186,746 research outputs found
Optimal sampling strategies for multiscale stochastic processes
In this paper, we determine which non-random sampling of fixed size gives the
best linear predictor of the sum of a finite spatial population. We employ
different multiscale superpopulation models and use the minimum mean-squared
error as our optimality criterion. In multiscale superpopulation tree models,
the leaves represent the units of the population, interior nodes represent
partial sums of the population, and the root node represents the total sum of
the population. We prove that the optimal sampling pattern varies dramatically
with the correlation structure of the tree nodes. While uniform sampling is
optimal for trees with ``positive correlation progression'', it provides the
worst possible sampling with ``negative correlation progression.'' As an
analysis tool, we introduce and study a class of independent innovations trees
that are of interest in their own right. We derive a fast water-filling
algorithm to determine the optimal sampling of the leaves to estimate the root
of an independent innovations tree.Comment: Published at http://dx.doi.org/10.1214/074921706000000509 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Homometric sets in trees
Let denote a simple graph with the vertex set and the edge
set . The profile of a vertex set denotes the multiset of
pairwise distances between the vertices of . Two disjoint subsets of
are \emph{homometric}, if their profiles are the same. If is a tree on
vertices we prove that its vertex sets contains a pair of disjoint homometric
subsets of size at least . Previously it was known that such a
pair of size at least roughly exists. We get a better result in case
of haircomb trees, in which we are able to find a pair of disjoint homometric
sets of size at least for a constant
Reconstruction Threshold for the Hardcore Model
In this paper we consider the reconstruction problem on the tree for the
hardcore model. We determine new bounds for the non-reconstruction regime on
the k-regular tree showing non-reconstruction when lambda < (ln
2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1.
This is almost tight as reconstruction is known to hold when lambda>
(e+o(1))ln^2(k). We discuss the relationship for finding large independent sets
in sparse random graphs and to the mixing time of Markov chains for sampling
independent sets on trees.Comment: 14 pages, 2 figure
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