从给定的二维分布函数中绘制随机数据

Question

我有一个理论分布，我想在2D空间中随机采样以进行以下分布：

def p(z,m):
    E = { 'ft':0.55, 'alpha': 2.99, 'z0':0.191, 'km':0.089, 'kt':0.25 }
    S = { 'ft':0.39, 'alpha': 2.15, 'z0':0.121, 'km':0.093, 'kt':-0.175 }
    I={ 'ft':0.06, 'alpha': 1.77, 'z0':0.045, 'km':0.096, 'kt':0.0 }
    Evalue=E['ft']*np.exp(-1*E['kt']*(m-20))*z**E['alpha']*np.exp(-1*(z/(E['z0']+E['km']*(m-20)))**E['alpha'])
    Svalue=S['ft']*np.exp(-1*S['kt']*(m-20))*z**S['alpha']*np.exp(-1*(z/(S['z0']+S['km']*(m-20)))**S['alpha'])
    Ivalue=I['ft']*np.exp(-1*I['kt']*(m-20))*z**I['alpha']*np.exp(-1*(z/(I['z0']+I['km']*(m-20)))**I['alpha'])
    value=Evalue+Svalue+Ivalue
    return value

更新： 我发现逆变换采样是从概率分布中采样数据的合适方法。 我如何在python中为2D数据编程这个方法，或者我可以使用任何库吗？

Answer 1

看看马尔可夫链蒙特卡洛（MCMC）的方法。基本上你会跳过（z，m）点的空间。无论你身在何处，你总是接受一个增加p（z，m）的跳跃。你接受一个以一定概率降低p（z，m）的跳跃。有一个Python库PyMC来执行该过程。

Answer 2

如果你想从p（z，m）中随机抽样一个值，那么实现它的一个简单方法就是在python中使用'random'模块。我使用numpy的随机版本来展示这个想法：

import numpy as np
import matplotlib.pyplot as plt

def p(z,m):
    E = { 'ft':0.55, 'alpha': 2.99, 'z0':0.191, 'km':0.089, 'kt':0.25 }
    S = { 'ft':0.39, 'alpha': 2.15, 'z0':0.121, 'km':0.093, 'kt':-0.175 }
    I={ 'ft':0.06, 'alpha': 1.77, 'z0':0.045, 'km':0.096, 'kt':0.0 }
    Evalue=E['ft']*np.exp(-1*E['kt']*(m-20))*z**E['alpha']*np.exp(-1*(z/(E['z0']+E['km']*(m-20)))**E['alpha'])
    Svalue=S['ft']*np.exp(-1*S['kt']*(m-20))*z**S['alpha']*np.exp(-1*(z/(S['z0']+S['km']*(m-20)))**S['alpha'])
    Ivalue=I['ft']*np.exp(-1*I['kt']*(m-20))*z**I['alpha']*np.exp(-1*(z/(I['z0']+I['km']*(m-20)))**I['alpha'])
    value=Evalue+Svalue+Ivalue
    return value

# Define the number of iterations you want for each variable    
num_iter_m = 50
num_iter_z = 50

# I then set rand_m to go from 20 to 30, as your function fails for <20
rand_m = (np.random.random(num_iter_m)*10)+20

# z goes from the range 0 - 1
rand_z = (np.random.random(num_iter_z))

# Note, I am sampling from a uniform distribution for m and z. You can use more complicated functions, i.e., Gaussian/Normal shapes or even user defined.

rand_p = np.zeros((len(rand_z), len(rand_m)))

# Fill a grid with the random p(z,m) values
for i in range(len(rand_z)):
  for j in range(len(rand_m)):
    rand_p[i][j] = p(rand_z[i], rand_m[j])

# Plot
fig = plt.figure(0)

ax1 = fig.add_subplot(211)
ax1.scatter(rand_z, rand_m)
ax1.set_xlabel("z")
ax1.set_ylabel("m")

ax2 = fig.add_subplot(212)
cf = ax2.contourf(rand_z, rand_m, rand_p)
ax2.set_xlabel("z")
ax2.set_ylabel("m")

colbar = plt.colorbar(cf)
colbar.set_label("p(z,m)")

plt.show()

以更复杂的方式使用的特定模块将是例如PyMC（ https://github.com/pymc-devs/pymc）或主持人（http://dan.iel.fm/emcee/current/）。

如果你想用2D函数p（z，m）对z和m加权进行采样，这有点复杂。