Timeline for How to sample uniformly from the surface of a hyper-ellipsoid (constant Mahalanobis distance)?
Current License: CC BY-SA 4.0
28 events
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| Sep 21, 2018 at 9:54 | comment | added | sachin vernekar | @Martin Weterings Got it! Thanks for the clarification. | |
| Sep 21, 2018 at 8:48 | comment | added | Sextus Empiricus | @sachinvernekar I was not explaining an approximate method. The rejection sampling method that I describe will give a sample according to the exact uniform distribution. It is just that the method is inefficient. Also possibly, depending on how you are applying the sample, you may not need a uniform sample and there could be other ways to solve your problem (I mean, not the sampling on an ellipsoid problem, but the underlying problem). | |
| Sep 21, 2018 at 6:09 | answer | added | Sextus Empiricus | timeline score: 5 | |
| Sep 21, 2018 at 5:58 | comment | added | sachin vernekar | @Alex R. The link gives a method to sample from inside of a hyperellipsoid and not from the surface. | |
| Sep 21, 2018 at 0:00 | history | tweeted | twitter.com/StackStats/status/1042926535831896064 | ||
| Sep 20, 2018 at 21:11 | history | edited | amoeba | CC BY-SA 4.0 |
edited tags
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| Sep 20, 2018 at 20:53 | history | reopened |
kjetil b halvorsen♦ mkt mdewey Sextus Empiricus whuber♦ |
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| Sep 20, 2018 at 20:26 | history | edited | sachin vernekar | CC BY-SA 4.0 |
added 141 characters in body
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| Sep 20, 2018 at 20:23 | comment | added | sachin vernekar | @Martinjin Weterings Yes, sample uniformly such that each area element dA of the hyper-surface contains the same probability mass. I am using it in the context of anomaly detection using a classifier. I understand that there is an approximate way to do this as you mentioned, but I just wanted to know if sampling uniformly was possible at all otherwise. | |
| Sep 20, 2018 at 11:54 | comment | added | Sextus Empiricus | Why, how and where are you going to apply this uniform sample? Such information may help to come with a best/sufficient strategy. For instance, when the different ellipsoid axes are not to much different then you can use rejection sampling by (1) sampling on a sphere, (2) squeezing it into an ellipsoid, (3) compute the rate by which the surface area was squeezed (4) reject samples according to the inverse of that rate. | |
| Sep 20, 2018 at 11:29 | comment | added | Sextus Empiricus | Do you want to have the sample uniform on the surface of an ellipsoid, in the sense that each area element dA of the hyper-surface contains the same probability mass? | |
| Sep 18, 2018 at 22:51 | comment | added | sachin vernekar | I understand that uniformly sampling from the surface of the sphere and then mapping it to ellipsoid won't give uniform samples on the ellipsoid. So I need a method that does sample uniformly from the surface of an ellipsoid. | |
| Sep 18, 2018 at 21:40 | comment | added | whuber♦ | These surfaces are rarely spheres. If you do want to sample from a sphere, see the generalization at stats.stackexchange.com/questions/310346. Note that sampling uniformly from a standardized sphere, as described at stats.stackexchange.com/questions/62092/…, and then transforming the sample back to the original units does not give a uniform sample of the ellipsoid. That's why your question still is unclear. | |
| Sep 18, 2018 at 21:28 | comment | added | sachin vernekar | @whuber I want to do exactly the same as the one mentioned in corysimon.github.io/articles/uniformdistn-on-sphere but for a hyper-ellipsoid. I am afraid I can't make it even more precise with my limited knowledge in statistics. | |
| Sep 18, 2018 at 20:58 | comment | added | whuber♦ | I'm afraid it's not clear, because--as I have attempted to describe--"uniformly" depends on how you represent the surface. Different forms of representation give rise to different probability distributions. | |
| Sep 18, 2018 at 20:39 | comment | added | sachin vernekar | @whuber I need to sample uniformly from the surface given by the equation in my question. This is similar to the case where you want to sample uniformly from the surface of a sphere. This means all the points in the surface have equal chances. Let me know if this is clear. | |
| Sep 18, 2018 at 13:58 | comment | added | whuber♦ | That's still unclear, though: the problem is that you haven't specified what density to use for sampling from that surface. "Uniformly" could mean various different things, as I tried to explain in the example I gave. | |
| Sep 18, 2018 at 7:30 | review | Reopen votes | |||
| Sep 18, 2018 at 13:58 | |||||
| Sep 18, 2018 at 7:24 | comment | added | sachin vernekar | @whuber I have made the question more clear now. I mean you fix the standardized distance (Mahalanobis distance) from the mean and sample from the surface of the hyper-ellipsoid. But note in my case, the hyper-ellipsoid is rotated. | |
| Sep 18, 2018 at 7:13 | history | edited | sachin vernekar | CC BY-SA 4.0 |
made the question more clear.
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| Sep 17, 2018 at 14:08 | history | closed | whuber♦ | Needs details or clarity | |
| Sep 17, 2018 at 14:07 | comment | added | whuber♦ | I think we will need a suitable definition of "uniformly." The reason is this: in two dimensions, this set of points lies along some ellipse. Is one supposed to sample from that ellipse in such a way that equal lengths have equal chances, or that equal angles have equal chances, or so that equal lengths when the variables are standardized have equal chances, or in some other way? If you could explain what this sampling aims to achieve, that might give us enough information to know what you are trying to ask. | |
| Sep 17, 2018 at 0:46 | comment | added | sachin vernekar | @Kevin Li Yes, that is right. | |
| Sep 16, 2018 at 22:17 | comment | added | Kevin Li | Correct me if I'm wrong: are you asking "given a random variable $ X $, how can I uniformly sample from the points that are a given Mahalanobis distance $ c $ away from $ \mathbb{E}[X] $?" | |
| S Sep 16, 2018 at 12:31 | history | suggested | abunickabhi | CC BY-SA 4.0 |
unclear problem addressal
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| Sep 16, 2018 at 10:54 | review | Suggested edits | |||
| S Sep 16, 2018 at 12:31 | |||||
| Sep 16, 2018 at 7:10 | review | First posts | |||
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| Sep 16, 2018 at 7:09 | history | asked | sachin vernekar | CC BY-SA 4.0 |