plt.pause(0.01)
if not os.path.exists(figure_save_path):
os.makedirs(figure_save_path)
else:
pass
plt.savefig(os.path.join(figure_save_path, "随机数生成分布" + ".jpg"))
return result
c = main()
多维随机向量的抽样
- 拒绝抽样方法
- 条件密度抽样方法