在 XY 网格中遵循 1/R 密度分布的点?
问题描述
我有一个 XY 网格,其中一些网格点具有分配给它们的某些值,在这种情况下,每个值都表示一定的质量,所以基本上是网格中的点质量.我现在想要获得一组遵循 1/R 密度分布的点,其中 R 是到中心的距离,所以 R = sqrt(x^2 + y^2).通过密度分布,我的意思是点的数量必须下降为 1/R.我将如何进行编码?
I have a XY grid with some gridpoints having certain values assigned to them, in this case, each value means a certain mass, so basically point masses in a grid. I now want to obtain a set of points which follow a density distribution of 1/R, where R is the distance from the center, so R = sqrt(x^2 + y^2). By density distribution, I mean the number of points has to fall off as 1/R. How would I go about coding this?
我的代码如下:
import numpy as np
x = np.linspace(-50,50,100)
y = np.linspace(-50,50,100)
X, Y = np.meshgrid(x,y)
zeta_a = (25,25)
zeta_b = (-10,5)
M_a = 150
M_b = 150
zeta_a 和 zeta_b 对应于 2 个质量为 150 单位的点质量.我还需要使用这些点进行后续计算,所以我也想知道如何使用更通用的格式,而不是使用 'a'、'b'、'c' 来表示 n 点质量.
The zeta_a and zeta_b correspond to 2 point masses having masses of 150 units. I also need to perform follow up calculations using these points, so i'd also like to know how to use a more general format rather than using 'a','b','c' for n-point masses.
感谢您的帮助.
解决方案
假设我理解你的问题(如果没有,欢迎评论):
Assuming I understood your question (if not comments are welcomed):
创建任何给定分布的方法是对分布 CDF 的逆进行插值.这是我的功能:
The way to create any given distribution is by interpolating over the inverse of the distribution CDF. This is my function to do it:
import numpy as np
import matplotlib.pyplot as plt
def randdist(PDF, x, n):
"""Create a distribution following PDF(x). PDF and x
must be of the same length. n is the number of samples."""
fp = np.random.rand(n,)
CDF = np.cumsum(PDF)
return np.interp(fp, CDF, x)
现在,在您的情况下,我们将在极坐标中工作,R 分布为 1/r,Theta 均匀分布:
Now, in your case we're going to work in polar coordinates with R distributed as 1/r and Theta uniformly distributed:
num = 1000 # The number of points
r = np.linspace(-50, 50, 100)
PDF = np.abs(1/r)
PDF = PDF/np.sum(PDF) # PDF should be normalized
R = randdist(PDF, r, num)
Theta = 2*np.pi*np.random.rand(num,)
现在让我们创建点 x 和 y 向量
Now let's create the points x and y vectors
x = [R[k]*np.cos(Theta[k]) for k in range(num)]
y = [R[k]*np.sin(Theta[k]) for k in range(num)]
绘制
plot(x,y,'.')
请注意,在我的回答中,在 r=50 处有一个硬截止.有办法克服这个问题,但现在我保持原样.
Note that in my answer there is a hard cutoff at r=50. There are ways to overcome this but for now I leave it as it is.
现在您似乎还想将点嵌入到 2D 网格中,就像直方图一样.您可以使用
Now you seem to also want to embed the points inside a 2D grid, much like a histogram. You can do that using
z, _, _ = np.histogram2d(x, y, [100, 100])
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