42,725 research outputs found
Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials
There are several common ways to encode a tree as a matrix, such as the
adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of
the natural random walk), and the matrix of pairwise distances between leaves.
Such representations involve a specific labeling of the vertices or at least
the leaves, and so it is natural to attempt to identify trees by some feature
of the associated matrices that is invariant under relabeling. An obvious
candidate is the spectrum of eigenvalues (or, equivalently, the characteristic
polynomial). We show for any of these choices of matrix that the fraction of
binary trees with a unique spectrum goes to zero as the number of leaves goes
to infinity. We investigate the rate of convergence of the above fraction to
zero using numerical methods. For the adjacency and Laplacian matrices, we show
that that the {\em a priori} more informative immanantal polynomials have no
greater power to distinguish between trees
Nested Archimedean copulas: a new class of nonparametric tree structure estimators
Any nested Archimedean copula is defined starting from a rooted phylogenetic
tree, for which a new class of nonparametric estimators is presented. An
estimator from this new class relies on a two-step procedure where first a
binary tree is built and second is collapsed if necessary to give an estimate
of the target tree structure. Several examples of estimators from this class
are given and the performance of each of these estimators, as well as of the
only known comparable estimator, is assessed by means of a simulation study
involving target structures in various dimensions, showing that the new
estimators, besides being faster, usually offer better performance as well.
Further, among the given examples of estimators from the new class, one of the
best performing one is applied on three datasets: 482 students and their
results to various examens, 26 European countries in 1979 and the percentage of
workers employed in different economic activities, and 104 countries in 2002
for which various health-related variables are available. The resulting
estimated trees offer valuable insights on the analyzed data. The future of
nested Archimedean copulas in general is also discussed
On joint detection and decoding of linear block codes on Gaussian vector channels
Optimal receivers recovering signals transmitted across noisy communication channels employ a maximum-likelihood (ML) criterion to minimize the probability of error. The problem of finding the most likely transmitted symbol is often equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In systems that employ error-correcting coding for data protection, the symbol space forms a sparse lattice, where the sparsity structure is determined by the code. In such systems, ML data recovery may be geometrically interpreted as a search for the closest point in the sparse lattice. In this paper, motivated by the idea of the "sphere decoding" algorithm of Fincke and Pohst, we propose an algorithm that finds the closest point in the sparse lattice to the given vector. This given vector is not arbitrary, but rather is an unknown sparse lattice point that has been perturbed by an additive noise vector whose statistical properties are known. The complexity of the proposed algorithm is thus a random variable. We study its expected value, averaged over the noise and over the lattice. For binary linear block codes, we find the expected complexity in closed form. Simulation results indicate significant performance gains over systems employing separate detection and decoding, yet are obtained at a complexity that is practically feasible over a wide range of system parameters
Trimmed trees and embedded particle systems
In a supercritical branching particle system, the trimmed tree consists of
those particles which have descendants at all times. We develop this concept in
the superprocess setting. For a class of continuous superprocesses with Feller
underlying motion on compact spaces, we identify the trimmed tree, which turns
out to be a binary splitting particle system with a new underlying motion that
is a compensated h-transform of the old one. We show how trimmed trees may be
estimated from above by embedded binary branching particle systems.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000009
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