Wikipedia:Reference desk/Archives/Mathematics/2007 August 24

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August 24

Fisher Information for Multivariate Gaussian Distribution

Hello,

I recently came across a useful expression for Fisher's Information matrix for the multivariate Gaussian distribution in the "Fisher Information" article on Wikipedia. The expression gives the elemets of the matrix as: ${\mathcal {I}}_{m,n}={\frac {\partial \mu }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \mu }{\partial \theta _{n}}}+{\frac {1}{2}}tr(\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{m}}}\Sigma ^{-1}{\frac {\partial \Sigma }{\partial \theta _{n}}})$

I would really like to see how this was derived or at least be able to cite a reference for this expression in a statistics book. Can anybody suggest a good reference or a good approach to obtaining this expression? I have searched extensively for a reference and found nothing.

Thanks,

Anon.

Grid Coloring

Take an NxN square grid (N>1), and color the vertices in two colors in any way, such that neither the square itself nor any subsquare has monochromatic corners. How would you find the greatest N for which this is possible? So far, I've found examples up to 11:

1 0 0 1 1 0 1 1 1 1 0

0 1 0 1 0 1 0 1 0 1 0

1 0 1 1 1 0 0 0 0 1 1

1 1 1 0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0 1 0 1

0 1 1 0 1 0 0 1 1 0 1

0 1 0 0 0 1 1 1 0 0 1

0 0 0 1 1 0 1 0 1 0 0

1 0 1 0 0 0 1 1 0 1 0

1 1 1 1 1 1 0 1 0 0 0

1 0 0 1 0 1 0 1 1 0 1 Black Carrot 10:00, 24 August 2007 (UTC)

I think the question to ask in preference to that is whether it is in fact possible to generate a grid in this way for arbitrary N rather than if there is an upper bound on N. (The preceding unsigned comment left by 203.49.213.95.)

I think the question to ask in preference to that is whether it is in fact possible to generate a grid in this way for arbitrary N rather than if there is an upper bound on N.

Are you sure?

1 0 0 1 1 0 1 1 1 1 0

0 1 0 1 0 1 0 1 0 1 0

1 0 1 1 1 0 0 0 0 1 1

1 1 1 0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0 1 0 1

0 1 1 0 1 0 0 1 1 0 1

0 1 0 0 0 1 1 1 0 0 1

0 0 0 1 1 0 1 0 1 0 0

1 0 1 0 0 0 1 1 0 1 0

1 1 1 1 1 1 0 1 0 0 0

1 0 0 1 0 1 0 1 1 0 1

--

1 0 0 1 1 0 1 1 1 1 0

0 1 0 1 0 1 0 1 0 1 0

1 0 1 1 1 0 0 0 0 1 1

1 1 1 0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0 1 0 1

0 1 1 0 1 0 0 1 1 0 1

0 1 0 0 0 1 1 1 0 0 1

0 0 0 1 1 0 1 0 1 0 0

1 0 1 0 0 0 1 1 0 1 0

1 1 1 1 1 1 0 1 0 0 0

1 0 0 1 0 1 0 1 1 0 1

– b_jonas 13:09, 24 August 2007 (UTC)

Oh, you said squares, not rectangles. – b_jonas 13:21, 24 August 2007 (UTC)

Yeah, squares. And a point you brought up with your first one - the squares are meant to be parallel to the grid. Black Carrot 13:52, 24 August 2007 (UTC)

In answer to the question if it is possible to have a colour assignment without corner-monochromatic subsquares for arbitrary (and therefore arbitrarily large) grids, the answer is no. This is a result of Ramsey theory; it is a special case of a multi-dimensional generalization of van der Waerden's theorem, which is a corollary (also in the multi-dimensional generalization) of the Hales–Jewett theorem. For more, see http://in-theory.blogspot.com/2007_01_21_archive.html. The sequence of the number of proper colourings on an n×n grid, for n = 1, 2, ... (sequence A018803 in the OEIS), is labelled "fini".  --Lambiam 14:49, 24 August 2007 (UTC)

So the answer isn't known then? Black Carrot 16:50, 25 August 2007 (UTC)

I think it is not known. If it was known, you'd expect someone to state it. Also, if the general pattern of Ramsey theory applies, it is a humungous number, as in wow, that's like really big, man. See for instance the Paris-Harrington theorem.  --Lambiam 18:14, 26 August 2007 (UTC)

A computer search shows that there is no 8x8 matrix in which the top row is all zeros. Therefore, the maximum dimension is less than V(2,8): if there were a sequence of 8 positions in arithmetic progression, all holding the same value, the square submatrix they define would have to contain a monochromatic square. I was also able to find an 11x11 by computer search but not a 12x12 yet. —David Eppstein 23:39, 26 August 2007 (UTC)

Found a bug in my program, fixed it. There are examples up to 9x9 with no 1 in the first row, but there are no such examples at 10x10. So, the maximum dimension is less than V(2,10): if there were a sequence of 10 positions in arithmetic progression, all holding the same value, the square submatrix they define would have to contain a monochromatic square. —David Eppstein 15:38, 27 August 2007 (UTC)

Interesting approach. How does that help? Black Carrot 09:53, 28 August 2007 (UTC)

A few of us have been looking at this. We have the following results. First generalize the problem to m x n rectangles:

. 13 x n is doable for all n (there is a nice tiling involved here with a period of length 13).

. 14 x 15 is NOT doable, but 14 x 14 is (courtesy of the SAT solver MiniSAT -- about a 100 minutes computation on my workstation).

We have a proof by hand that n x n is not doable for sufficiently large n (currently the best n we have is around 4500).

VictorSMiller 19:44, 3 October 2007 (UTC)

Continuity of solution space

This is somewhat related to a question I asked a few days ago. I have a nonlinear boundary value problem defined as follows:

u'(s) = F(u(s)) ,

With boundary conditions g(u(0),u(1)) = k

Let the solution of this be u₀(s) . I have to find whether every infinitesimal change in the boundary conditions g(u(0),u(1)) = k produces only an infinitesimal change in the solution u₀(s). If any small disturbance in g can produce a finite change in solution, then the solution u0 is unstable because any disturbance can take it to u1. Is there a way to do this numerically? Thanks, deeptrivia (talk) 14:46, 24 August 2007 (UTC)

WP:RD/MA

Why in classic skin, do I get a link to Wikipedia:Reference desk/Mathematics using WP:RD/MA when I am on the Wikipedia:Reference desk/Mathematics page? And why is it in the top right overprinting the "My preferences" | "Help" links? Seems a dumb thing to do but there must be a reason... -- SGBailey 22:35, 24 August 2007 (UTC)

Its in the transcluded page Wikipedia:Reference desk/header. The links there to indicate the shortcut to refer to the page and is placed in a span tag to position it. You may want to discuss this with Froth (talk · contribs) who's been modifying the header. --Salix alba (talk) 10:48, 25 August 2007 (UTC)

That link appears up there as a replacement "shortcut" box since there's nowhere to put that shortcut box. this is how it's supposed to look. What browser are you using? Look at the top right of Wikipedia_talk:Reference_desk. Does that color wheel overlap your links at the top too? How about the lock on a semi protected article like Jew? Or the star on any featured page? They all use the same positioning method, so it's fairly standard on WP. It could be that your browser is incorrectly interpreting the "em" unit, which would cause it to appear very close to the top, but (despite convention on WP to the contrary) it's important to use "em"s instead of "px"s in case the browser has a non standard text size. If anyone else complains (or you really take exception) I'll try to find somewhere else for it. Of course if anyone has any ideas.. --froth^t 20:40, 25 August 2007 (UTC)