Author(s): |
C. Lantuéjoul, MinesParisTech (FR)
F. MAISONNEUVE, Mines-ParisTech (FR)
J.N. BACRO, Université de Montpellier (FR)
L. BEL, AgroParisTech (FR)
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Abstract: |
Let $X = ( X_1,X_2,...,X_p )$ be a random vector with positive, independent and continuous components. The problem addressed in this presentation is the conditional simulation of $X$ subject to the fact that its components satisfy a set of $n$ max-linear constraints $\max_j a_{ij} X_j = b_i$ for $i=1, \ldots, n$. The coefficients $a_{ij}$ are nonnegative and the $b_i$'s are strictly positive.
This problem is encountered in simulation studies of object-based processes and max-stable processes. It has been investigated by Wang and Stoev (2010, arXiv:1005.0312v1) using a purely algebraic approach. Our approach is more geometric and provides a number of complementary and sometimes unexpected results. It turns out that the set of vectors $b=(b_1,...,b_n)$ that make the $n$ constraints mutually compatible for a given matrix $(a_ij)$ has a very peculiar geometry. A consequence is that the set of conditional simulations does not vary continuously as a function of $b$.
Suppose that the constraints are mutually compatible. Then it can be shown that each conditional simulation $x=(x_1,...,x_p)$ is necessarily located in the union of a number of faces of a certain polytope. An explicit expression of the conditional p.d.f. $f(x)$ is established that corroborates the results by Wang and Stoev. In particular $f(x)=0$ unless $x$ belongs to one of the faces of maximal dimension among the family of allowed faces. A critical problem is therefore the efficient identification of all allowed faces of maximal dimension, which is currently undergone using operational research techniques. |
