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 Post subject: Orthogonal projection of a random vector on the spherePosted: Thu, 7 Jun 2012 11:25:56 UTC
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Joined: Thu, 7 Jun 2012 11:13:33 UTC
Posts: 4
Hey there,

I have a fixed orthogonal projection P on R^d and a uniformly distributed random vector X on the unit sphere in R^d. Now I am interested in the distribution of the norm (or squared norm) of the orthogonal projection of X, i.e., I am looking for the distribution of \|P(X)\| or \|P(X)\|^2.

If each entry of X is independent and standard normally distributed, then \|P(X)\|^2 is chi-square distributed. Moreover, if I normalized such X, i.e., use Y=X/\|X\|, then Y is uniformly distributed on the sphere. However, I do not know how this may change \|P(X)\|^2.

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 Post subject: Re: Orthogonal projection of a random vector on the spherePosted: Thu, 7 Jun 2012 12:27:12 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6010
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Lucky Hans wrote:
Hey there,

I have a fixed orthogonal projection P on R^d and a uniformly distributed random vector X on the unit sphere in R^d. Now I am interested in the distribution of the norm (or squared norm) of the orthogonal projection of X, i.e., I am looking for the distribution of \|P(X)\| or \|P(X)\|^2.

If each entry of X is independent and standard normally distributed, then \|P(X)\|^2 is chi-square distributed. Moreover, if I normalized such X, i.e., use Y=X/\|X\|, then Y is uniformly distributed on the sphere. However, I do not know how this may change \|P(X)\|^2.

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 Post subject: Re: Orthogonal projection of a random vector on the spherePosted: Thu, 7 Jun 2012 12:41:15 UTC
 S.O.S. Newbie

Joined: Thu, 7 Jun 2012 11:13:33 UTC
Posts: 4
well, I am sure that is changes a lot. In fact, I know that the expectation is k/d, where d is the dimension of the ambient space and k the rank of the orthogonal projection. I also have some deviation estimates that hold with high probability. Nevertheless, I am interested in the actual distribution and was wondering if it is possible to determine it analytically.

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 Post subject: Re: Orthogonal projection of a random vector on the spherePosted: Thu, 7 Jun 2012 13:57:02 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6010
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Lucky Hans wrote:
well, I am sure that is changes a lot. In fact, I know that the expectation is k/d, where d is the dimension of the ambient space and k the rank of the orthogonal projection. I also have some deviation estimates that hold with high probability. Nevertheless, I am interested in the actual distribution and was wondering if it is possible to determine it analytically.

You can.

Without loss of generality, we can assume P is projection ontothe first k components of . So changing to spherical polars by

Since r=1, the area element is , now integrate the coordinates, remembering except parametrises almost every point on the unit sphere. Finally, change coordinates back.

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