1,832 research outputs found
Trees under attack: a Ray-Knight representation of Feller's branching diffusion with logistic growth
We obtain a representation of Feller's branching diffusion with logistic
growth in terms of the local times of a reflected Brownian motion with a
drift that is affine linear in the local time accumulated by at its current
level. As in the classical Ray-Knight representation, the excursions of are
the exploration paths of the trees of descendants of the ancestors at time
, and the local time of at height measures the population size at
time (see e.g. \cite{LG4}). We cope with the dependence in the reproduction
by introducing a pecking order of individuals: an individual explored at time
and living at time is prone to be killed by any of its
contemporaneans that have been explored so far. The proof of our main result
relies on approximating with a sequence of Harris paths which figure
in a Ray-Knight representation of the total mass of a branching particle
system. We obtain a suitable joint convergence of together with its local
times {\em and} with the Girsanov densities that introduce the dependence in
the reproduction
Time reversal dualities for some random forests
We consider a random forest , defined as a sequence of i.i.d.
birth-death (BD) trees, each started at time 0 from a single ancestor, stopped
at the first tree having survived up to a fixed time . We denote by
the population size process associated
to this forest, and we prove that if the BD trees are supercritical, then the
time-reversed process , has the same
distribution as , the
corresponding population size process of an equally defined forest
, but where the underlying BD trees are subcritical,
obtained by swapping birth and death rates or equivalently, conditioning on
ultimate extinction.
We generalize this result to splitting trees (i.e. life durations of
individuals are not necessarily exponential), provided that the i.i.d.
lifetimes of the ancestors have a specific explicit distribution, different
from that of their descendants. The results are based on an identity between
the contour of these random forests truncated up to and the duality
property of L\'evy processes. This identity allows us to also derive other
useful properties such as the distribution of the population size process
conditional on the reconstructed tree of individuals alive at , which has
potential applications in epidemiology.Comment: 28 pages, 3 figure
Uniform multifractal structure of stable trees
In this work, we investigate the spectrum of singularities of random stable
trees with parameter . We consider for that purpose the scaling
exponents derived from two natural measures on stable trees: the local time
and the mass measure , providing as well a purely
geometrical interpretation of the latter exponent. We first characterise the
uniform component of the multifractal spectrum which exists at every level
of stable trees and corresponds to large masses with scaling index
for the mass measure
(or equivalently for the local
time). In addition, we investigate the distribution of vertices appearing at
random levels with exceptionally large masses of index
. Finally, we discuss more precisely the
order of the largest mass existing on any subset of a stable
tree, characterising the former with the packing dimension of the set .Comment: 50 pages. Major overhaul of the paper, correcting Theorem 4 and
adding the study of the mass measure spectru
Non-binary branching process and non-Markovian exploration process
International audienceWe study both a continuous time non-binary Galton−Watson random tree and its explo- ration (or height) process in the subcritical, critical and supercritical cases. We then renormalize our branching process and exploration process, and take the weak limit as the size of the population tends to infinity
Machine learning and its applications in reliability analysis systems
In this thesis, we are interested in exploring some aspects of Machine Learning (ML) and its application in the Reliability Analysis systems (RAs). We begin by investigating some ML paradigms and their- techniques, go on to discuss the possible applications of ML in improving RAs performance, and lastly give guidelines of the architecture of learning RAs. Our survey of ML covers both levels of Neural Network learning and Symbolic learning. In symbolic process learning, five types of learning and their applications are discussed: rote learning, learning from instruction, learning from analogy, learning from examples, and learning from observation and discovery. The Reliability Analysis systems (RAs) presented in this thesis are mainly designed for maintaining plant safety supported by two functions: risk analysis function, i.e., failure mode effect analysis (FMEA) ; and diagnosis function, i.e., real-time fault location (RTFL). Three approaches have been discussed in creating the RAs. According to the result of our survey, we suggest currently the best design of RAs is to embed model-based RAs, i.e., MORA (as software) in a neural network based computer system (as hardware). However, there are still some improvement which can be made through the applications of Machine Learning. By implanting the 'learning element', the MORA will become learning MORA (La MORA) system, a learning Reliability Analysis system with the power of automatic knowledge acquisition and inconsistency checking, and more. To conclude our thesis, we propose an architecture of La MORA
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