43,203 research outputs found
Extreme Value Theory and the Solar Cycle
We investigate the statistical properties of the extreme events of the solar
cycle as measured by the sunspot number. The recent advances in the methodology
of the theory of extreme values is applied to the maximal extremes of the time
series of sunspots. We focus on the extreme events that exceed a carefully
chosen threshold and a generalized Pareto distribution is fitted to the tail of
the empirical cumulative distribution. A maximum likelihood method is used to
estimate the parameters of the generalized Pareto distribution and confidence
levels are also given to the parameters. Due to the lack of an automatic
procedure for selecting the threshold, we analyze the sensitivity of the fitted
generalized Pareto distribution to the exact value of the threshold. According
to the available data, that only spans the previous ~250 years, the cumulative
distribution of the time series is bounded, yielding an upper limit of 324 for
the sunspot number. We also estimate that the return value for each solar cycle
is ~188, while the return value for a century increases to ~228. Finally, the
results also indicate that the most probable return time for a large event like
the maximum at solar cycle 19 happens once every ~700 years and that the
probability of finding such a large event with a frequency smaller than ~50
years is very small. In spite of the essentially extrapolative character of
these results, their statistical significance is very large.Comment: 6 pages, 4 figures, accepted for publication in A&
Applications and identification of surface correlations
We compare theoretical, experimental, and computational approaches to random
rough surfaces. The aim is to produce rough surfaces with desirable
correlations and to analyze the correlation functions extracted from the
surface profiles. Physical applications include ultracold neutrons in a rough
waveguide, lateral electronic transport, and scattering of longwave particles
and waves. Results provide guidance on how to deal with experimental and
computational data on rough surfaces. A supplemental goal is to optimize the
neutron waveguide for GRANIT experiments. The measured correlators are
identified by fitting functions or by direct spectral analysis. The results are
used to compare the calculated observables with theoretical values. Because of
fluctuations, the fitting procedures lead to inaccurate physical results even
if the quality of the fit is very good unless one guesses the right shape of
the fitting function. Reliable extraction of the correlation function from the
measured surface profile seems virtually impossible without independent
information on the structure of the correlation function. Direct spectral
analysis of raw data rarely works better than the use of a "wrong" fitting
function. Analysis of surfaces with a large correlation radius is hindered by
the presence of domains and interdomain correlations
Rank-normalization, folding, and localization: An improved for assessing convergence of MCMC
Markov chain Monte Carlo is a key computational tool in Bayesian statistics,
but it can be challenging to monitor the convergence of an iterative stochastic
algorithm. In this paper we show that the convergence diagnostic
of Gelman and Rubin (1992) has serious flaws. Traditional will
fail to correctly diagnose convergence failures when the chain has a heavy tail
or when the variance varies across the chains. In this paper we propose an
alternative rank-based diagnostic that fixes these problems. We also introduce
a collection of quantile-based local efficiency measures, along with a
practical approach for computing Monte Carlo error estimates for quantiles. We
suggest that common trace plots should be replaced with rank plots from
multiple chains. Finally, we give recommendations for how these methods should
be used in practice.Comment: Minor revision for improved clarit
Spreading of waves in nonlinear disordered media
We analyze mechanisms and regimes of wave packet spreading in nonlinear
disordered media. We predict that wave packets can spread in two regimes of
strong and weak chaos. We discuss resonance probabilities, nonlinear diffusion
equations, and predict a dynamical crossover from strong to weak chaos. The
crossover is controlled by the ratio of nonlinear frequency shifts and the
average eigenvalue spacing of eigenstates of the linear equations within one
localization volume. We consider generalized models in higher lattice
dimensions and obtain critical values for the nonlinearity power, the
dimension, and norm density, which influence possible dynamical outcomes in a
qualitative way.Comment: 24 pages, 3 figures. arXiv admin note: text overlap with
arXiv:0901.441
Rank-normalization, folding, and localization: An improved for assessing convergence of MCMC
Markov chain Monte Carlo is a key computational tool in Bayesian statistics,
but it can be challenging to monitor the convergence of an iterative stochastic
algorithm. In this paper we show that the convergence diagnostic
of Gelman and Rubin (1992) has serious flaws. Traditional will
fail to correctly diagnose convergence failures when the chain has a heavy tail
or when the variance varies across the chains. In this paper we propose an
alternative rank-based diagnostic that fixes these problems. We also introduce
a collection of quantile-based local efficiency measures, along with a
practical approach for computing Monte Carlo error estimates for quantiles. We
suggest that common trace plots should be replaced with rank plots from
multiple chains. Finally, we give recommendations for how these methods should
be used in practice.Comment: Minor revision for improved clarit
Non-Gaussian Geostatistical Modeling using (skew) t Processes
We propose a new model for regression and dependence analysis when addressing
spatial data with possibly heavy tails and an asymmetric marginal distribution.
We first propose a stationary process with marginals obtained through scale
mixing of a Gaussian process with an inverse square root process with Gamma
marginals. We then generalize this construction by considering a skew-Gaussian
process, thus obtaining a process with skew-t marginal distributions. For the
proposed (skew) process we study the second-order and geometrical
properties and in the case, we provide analytic expressions for the
bivariate distribution. In an extensive simulation study, we investigate the
use of the weighted pairwise likelihood as a method of estimation for the
process. Moreover we compare the performance of the optimal linear predictor of
the process versus the optimal Gaussian predictor. Finally, the
effectiveness of our methodology is illustrated by analyzing a georeferenced
dataset on maximum temperatures in Australi
A continuous time random walk model for financial distributions
We apply the formalism of the continuous time random walk to the study of
financial data. The entire distribution of prices can be obtained once two
auxiliary densities are known. These are the probability densities for the
pausing time between successive jumps and the corresponding probability density
for the magnitude of a jump. We have applied the formalism to data on the US
dollar/Deutsche Mark future exchange, finding good agreement between theory and
the observed data.Comment: 14 pages, 5 figures, revtex4, submitted for publicatio
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