Let X be a centered Gaussian vector of Rn and Γ its covariance matrix.
I assume that the diagonal terms of Γ are all equal to 1.
For α ∈]0,1[, I am looking for a rectangular parallelepiped [−a1,a1]×⋯×[−an,an] with a minimal volume such that P(X∈[−a1,a1]×⋯×[−an,an]) = α.
In other term, I want to solve
arginf a1 x ... x an among the (a_1,...,a_n) \in [R^+]^n such that P(X∈[−a1,a1]×⋯×[−an,an]) = α.
@ Paul
1o [-1,1] x [-2,2] is a rectangle (obviously in the rectangular coordinates' system)
2o The problem by Didier is perfectly formulated and says that he wanted to impose the additional constrain. But was it really what he wanted - who knows but him?
3o Your problem to seek the optimal (multidimensional) rectangular parallelograms within differently oriented is also interesting! Perhaps you know whether the faces then have to be alligned to the main axis of the covariance matrix; the proof whould be very interesting for itself.
4o And in general, when extending then why not to look for the optimal sets within triangles, for instance, or stars:-)
Best regards
Thank you for your comments.
The question I have to solve is the one written.
Looking for the optimal rectangle in the principal component basis is obvious. A simple change of variables allows one to express this problem with a diagonal Γ (independent case).
This problem can be easily solved for n=2 because (1,1) is always an eigenvector of Γ (remember that Γ is actually a correlation matrix).
I guess that the answer relies on general concentration properties of the Gaussian distribution but which ones ?
Joachim's extension 4° is a much more difficult problem.
Another way to write this problem can be found by following the link mathoverflow.
Kind regards
http://mathoverflow.net/questions/259501/for-a-given-gaussian-vector-which-rectangular-parallelepiped-with-unit-volum-ha
@ Joachim, sorry you are right, I was stupidly reading the notation incorrectly and interpreting those points as the vertices. I deleted my comment so as to avoid confusion. So I approached this problem in the simplified case of only two gaussian variables with correlation \rho. I started out using a Lagrangian formulation, which then required the evaluation of the marginal distributions of the bivariate normal distribution in order to get the gradients. I ended up with a system of equations that looked like this (attached). It seems to me that there is not a nice closed-form solution even for the case of two dimensions, and even if we are using axis-aligned rectangles. And evaluating the integrals of the marginal distributions would have been much harder with >2 dimensions -- I could only do this because of the completing the square trick. I did this pretty quickly so I wouldn't be surprised if there is a constant or sign off, but I hopefully it gives a general idea of what the solution would look like.
This is a constraint optimization problems with the probability constraints. If the value of n is fixed to 2 or 3, then it will have to be converted to an unconstrained optimization problems with the help of Lagrangian multipliers. That equation can be solved.
Thanks Mohammad.
The problem we have is with n=450.
For a fixed value of a1,..., a450, I do not even know how to evaluate P(X∈[−a1,a1]×⋯×[−an,an]) in a reasonnable time.
Maybe simulations could do an acceptable job ?
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