693,549 research outputs found
Total variation error bounds for geometric approximation
We develop a new formulation of Stein's method to obtain computable upper
bounds on the total variation distance between the geometric distribution and a
distribution of interest. Our framework reduces the problem to the construction
of a coupling between the original distribution and the "discrete equilibrium"
distribution from renewal theory. We illustrate the approach in four
non-trivial examples: the geometric sum of independent, non-negative,
integer-valued random variables having common mean, the generation size of the
critical Galton-Watson process conditioned on non-extinction, the in-degree of
a randomly chosen node in the uniform attachment random graph model and the
total degree of both a fixed and randomly chosen node in the preferential
attachment random graph model.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ406 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Anytime system level verification via parallel random exhaustive hardware in the loop simulation
System level verification of cyber-physical systems has the goal of verifying that the whole (i.e., software + hardware) system meets the given specifications. Model checkers for hybrid systems cannot handle system level verification of actual systems. Thus, Hardware In the Loop Simulation (HILS) is currently the main workhorse for system level verification. By using model checking driven exhaustive HILS, System Level Formal Verification (SLFV) can be effectively carried out for actual systems.
We present a parallel random exhaustive HILS based model checker for hybrid systems that, by simulating all operational scenarios exactly once in a uniform random order, is able to provide, at any time during the verification process, an upper bound to the probability that the System Under Verification exhibits an error in a yet-to-be-simulated scenario (Omission Probability).
We show effectiveness of the proposed approach by presenting experimental results on SLFV of the Inverted Pendulum on a Cart and the Fuel Control System examples in the Simulink distribution. To the best of our knowledge, no previously published model checker can exhaustively verify hybrid systems of such a size and provide at any time an upper bound to the Omission Probability
SDEs with uniform distributions: Peacocks, Conic martingales and mean reverting uniform diffusions
It is known since Kellerer (1972) that for any peacock process there exist mar-tingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support [a(t), b(t)]. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries a(t) and b(t), they admit a unique strong solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds strong uniqueness, and study the local time and activity of the solution processes. We then study the peacock with uniform law at all times on a constant support [−1, 1] and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a solution in [0, T ]. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations
Poisson point process models solve the "pseudo-absence problem" for presence-only data in ecology
Presence-only data, point locations where a species has been recorded as
being present, are often used in modeling the distribution of a species as a
function of a set of explanatory variables---whether to map species occurrence,
to understand its association with the environment, or to predict its response
to environmental change. Currently, ecologists most commonly analyze
presence-only data by adding randomly chosen "pseudo-absences" to the data such
that it can be analyzed using logistic regression, an approach which has
weaknesses in model specification, in interpretation, and in implementation. To
address these issues, we propose Poisson point process modeling of the
intensity of presences. We also derive a link between the proposed approach and
logistic regression---specifically, we show that as the number of
pseudo-absences increases (in a regular or uniform random arrangement),
logistic regression slope parameters and their standard errors converge to
those of the corresponding Poisson point process model. We discuss the
practical implications of these results. In particular, point process modeling
offers a framework for choice of the number and location of pseudo-absences,
both of which are currently chosen by ad hoc and sometimes ineffective methods
in ecology, a point which we illustrate by example.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS331 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
Existence and uniqueness of the stationary measure in the continuous Abelian sandpile
Let \Lambda be a finite subset of Z^d. We study the following sandpile model
on \Lambda. The height at any given vertex x of \Lambda is a positive real
number, and additions are uniformly distributed on some interval [a,b], which
is a subset of [0,1]. The threshold value is 1; when the height at a given
vertex exceeds 1, it topples, that is, its height is reduced by 1, and the
heights of all its neighbours in \Lambda increase by 1/2d. We first establish
that the uniform measure \mu on the so called "allowed configurations" is
invariant under the dynamics. When a < b, we show with coupling ideas that
starting from any initial configuration of heights, the process converges in
distribution to \mu, which therefore is the unique invariant measure for the
process. When a = b, that is, when the addition amount is non-random, and a is
rational, it is still the case that \mu is the unique invariant probability
measure, but in this case we use random ergodic theory to prove this; this
proof proceeds in a very different way. Indeed, the coupling approach cannot
work in this case since we also show the somewhat surprising fact that when a =
b is rational, the process does not converge in distribution at all starting
from any initial configuration.Comment: 22 page
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