13,438 research outputs found
On the limiting distribution of the metric dimension for random forests
The metric dimension of a graph G is the minimum size of a subset S of
vertices of G such that all other vertices are uniquely determined by their
distances to the vertices in S. In this paper we investigate the metric
dimension for two different models of random forests, in each case obtaining
normal limit distributions for this parameter.Comment: 22 pages, 5 figure
Global and Local Two-Sample Tests via Regression
Two-sample testing is a fundamental problem in statistics. Despite its long
history, there has been renewed interest in this problem with the advent of
high-dimensional and complex data. Specifically, in the machine learning
literature, there have been recent methodological developments such as
classification accuracy tests. The goal of this work is to present a regression
approach to comparing multivariate distributions of complex data. Depending on
the chosen regression model, our framework can efficiently handle different
types of variables and various structures in the data, with competitive power
under many practical scenarios. Whereas previous work has been largely limited
to global tests which conceal much of the local information, our approach
naturally leads to a local two-sample testing framework in which we identify
local differences between multivariate distributions with statistical
confidence. We demonstrate the efficacy of our approach both theoretically and
empirically, under some well-known parametric and nonparametric regression
methods. Our proposed methods are applied to simulated data as well as a
challenging astronomy data set to assess their practical usefulness
Genealogy of catalytic branching models
We consider catalytic branching populations. They consist of a catalyst
population evolving according to a critical binary branching process in
continuous time with a constant branching rate and a reactant population with a
branching rate proportional to the number of catalyst individuals alive. The
reactant forms a process in random medium. We describe asymptotically the
genealogy of catalytic branching populations coded as the induced forest of
-trees using the many individuals--rapid branching continuum limit.
The limiting continuum genealogical forests are then studied in detail from
both the quenched and annealed points of view. The result is obtained by
constructing a contour process and analyzing the appropriately rescaled version
and its limit. The genealogy of the limiting forest is described by a point
process. We compare geometric properties and statistics of the reactant limit
forest with those of the "classical" forest.Comment: Published in at http://dx.doi.org/10.1214/08-AAP574 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Scaling Limits for Random Quadrangulations of Positive Genus
We discuss scaling limits of large bipartite quadrangulations of positive
genus. For a given , we consider, for every , a random
quadrangulation \q_n uniformly distributed over the set of all rooted
bipartite quadrangulations of genus with faces. We view it as a metric
space by endowing its set of vertices with the graph distance. We show that, as
tends to infinity, this metric space, with distances rescaled by the factor
, converges in distribution, at least along some subsequence, toward
a limiting random metric space. This convergence holds in the sense of the
Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of
the choice of the subsequence, the Hausdorff dimension of the limiting space is
almost surely equal to 4. Our main tool is a bijection introduced by Chapuy,
Marcus, and Schaeffer between the quadrangulations we consider and objects they
call well-labeled -trees. An important part of our study consists in
determining the scaling limits of the latter
Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions
A general formulation is presented for continuum scaling limits of stochastic
spanning trees. A spanning tree is expressed in this limit through a consistent
collection of subtrees, which includes a tree for every finite set of endpoints
in . Tightness of the distribution, as , is established for
the following two-dimensional examples: the uniformly random spanning tree on
, the minimal spanning tree on (with random edge
lengths), and the Euclidean minimal spanning tree on a Poisson process of
points in with density . In each case, sample trees are
proven to have the following properties, with probability one with respect to
any of the limiting measures: i) there is a single route to infinity (as was
known for ), ii) the tree branches are given by curves which are
regular in the sense of H\"older continuity, iii) the branches are also rough,
in the sense that their Hausdorff dimension exceeds one, iv) there is a random
dense subset of , of dimension strictly between one and two, on the
complement of which (and only there) the spanning subtrees are unique with
continuous dependence on the endpoints, v) branching occurs at countably many
points in , and vi) the branching numbers are uniformly bounded. The
results include tightness for the loop erased random walk (LERW) in two
dimensions. The proofs proceed through the derivation of scale-invariant power
bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
Localized Regression
The main problem with localized discriminant techniques is the curse of dimensionality, which seems to restrict their use to the case of few variables. This restriction does not hold if localization is combined with a reduction of dimension. In particular it is shown that localization yields powerful classifiers even in higher dimensions if localization is combined with locally adaptive selection of predictors. A robust localized logistic regression (LLR) method is developed for which all tuning parameters are chosen dataÂĄadaptively. In an extended simulation study we evaluate the potential of the proposed procedure for various types of data and compare it to other classification procedures. In addition we demonstrate that automatic choice of localization, predictor selection and penalty parameters based on cross validation is working well. Finally the method is applied to real data sets and its real world performance is compared to alternative procedures
Compact Brownian surfaces I. Brownian disks
We show that, under certain natural assumptions, large random plane bipartite
maps with a boundary converge after rescaling to a one-parameter family
(, ) of random metric spaces homeomorphic to the
closed unit disk of , the space being called the
Brownian disk of perimeter and unit area. These results can be seen as an
extension of the convergence of uniform plane quadrangulations to the Brownian
map, which intuitively corresponds to the limit case where . Similar
results are obtained for maps following a Boltzmann distribution, in which the
perimeter is fixed but the area is random
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