20,230 research outputs found
Matrix Analytic Methods in Branching processes
We examine the question of solving the extinction
probability of a particular class of continuous-time multi-type
branching processes, named Markovian binary trees (MBT). The
extinction probability is the minimal nonnegative solution of a
fixed point equation that turns out to be quadratic, which makes its
resolution particularly clear.
We analyze first two linear algorithms to compute the extinction
probability of an MBT, of which one is new, and, we propose a
quadratic algorithm arising from Newton\u27s iteration method for
fixed-point equations.
Finally, we add a catastrophe process to the
initial MBT, and we analyze the resulting system. The extinction
probability turns out to be much more difficult to compute; we use a
-type Markovian process approach to approximate this
probability
A low-rank technique for computing the quasi-stationary distribution of subcritical Galton-Watson processes
We present a new algorithm for computing the quasi-stationary distribution of
subcritical Galton--Watson branching processes. This algorithm is based on a
particular discretization of a well-known functional equation that
characterizes the quasi-stationary distribution of these processes. We provide
a theoretical analysis of the approximate low-rank structure that stems from
this discretization, and we extend the procedure to multitype branching
processes. We use numerical examples to demonstrate that our algorithm is both
more accurate and more efficient than other approaches
Birth/birth-death processes and their computable transition probabilities with biological applications
Birth-death processes track the size of a univariate population, but many
biological systems involve interaction between populations, necessitating
models for two or more populations simultaneously. A lack of efficient methods
for evaluating finite-time transition probabilities of bivariate processes,
however, has restricted statistical inference in these models. Researchers rely
on computationally expensive methods such as matrix exponentiation or Monte
Carlo approximation, restricting likelihood-based inference to small systems,
or indirect methods such as approximate Bayesian computation. In this paper, we
introduce the birth(death)/birth-death process, a tractable bivariate extension
of the birth-death process. We develop an efficient and robust algorithm to
calculate the transition probabilities of birth(death)/birth-death processes
using a continued fraction representation of their Laplace transforms. Next, we
identify several exemplary models arising in molecular epidemiology,
macro-parasite evolution, and infectious disease modeling that fall within this
class, and demonstrate advantages of our proposed method over existing
approaches to inference in these models. Notably, the ubiquitous stochastic
susceptible-infectious-removed (SIR) model falls within this class, and we
emphasize that computable transition probabilities newly enable direct
inference of parameters in the SIR model. We also propose a very fast method
for approximating the transition probabilities under the SIR model via a novel
branching process simplification, and compare it to the continued fraction
representation method with application to the 17th century plague in Eyam.
Although the two methods produce similar maximum a posteriori estimates, the
branching process approximation fails to capture the correlation structure in
the joint posterior distribution
Stability analysis of financial contagion due to overlapping portfolios
Common asset holdings are widely believed to have been the primary vector of
contagion in the recent financial crisis. We develop a network approach to the
amplification of financial contagion due to the combination of overlapping
portfolios and leverage, and we show how it can be understood in terms of a
generalized branching process. By studying a stylized model we estimate the
circumstances under which systemic instabilities are likely to occur as a
function of parameters such as leverage, market crowding, diversification, and
market impact. Although diversification may be good for individual
institutions, it can create dangerous systemic effects, and as a result
financial contagion gets worse with too much diversification. Under our model
there is a critical threshold for leverage; below it financial networks are
always stable, and above it the unstable region grows as leverage increases.
The financial system exhibits "robust yet fragile" behavior, with regions of
the parameter space where contagion is rare but catastrophic whenever it
occurs. Our model and methods of analysis can be calibrated to real data and
provide simple yet powerful tools for macroprudential stress testing.Comment: 25 pages, 8 figure
Extracting the resonance parameters from experimental data on scattering of charged particles
A new parametrization of the multi-channel S-matrix is used to fit scattering
data and then to locate the resonances as its poles. The S-matrix is written in
terms of the corresponding "in" and "out" Jost matrices which are expanded in
the Taylor series of the collision energy E around an appropriately chosen
energy E0. In order to do this, the Jost matrices are written in a
semi-analytic form where all the factors (involving the channel momenta and
Sommerfeld parameters) responsible for their "bad behaviour" (i.e. responsible
for the multi-valuedness of the Jost matrices and for branching of the Riemann
surface of the energy) are given explicitly. The remaining unknown factors in
the Jost matrices are analytic and single-valued functions of the variable E
and are defined on a simple energy plane. The expansion is done for these
analytic functions and the expansion coefficients are used as the fitting
parameters. The method is tested on a two-channel model, using a set of
artificially generated data points with typical error bars and a typical random
noise in the positions of the points.Comment: 15 pages, 7 figures, 2 table
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