I have many vectors $a_i\in\mathbb{R}^p:||a||=1,1\leq i \leq n$. I would like to test whether the $a_i$'s are randomly spread out on $S^{p-1}$ (the $p$ variate unit circle). Can anyone point to a test for that?
Thanks in advance,
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1$\begingroup$ How is what you're wanting to do different from a goodness-of-fit test that your data come from the uniform measure on the $S^{p-1}$ sphere? $\endgroup$– cardinalJul 1, 2012 at 23:59
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$\begingroup$ Could work. How would you test this in practice? $\endgroup$– user603Jul 2, 2012 at 7:41
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$\begingroup$ I offer a diagnostic plot at stats.stackexchange.com/a/7984/919. $\endgroup$– whuber ♦Nov 24, 2022 at 18:58
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1 Answer
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You could transform your data points to spherical coordinates so that you get $p-1$ angles. Your null hypothesis is equivalent to the fact that those angles are independent and uniformly distributed. So you can do a goodness of fit test.
Now there is the complication that you have a $p-1$-dimensional distribution. This paper shows a multidimensional version of the Smirnov statistic that may come in handy. Otherwise, another idea that comes to my mind woud be to scale the angles to $(0,1)$, apply the inverse erf function and do a test for multivariate normality.
Erratum: it is not true that the angles will be uniform as shown in this answer. One angle is uniform between $0$ and $2\pi$, and the cosine of the others is uniform between $-1$ and $1$.
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$\begingroup$ in your second paragraph, by scaling the angles to 0-1, you mean dividing each component of the p-1 dimensional angle vector by pi? $\endgroup$– user603Jun 3, 2012 at 9:26
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$\begingroup$ Depends on your parametrization. For example in 3D, spherical coordinates are usually given as an angle between $-\pi$ and $\pi$ (longitude) and another one between $-\pi/2$ and $\pi/2$ (latitude). For longitude add $\pi$ and divide by $2\pi$ and for latitude add $\pi/2$ and divide by $\pi$. $\endgroup$ Jun 3, 2012 at 10:42