20,170 research outputs found
The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions
The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis.
This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case.
More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0.
For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors
Statistical properties of a filtered Poisson process with additive random noise: Distributions, correlations and moment estimation
Filtered Poisson processes are often used as reference models for
intermittent fluc- tuations in physical systems. Such a process is here
extended by adding a noise term, either as a purely additive term to the
process or as a dynamical term in a stochastic differential equation. The
lowest order moments, probability density function, auto-correlation function
and power spectral density are derived and used to identify and compare the
effects of the two different noise terms. Monte-Carlo studies of synthetic time
series are used to investigate the accuracy of model pa- rameter estimation and
to identify methods for distinguishing the noise types. It is shown that the
probability density function and the three lowest order moments provide
accurate estimations of the parameters, but are unable to separate the noise
types. The auto-correlation function and the power spectral density also
provide methods for estimating the model parameters, as well as being capable
of identifying the noise type. The number of times the signal crosses a
prescribed threshold level in the positive direction also promises to be able
to differentiate the noise type.Comment: 34 pages, 25 figure
Maximum Significance at the LHC and Higgs Decays to Muons
We present a new way to define and compute the maximum significance
achievable for signal and background processes at the LHC, using all available
phase space information. As an example, we show that a light Higgs boson
produced in weak--boson fusion with a subsequent decay into muons can be
extracted from the backgrounds. The method, aimed at phenomenological studies,
can be incorporated in parton--level event generators and accommodate
parametric descriptions of detector effects for selected observables.Comment: 7 pages, 2 figures, changes to wording and new references, published
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