如何生成圆周内的齐次泊松点过程?
How to Generate a Homogeneous Poisson Point Process in a circle?
我想在圆心 (0,0) 和半径 R=200 的圆 C 中生成 N 个点。这些点服从泊松分布。换句话说,我想在C中生成N个齐次泊松点过程(HPPP)。
我找到这篇论文 Generating Homogeneous Poisson Processes 。在第 2 节中,正是我想要的。具体来说,在第 4 页中,算法 3 生成 C 内部的点 HPPP。
我在 Python 中实现了如下代码:
""" Main Codes """
import matplotlib.pyplot as plt
import numpy as np
lamb = 0.0005 # the rate
pi = np.pi # pi = 3.14...
r = 200 # the radius of the circle C
mean = lamb * pi * r ** 2 # the mean of the Poisson random variable n
n = np.random.poisson(mean) # the Poisson random variable (i.e., the number of points inside C)
u_1 = np.random.uniform(0.0, 1.0, n) # generate n uniformly distributed points
radii = np.zeros(n) # the radial coordinate of the points
for i in range(n):
radii[i] = r * (np.sqrt(u_1[i]))
u_2 = np.random.uniform(0.0, 1.0, n) # generate another n uniformly distributed points
angle = np.zeros(n) # the angular coordinate of the points
for i in range(n):
angle[i] = 2 * pi * u_2[i]
""" Plots """
fig = plt.gcf()
ax = fig.gca()
plt.xlim(-300, 300)
plt.ylim(-300, 300)
circ = plt.Circle((0, 0), radius=200, color='r', linewidth=2, fill=False)
plt.polar(angle, radii, 'bo')
ax.add_artist(circ)
plt.show()
首先,我看不到圆圈内的点。其次,我不知道为什么点不能在圆内正确生成。我的代码有问题吗?
输出如下:圆圈 C 为红色。
我找到了答案。我只是将极坐标转换为笛卡尔坐标,然后我用 plt.plot()
绘图而不是 plt.polar()
.
# Cartesian Coordinates
x = np.zeros(n)
y = np.zeros(n)
for i in range(n):
x[i] = radii[i] * np.cos(angle[i])
y[i] = radii[i] * np.sin(angle[i])
plt.plot(x,y,'bo')
所以我得到了想要的输出。
晚了几年,但我几个月前就写过这个问题;看到这个 post.
对于未来的读者,这是我的代码:
import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
#Simulation window parameters
r=1;
xx0=0; yy0=0; #centre of disk
areaTotal=np.pi*r**2; #area of disk
#Point process parameters
lambda0=100; #intensity (ie mean density) of the Poisson process
#Simulate Poisson point process
numbPoints = scipy.stats.poisson( lambda0*areaTotal ).rvs()#Poisson number of points
theta = 2*np.pi*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))#angular coordinates of Poisson points
rho = r*np.sqrt(scipy.stats.uniform.rvs(0,1,((numbPoints,1))))#radial coordinates of Poisson points
#Convert from polar to Cartesian coordinates
xx = rho * np.cos(theta)
yy = rho * np.sin(theta)
#Shift centre of disk to (xx0,yy0)
xx=xx+xx0; yy=yy+yy0;
#Plotting
plt.scatter(xx,yy, edgecolor='b', facecolor='none', alpha=0.5 )
plt.xlabel("x"); plt.ylabel("y")
plt.axis('equal')
一个例子:
A realization of a Poisson point process on a disk
我想在圆心 (0,0) 和半径 R=200 的圆 C 中生成 N 个点。这些点服从泊松分布。换句话说,我想在C中生成N个齐次泊松点过程(HPPP)。
我找到这篇论文 Generating Homogeneous Poisson Processes 。在第 2 节中,正是我想要的。具体来说,在第 4 页中,算法 3 生成 C 内部的点 HPPP。
我在 Python 中实现了如下代码:
""" Main Codes """
import matplotlib.pyplot as plt
import numpy as np
lamb = 0.0005 # the rate
pi = np.pi # pi = 3.14...
r = 200 # the radius of the circle C
mean = lamb * pi * r ** 2 # the mean of the Poisson random variable n
n = np.random.poisson(mean) # the Poisson random variable (i.e., the number of points inside C)
u_1 = np.random.uniform(0.0, 1.0, n) # generate n uniformly distributed points
radii = np.zeros(n) # the radial coordinate of the points
for i in range(n):
radii[i] = r * (np.sqrt(u_1[i]))
u_2 = np.random.uniform(0.0, 1.0, n) # generate another n uniformly distributed points
angle = np.zeros(n) # the angular coordinate of the points
for i in range(n):
angle[i] = 2 * pi * u_2[i]
""" Plots """
fig = plt.gcf()
ax = fig.gca()
plt.xlim(-300, 300)
plt.ylim(-300, 300)
circ = plt.Circle((0, 0), radius=200, color='r', linewidth=2, fill=False)
plt.polar(angle, radii, 'bo')
ax.add_artist(circ)
plt.show()
首先,我看不到圆圈内的点。其次,我不知道为什么点不能在圆内正确生成。我的代码有问题吗?
输出如下:圆圈 C 为红色。
我找到了答案。我只是将极坐标转换为笛卡尔坐标,然后我用 plt.plot()
绘图而不是 plt.polar()
.
# Cartesian Coordinates
x = np.zeros(n)
y = np.zeros(n)
for i in range(n):
x[i] = radii[i] * np.cos(angle[i])
y[i] = radii[i] * np.sin(angle[i])
plt.plot(x,y,'bo')
所以我得到了想要的输出。
晚了几年,但我几个月前就写过这个问题;看到这个 post.
对于未来的读者,这是我的代码:
import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
#Simulation window parameters
r=1;
xx0=0; yy0=0; #centre of disk
areaTotal=np.pi*r**2; #area of disk
#Point process parameters
lambda0=100; #intensity (ie mean density) of the Poisson process
#Simulate Poisson point process
numbPoints = scipy.stats.poisson( lambda0*areaTotal ).rvs()#Poisson number of points
theta = 2*np.pi*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))#angular coordinates of Poisson points
rho = r*np.sqrt(scipy.stats.uniform.rvs(0,1,((numbPoints,1))))#radial coordinates of Poisson points
#Convert from polar to Cartesian coordinates
xx = rho * np.cos(theta)
yy = rho * np.sin(theta)
#Shift centre of disk to (xx0,yy0)
xx=xx+xx0; yy=yy+yy0;
#Plotting
plt.scatter(xx,yy, edgecolor='b', facecolor='none', alpha=0.5 )
plt.xlabel("x"); plt.ylabel("y")
plt.axis('equal')
一个例子: A realization of a Poisson point process on a disk