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I have an algorithm that embeds data points into Euclidean space. If I norm these points then they will lie on the unit $n$-sphere, where $n+1$ is the dimensionality of the embedding space (generally 512 in my case). My problem is: I want to determine if a certain subset of these points is distributed as a uniform Poisson process. The idea here is I know a priori that a subset of points belongs to a certain class and if I can show that these points are not uniformly distributed then this suggests my algorithm is properly embedding similar points close together.

I'm not sure how to formulate this problem, and in general the literature on Poission processes in $n$-spherical spaces is minimal. Baddeley gives an example here (example 1.3 and figure 13) but it is only on the $2$-sphere and doesn't generalize to higher dimensional spaces. I understand that I could generate a Poisson process on the $n$-sphere by changing his measure function $$\Lambda(S)=4\pi\beta$$ to $$\Lambda(S)=\frac{2\pi^\frac{n+1}{2}}{\Gamma(\frac{n+1}{2})}\beta$$ (i.e. the surface area of a unit $n$-sphere with Poisson process intensity $\beta$). However, I don't know how to actually test the uniform Poisson process null hypothesis given a set of data points on the $n$-sphere. I'm not testing whether the number of points is expected under a uniform Poisson process, but rather if the spatial distribution of these points on the $n$-sphere follows a uniform Poisson process.

Any help is appreciated!

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    $\begingroup$ As described at stats.stackexchange.com/a/7984/919, you might consider examining Ripley's $K$ function. Although my explanation in that post focuses on the two-sphere, it is stated in a way that makes its generalization to higher dimensions obvious. A quick calculation suggests the expected number of points within spherical distance $\rho$ of a point on the unit $n-1$-sphere is given in terms of an incomplete Beta$((n-1)/2,1/2)$ function. BTW, the sphere in $n$ space is called the $n-1$-sphere, not the $n$-sphere. $\endgroup$ – whuber Dec 1 '19 at 19:12
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    $\begingroup$ Is the number of points random? Or is it fixed? It seems you may want to assume the points are a fixed-size sample of iid uniform draws, which is similar to a homogeneous Poisson process but it requires one to condition on the total count. Reference: maths.qmul.ac.uk/~ig/MAS338/PP%20and%20uniform%20d-n.pdf $\endgroup$ – eric_kernfeld Dec 1 '19 at 19:34
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Your problem can be more simply phrased as testing for a uniform distribution on the sphere. (Working with point processes is not necessary.) Here's a way to test uniformity against a simple parametric alternative. Compute the likelihoods of your points assuming a uniform distribution and then again assuming a Von Mises-Fisher distribution, which is a unimodal family that allows higher density somewhere on the sphere and includes the uniform as a special case. Use a likelihood ratio test between those two models.

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One of the simplest ways to detect departure from CSR (complete spatial randomness -- uniform Poisson process) is by a quadrat test which extends straightforwardly to the case of the n-sphere. You can simply cut the sphere into smaller regions and calculate the expected versus observed number of points in each region and evaluate the test statistic. However, this test will fail to reveal departure from CSR if the points are fairly uniformly distributed, and the departure is in terms of the finer scale interpoint distances. In that case using a functional summary statistics such as the $K$-function suggested by @whuber may help you. This will of course be more involved to implement than the quadrat test.

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