Timeline for Bivariate normal covering circles and ellipses
Current License: CC BY-SA 4.0
21 events
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| S Nov 10, 2023 at 0:49 | history | bounty ended | feetwet | ||
| S Nov 10, 2023 at 0:49 | history | notice removed | feetwet | ||
| Nov 4, 2023 at 0:12 | vote | accept | feetwet | ||
| Nov 3, 2023 at 17:18 | history | edited | feetwet | CC BY-SA 4.0 |
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| Nov 3, 2023 at 3:44 | answer | added | dherrera | timeline score: 7 | |
| Nov 3, 2023 at 0:20 | history | edited | feetwet | CC BY-SA 4.0 |
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| S Nov 3, 2023 at 0:18 | history | bounty started | feetwet | ||
| S Nov 3, 2023 at 0:18 | history | notice added | feetwet | Draw attention | |
| Nov 3, 2023 at 0:09 | history | edited | feetwet | CC BY-SA 4.0 |
Added new discovery from research!
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| Oct 26, 2023 at 20:16 | history | edited | kjetil b halvorsen♦ |
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| Oct 24, 2023 at 15:41 | comment | added | feetwet | @MattF. even if we have to use numerical methods to produce values for a formula, having a formal expression would be helpful! | |
| Oct 24, 2023 at 12:46 | comment | added | user225256 | It looks like you can express the proportion in terms of the radius and a modified Bessel function ($I_0$), but there’s no nice formula to invert that function. | |
| Oct 19, 2023 at 19:12 | history | edited | feetwet | CC BY-SA 4.0 |
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| Oct 18, 2023 at 15:42 | history | edited | feetwet | CC BY-SA 4.0 |
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| Oct 18, 2023 at 15:34 | history | edited | feetwet | CC BY-SA 4.0 |
Checked and added observation from comments
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| Oct 18, 2023 at 14:28 | comment | added | PBulls | I played around with some simulations over the parameter space: imgur.com/a/woF0TGR. The different green lines are increasing ratios of variances away from the identity line, it goes 1.5, 2, 3, 5, 10 up to 100 but past 10 there's little absolute difference. Through sheer luck $p=0.8$ is roughly the point where it goes from over to under, but even that point depends on the ratio (it decreases slightly as ratio increases). Anyway, to me this proves that you'll definitely have to account for this variance ratio in some way. | |
| Oct 18, 2023 at 14:00 | comment | added | PBulls | Hmm, but for the same case R(0.8) covers 0.795 here. Increasing $p$ also increases the deficit. | |
| Oct 18, 2023 at 13:54 | comment | added | feetwet | @PBulls: I can't find an example where that circle undercovers. For example: if $\sigma_x=1000, \sigma_y=1$ then R(0.5, 707) covers 60% of samples, not 50%. | |
| Oct 18, 2023 at 4:58 | comment | added | PBulls | I'm not so sure that circle always overcovers? At least empirically it seems to slightly undercover if $\sigma_x$ and $\sigma_y$ differ by >2 orders of magnitude. I'm also not so convinced there would be a straightforward generalization, because the two axes are arbitrarily independent - solved by the ellipse - and the circle re-introduces a fixed dependence. | |
| Oct 17, 2023 at 23:36 | history | edited | feetwet | CC BY-SA 4.0 |
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| Oct 17, 2023 at 23:21 | history | asked | feetwet | CC BY-SA 4.0 |