3 ms·

> So I would anticipate the clumping to even out as the sample size increases. Clumping won't even out. A useful way of thinking about it is in terms of the di

by Osmium 11y ago

> So I would anticipate the clumping to even out as the sample size increases. Clumping won't even out. A useful way of thinking about it is in terms of the distance between samples (e.g. a radial distribution function). If you don't have clumping, that distribution of distances is decidedly non-random and has peaks. Additionally, there's no specific length-scale for the clumping to happen on, it's essentially fractal in nature. (Happy to be corrected if I'm wrong on this.)

eru 11y ago
I think in order to pass any judgement on the correctness, you'd have to make your remarks more precise. Are you talking about a random distribution of points on a line / square, or about rolling dice?
- Osmium 11y ago
  > Are you talking about a random distribution of points on a line / square, or about rolling dice? Right, exactly. I made my comment in reply to the grandparent problem talking about dots on a screen. For a discrete distribution like a dice roll in the parent post it'd be a different matter. (I was going to edit my comment to clarify, but the edit window had expired.) Which serves to illustrate why it's important to be precise when talking about what distribution you're sampling. Also, trying to run this experiment in practice is also problem even for dots on a screen, because eventually the finite display size of your dots will become an issue, as will floating point arithmetic (since you only really have a finite number of floats in a given range).
  - eru 11y ago
    Thanks for clarifying. > Also, trying to run this experiment in practice is also problem even for dots on a screen, because eventually the finite display size of your dots will become an issue, as will floating point arithmetic (since you only really have a finite number of floats in a given range). Yes, but that's not a problem for the theoretical analysis of fractal dimension. Uniformly placed dots do have a characteristic density and a relatively simple distribution of nearest neighbour distances. Even the clumpiness doesn't lead to scale invariant lumps. The lumps get absolutely bigger on bigger scales, but relatively smaller and smoother. It would be useful to do the analysis properly, though.

by Osmium 11y ago

eru 11y ago

I think in order to pass any judgement on the correctness, you'd have to make your remarks more precise. Are you talking about a random distribution of points on a line / square, or about rolling dice?

Osmium 11y ago
> Are you talking about a random distribution of points on a line / square, or about rolling dice? Right, exactly. I made my comment in reply to the grandparent problem talking about dots on a screen. For a discrete distribution like a dice roll in the parent post it'd be a different matter. (I was going to edit my comment to clarify, but the edit window had expired.) Which serves to illustrate why it's important to be precise when talking about what distribution you're sampling. Also, trying to run this experiment in practice is also problem even for dots on a screen, because eventually the finite display size of your dots will become an issue, as will floating point arithmetic (since you only really have a finite number of floats in a given range).
- eru 11y ago
  Thanks for clarifying. > Also, trying to run this experiment in practice is also problem even for dots on a screen, because eventually the finite display size of your dots will become an issue, as will floating point arithmetic (since you only really have a finite number of floats in a given range). Yes, but that's not a problem for the theoretical analysis of fractal dimension. Uniformly placed dots do have a characteristic density and a relatively simple distribution of nearest neighbour distances. Even the clumpiness doesn't lead to scale invariant lumps. The lumps get absolutely bigger on bigger scales, but relatively smaller and smoother. It would be useful to do the analysis properly, though.