46,668 research outputs found
Probabilistic Reachability Analysis for Large Scale Stochastic Hybrid Systems
This paper studies probabilistic reachability analysis for large scale stochastic hybrid systems (SHS) as a problem of rare event estimation. In literature, advanced rare event estimation theory has recently been embedded within a stochastic analysis framework, and this has led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This paper presents this rare event estimation theory directly in terms of probabilistic reachability analysis of an SHS, and develops novel theory which allows to extend the novel results for application to a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the reach probability. Essentially, the approach taken is to introduce an aggregation of the discrete modes, and to develop importance sampling relative to the rare switching between the aggregation modes. The practical working of this approach is demonstrated for the safety verification of an advanced air traffic control example
Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks
We consider a standard splitting algorithm for the rare-event simulation of
overflow probabilities in any subset of stations in a Jackson network at level
n, starting at a fixed initial position. It was shown in DeanDup09 that a
subsolution to the Isaacs equation guarantees that a subexponential number of
function evaluations (in n) suffice to estimate such overflow probabilities
within a given relative accuracy. Our analysis here shows that in fact
O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative
precision, where {\beta} is the number of bottleneck stations in the network.
This is the first rigorous analysis that allows to favorably compare splitting
against directly computing the overflow probability of interest, which can be
evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page
Biological applications of the theory of birth-and-death processes
In this review, we discuss the applications of the theory of birth-and-death
processes to problems in biology, primarily, those of evolutionary genomics.
The mathematical principles of the theory of these processes are briefly
described. Birth-and-death processes, with some straightforward additions such
as innovation, are a simple, natural formal framework for modeling a vast
variety of biological processes such as population dynamics, speciation, genome
evolution, including growth of paralogous gene families and horizontal gene
transfer, and somatic evolution of cancers. We further describe how empirical
data, e.g., distributions of paralogous gene family size, can be used to choose
the model that best reflects the actual course of evolution among different
versions of birth-death-and-innovation models. It is concluded that
birth-and-death processes, thanks to their mathematical transparency,
flexibility and relevance to fundamental biological process, are going to be an
indispensable mathematical tool for the burgeoning field of systems biology.Comment: 29 pages, 4 figures; submitted to "Briefings in Bioinformatics
Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis
Particle splitting methods are considered for the estimation of rare events.
The probability of interest is that a Markov process first enters a set
before another set , and it is assumed that this probability satisfies a
large deviation scaling. A notion of subsolution is defined for the related
calculus of variations problem, and two main results are proved under mild
conditions. The first is that the number of particles generated by the
algorithm grows subexponentially if and only if a certain scalar multiple of
the importance function is a subsolution. The second is that, under the same
condition, the variance of the algorithm is characterized (asymptotically) in
terms of the subsolution. The design of asymptotically optimal schemes is
discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application
Actions Speak Louder Than Goals: Valuing Player Actions in Soccer
Assessing the impact of the individual actions performed by soccer players
during games is a crucial aspect of the player recruitment process.
Unfortunately, most traditional metrics fall short in addressing this task as
they either focus on rare actions like shots and goals alone or fail to account
for the context in which the actions occurred. This paper introduces (1) a new
language for describing individual player actions on the pitch and (2) a
framework for valuing any type of player action based on its impact on the game
outcome while accounting for the context in which the action happened. By
aggregating soccer players' action values, their total offensive and defensive
contributions to their team can be quantified. We show how our approach
considers relevant contextual information that traditional player evaluation
metrics ignore and present a number of use cases related to scouting and
playing style characterization in the 2016/2017 and 2017/2018 seasons in
Europe's top competitions.Comment: Significant update of the paper. The same core idea, but with a
clearer methodology, applied on a different data set, and more extensive
experiments. 9 pages + 2 pages appendix. To be published at SIGKDD 201
- …