4,475 research outputs found
Effects of memory on the shapes of simple outbreak trees
Genomic tools, including phylogenetic trees derived from sequence data, are increasingly used to understand outbreaks of infectious diseases. One challenge is to link phylogenetic trees to patterns of transmission. Particularly in bacteria that cause chronic infections, this inference is affected by variable infectious periods and infectivity over time. It is known that non-exponential infectious periods can have substantial effects on pathogens’ transmission dynamics. Here we ask how this non-Markovian nature of an outbreak process affects the branching trees describing that process, with particular focus on tree shapes. We simulate Crump-Mode-Jagers branching processes and compare different patterns of infectivity over time. We find that memory (non-Markovian-ness) in the process can have a pronounced effect on the shapes of the outbreak’s branching pattern. However, memory also has a pronounced effect on the sizes of the trees, even when the duration of the simulation is fixed. When the sizes of the trees are constrained to a constant value, memory in our processes has little direct effect on tree shapes, but can bias inference of the birth rate from trees. We compare simulated branching trees to phylogenetic trees from an outbreak of tuberculosis in Canada, and discuss the relevance of memory to this dataset
Data-Efficient Quickest Outlying Sequence Detection in Sensor Networks
A sensor network is considered where at each sensor a sequence of random
variables is observed. At each time step, a processed version of the
observations is transmitted from the sensors to a common node called the fusion
center. At some unknown point in time the distribution of observations at an
unknown subset of the sensor nodes changes. The objective is to detect the
outlying sequences as quickly as possible, subject to constraints on the false
alarm rate, the cost of observations taken at each sensor, and the cost of
communication between the sensors and the fusion center. Minimax formulations
are proposed for the above problem and algorithms are proposed that are shown
to be asymptotically optimal for the proposed formulations, as the false alarm
rate goes to zero. It is also shown, via numerical studies, that the proposed
algorithms perform significantly better than those based on fractional
sampling, in which the classical algorithms from the literature are used and
the constraint on the cost of observations is met by using the outcome of a
sequence of biased coin tosses, independent of the observation process.Comment: Submitted to IEEE Transactions on Signal Processing, Nov 2014. arXiv
admin note: text overlap with arXiv:1408.474
Uniform Markov Renewal Theory and Ruin Probabilities in Markov Random Walks
Let {X_n,n\geq0} be a Markov chain on a general state space X with transition
probability P and stationary probability \pi. Suppose an additive component
S_n takes values in the real line R and is adjoined to the chain such that
{(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform
Markov renewal theorem with an estimate on the rate of convergence. This
result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}.
To be more precise, for given b\geq0, define the stopping time
\tau=\tau(b)=inf{n:S_n>b}.
When a drift \mu of the random walk S_n is 0, we derive a one-term Edgeworth
type asymptotic expansion for the first passage probabilities P_{\pi}{\tau<m}
and P_{\pi}{\tau<m,S_m<c}, where m\leq\infty, c\leq b and P_{\pi} denotes the
probability under the initial distribution \pi. When \mu\neq0, Brownian
approximations for the first passage probabilities with correction terms are
derived
Modulated Branching Processes, Origins of Power Laws and Queueing Duality
Power law distributions have been repeatedly observed in a wide variety of
socioeconomic, biological and technological areas. In many of the observations,
e.g., city populations and sizes of living organisms, the objects of interest
evolve due to the replication of their many independent components, e.g.,
births-deaths of individuals and replications of cells. Furthermore, the rates
of the replication are often controlled by exogenous parameters causing periods
of expansion and contraction, e.g., baby booms and busts, economic booms and
recessions, etc. In addition, the sizes of these objects often have reflective
lower boundaries, e.g., cities do not fall bellow a certain size, low income
individuals are subsidized by the government, companies are protected by
bankruptcy laws, etc.
Hence, it is natural to propose reflected modulated branching processes as
generic models for many of the preceding observations. Indeed, our main results
show that the proposed mathematical models result in power law distributions
under quite general polynomial Gartner-Ellis conditions, the generality of
which could explain the ubiquitous nature of power law distributions. In
addition, on a logarithmic scale, we establish an asymptotic equivalence
between the reflected branching processes and the corresponding multiplicative
ones. The latter, as recognized by Goldie (1991), is known to be dual to
queueing/additive processes. We emphasize this duality further in the
generality of stationary and ergodic processes.Comment: 36 pages, 2 figures; added references; a new theorem in Subsection
4.
Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder
We study the distribution of the -th energy level for two different
one-dimensional random potentials. This distribution is shown to be related to
the distribution of the distance between two consecutive nodes of the wave
function.
We first consider the case of a white noise potential and study the
distributions of energy level both in the positive and the negative part of the
spectrum. It is demonstrated that, in the limit of a large system
(), the distribution of the -th energy level is given by a
scaling law which is shown to be related to the extreme value statistics of a
set of independent variables.
In the second part we consider the case of a supersymmetric random
Hamiltonian (potential ). We study first the case of
being a white noise with zero mean. It is in particular shown that
the ground state energy, which behaves on average like in
agreement with previous work, is not a self averaging quantity in the limit
as is seen in the case of diagonal disorder. Then we consider the
case when has a non zero mean value.Comment: LaTeX, 33 pages, 9 figure
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde
- …