Timeline for Estimate the expected distance between two random points on the unit $n$-sphere
Current License: CC BY-SA 4.0
6 events
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| Nov 21, 2020 at 16:19 | comment | added | Penelope Benenati | Thank you @flinty . | |
| Nov 21, 2020 at 16:19 | comment | added | flinty |
@PenelopeBenenati for $n>3$ to create a point on the $n$-sphere, just generate $n$ variates from a normal distribution, and Normalize the point: points = Partition[ Normalize /@ RandomVariate[NormalDistribution[0, 1], {1000000, n}], 2]; distances = EuclideanDistance @@@ points;
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| Nov 21, 2020 at 16:17 | comment | added | flinty |
For other dimensions, the approximate distribution looks a bit like TransformedDistribution[2 x, x \[Distributed] BetaDistribution[a, b]] over domain $0<x<2$, where for $n=2, \alpha=1, \beta=1/2$, $n=3, \alpha=2, \beta=1$, $n=4, \alpha=3, \beta=3/2$, ... However, these are approximate - I'm not sure if those parameters are exact.
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| Nov 21, 2020 at 16:17 | comment | added | Penelope Benenati | Thank you @flinty ! My problem is to find an estimation when the number of dimensions $n$ is very large, without writing a list of constraints, i.e., one constraint per dimension when $n\gg1$. How can we solve this problem? Maybe it is worth to create a list of constraints first, and then use it with a wolfram command. | |
| Nov 21, 2020 at 15:40 | history | edited | flinty | CC BY-SA 4.0 |
added 220 characters in body
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| Nov 21, 2020 at 15:21 | history | answered | flinty | CC BY-SA 4.0 |