多维环中无排斥的均匀抽样

Question

这个question中的算法告诉我们如何有效地从多维球中采样。有没有一种方法可以同样有效地从多维环中进行采样，例如，r1<r<r2

我希望对这个标度函数做一个不太复杂的修改 r*(gammainc(s2/2,n/2).^(1/n))./sqrt(s2)是可能的。（平庸免责声明：甚至还没有为原始缩放函数计算代数/几何）。在

原始matlab代码复制：

function X = randsphere(m,n,r)

% This function returns an m by n array, X, in which 
% each of the m rows has the n Cartesian coordinates 
% of a random point uniformly-distributed over the 
% interior of an n-dimensional hypersphere with 
% radius r and center at the origin.  The function 
% 'randn' is initially used to generate m sets of n 
% random variables with independent multivariate 
% normal distribution, with mean 0 and variance 1.
% Then the incomplete gamma function, 'gammainc', 
% is used to map these points radially to fit in the 
% hypersphere of finite radius r with a uniform % spatial distribution.
% Roger Stafford - 12/23/05

X = randn(m,n);
s2 = sum(X.^2,2);
X = X.*repmat(r*(gammainc(s2/2,n/2).^(1/n))./sqrt(s2),1,n);

包含来自Daniel's answer的演示的等效python代码：

^{pr2}$

Answer 1

实际上，我最后使用了反cdf方法来处理均匀分布在球体上的点

像这样

def random_uniform_ring(center=np.array([0,0]),R=1,r=0,nsamples=1):
    """
    generate point uniformly distributed in a ring
    """
    nd = len(center)
    x = np.random.normal(size = (nsamples,nd))
    x /=np.linalg.norm(x,axis=1)[:,np.newaxis] #generate on unit sphere
    # using the inverse cdf method
    u = np.random.uniform(size=(nsamples))
    sc = (u*(R**nd-r**nd)+r**nd)**(1/nd) #this is inverse the cdf of ring volume as a function of radius
    return x*sc[:,None]+center

测试

^{pr2}$

这可能相当于@Mbo的答案，但不幸的是我没有时间去测试。如果有人能验证他的答案，我很乐意接受。在