The "north pole problem" and random orthogonal matrices
This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $Γ$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=Γx_0$ that is uniformly distributed on the unit sphere. Now use the...
Main Authors: | , |
---|---|
Format: | Report |
Language: | unknown |
Published: |
arXiv
2008
|
Subjects: | |
Online Access: | https://dx.doi.org/10.48550/arxiv.0811.2678 https://arxiv.org/abs/0811.2678 |
id |
ftdatacite:10.48550/arxiv.0811.2678 |
---|---|
record_format |
openpolar |
spelling |
ftdatacite:10.48550/arxiv.0811.2678 2023-05-15T17:39:40+02:00 The "north pole problem" and random orthogonal matrices Eaton, Morris L. Muirhead, Robb J. 2008 https://dx.doi.org/10.48550/arxiv.0811.2678 https://arxiv.org/abs/0811.2678 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Statistics Theory math.ST Probability math.PR Computation stat.CO FOS Mathematics FOS Computer and information sciences Preprint Article article CreativeWork 2008 ftdatacite https://doi.org/10.48550/arxiv.0811.2678 2022-04-01T15:05:49Z This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $Γ$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=Γx_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=Γu=Γ^2 x_0$. Simulations reported in Marzetta et al (2002) suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, morever, that $w=Γ^3 x_0$ has higher probability of being closer to the poles $\pm x_0$ than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension $p\ge 3$, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in $R^p$, where $p\ge 3$. What are the distributions of $U_2=x'Γ^2 x$ and $U_3=x'Γ^3 x$? It is clear by orthogonal invariance that these distribution do not depend on x, so that we can, without loss of generality, take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then $x_0'Γ^ k x_0$ is the first component of the vector $Γ^k x_0$. We derive stochastic representations for the exact distributions of $U_2$ and $U_3$ in terms of random variables with known distributions. Report North Pole DataCite Metadata Store (German National Library of Science and Technology) North Pole |
institution |
Open Polar |
collection |
DataCite Metadata Store (German National Library of Science and Technology) |
op_collection_id |
ftdatacite |
language |
unknown |
topic |
Statistics Theory math.ST Probability math.PR Computation stat.CO FOS Mathematics FOS Computer and information sciences |
spellingShingle |
Statistics Theory math.ST Probability math.PR Computation stat.CO FOS Mathematics FOS Computer and information sciences Eaton, Morris L. Muirhead, Robb J. The "north pole problem" and random orthogonal matrices |
topic_facet |
Statistics Theory math.ST Probability math.PR Computation stat.CO FOS Mathematics FOS Computer and information sciences |
description |
This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $Γ$, and use it to "rotate" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=Γx_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=Γu=Γ^2 x_0$. Simulations reported in Marzetta et al (2002) suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, morever, that $w=Γ^3 x_0$ has higher probability of being closer to the poles $\pm x_0$ than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension $p\ge 3$, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in $R^p$, where $p\ge 3$. What are the distributions of $U_2=x'Γ^2 x$ and $U_3=x'Γ^3 x$? It is clear by orthogonal invariance that these distribution do not depend on x, so that we can, without loss of generality, take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then $x_0'Γ^ k x_0$ is the first component of the vector $Γ^k x_0$. We derive stochastic representations for the exact distributions of $U_2$ and $U_3$ in terms of random variables with known distributions. |
format |
Report |
author |
Eaton, Morris L. Muirhead, Robb J. |
author_facet |
Eaton, Morris L. Muirhead, Robb J. |
author_sort |
Eaton, Morris L. |
title |
The "north pole problem" and random orthogonal matrices |
title_short |
The "north pole problem" and random orthogonal matrices |
title_full |
The "north pole problem" and random orthogonal matrices |
title_fullStr |
The "north pole problem" and random orthogonal matrices |
title_full_unstemmed |
The "north pole problem" and random orthogonal matrices |
title_sort |
"north pole problem" and random orthogonal matrices |
publisher |
arXiv |
publishDate |
2008 |
url |
https://dx.doi.org/10.48550/arxiv.0811.2678 https://arxiv.org/abs/0811.2678 |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.0811.2678 |
_version_ |
1766140452521639936 |