Timeline for What is the nearest-neighbor distribution in this picture?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 27, 2018 at 21:44 | vote | accept | Tom Solberg | ||
| Aug 27, 2018 at 5:29 | comment | added | Anthony Quas | I see. Yes they are dependent in a single realization (=choice of points). For example if you take the closest pair of points, then the mutual distance is the minimum distance for both of them. But: since you are taking a large number of points, what you are really looking at is the distribution of the nearest neighbour distance from a single point. (This is essentially the strong law of large numbers: the number of points for which the nearest point is at distance in the range $[a/\sqrt n,b/\sqrt n]$ is $n$ times the probability that a single point has nearest point in that range). | |
| Aug 27, 2018 at 4:05 | comment | added | Tom Solberg | Sorry, this is kind of unfamiliar territory for me. What I meant was, when I built my histogram, I sampled $n$ points and did a histogram of each of their nearest-neighbor distances. I'm having difficulty resolving the fact that those $n$ distances are dependent, and I think they continue to be dependent even if we take the Poisson process approximation, no? | |
| Aug 27, 2018 at 2:47 | comment | added | Anthony Quas | The dependency between points went away when I moved from having exactly $n$ points in the square to a Poisson process with intensity $n$ in the square. This is a very mild change. Things got even better when I moved to a Poisson process with intensity $n$ in the plane as that made the edge effects go away. | |
| Aug 27, 2018 at 2:33 | comment | added | Tom Solberg | I see, is there a standard way to resolve the dependency between the samples? In other words, I understand that if I choose a single point and look at the nearest neighbor distance, I get the function you wrote, but does the fact that I took the nearest-neighbor distances from all $n$ points change anything? | |
| Aug 27, 2018 at 2:19 | history | edited | Anthony Quas | CC BY-SA 4.0 |
added 123 characters in body
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| Aug 27, 2018 at 2:12 | history | answered | Anthony Quas | CC BY-SA 4.0 |