656 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&
Extended Generalised Pareto Models for Tail Estimation
The most popular approach in extreme value statistics is the modelling of
threshold exceedances using the asymptotically motivated generalised Pareto
distribution. This approach involves the selection of a high threshold above
which the model fits the data well. Sometimes, few observations of a
measurement process might be recorded in applications and so selecting a high
quantile of the sample as the threshold leads to almost no exceedances. In this
paper we propose extensions of the generalised Pareto distribution that
incorporate an additional shape parameter while keeping the tail behaviour
unaffected. The inclusion of this parameter offers additional structure for the
main body of the distribution, improves the stability of the modified scale,
tail index and return level estimates to threshold choice and allows a lower
threshold to be selected. We illustrate the benefits of the proposed models
with a simulation study and two case studies.Comment: 18 pages, 7 figure
Modelling extreme concentration from a source in a turbulent flow over rough wall
The concentration fluctuations in passive plumes from an elevated and a groundlevel
source in a turbulent boundary layer over a rough wall were studied using
large eddy simulation and wind tunnel experiment. The predictions of statistics
up to second order moments were thereby validated. In addition, the trend of relative
fluctuations far downstream for a ground level source was estimated using
dimensional analysis. The techniques of extreme value theory were then applied
to predict extreme concentrations by modelling the upper tail of the probability
density function of the concentration time series by the Generalised Pareto Distribution.
Data obtained from both the simulations and experiments were analysed in
this manner. The predicted maximum concentration (?0) normalized by the local
mean concentration (Cm) or by the local r.m.s of concentration fluctuation (crms),
was extensively investigated. Values for ?0/Cm and ?0/crms as large as 50 and 20
respectively were found for the elevated source and 10 and 15 respectively for the
ground-level source
Functional kernel estimators of conditional extreme quantiles
We address the estimation of "extreme" conditional quantiles i.e. when their
order converges to one as the sample size increases. Conditions on the rate of
convergence of their order to one are provided to obtain asymptotically
Gaussian distributed kernel estimators. A Weissman-type estimator and kernel
estimators of the conditional tail-index are derived, permitting to estimate
extreme conditional quantiles of arbitrary order.Comment: arXiv admin note: text overlap with arXiv:1107.226
Using Extreme Value Theory for Determining the Probability of Carrington-Like Solar Flares
Space weather events can negatively affect satellites, the electricity grid,
satellite navigation systems and human health. As a consequence, extreme space
weather has been added to the UK and other national risk registers. By their
very nature, extreme space weather events occur rarely and, therefore,
statistical methods are required to determine the probability of their
occurrence. Space weather events can be characterised by a number of natural
phenomena such as X-ray (solar) flares, solar energetic particle (SEP) fluxes,
coronal mass ejections and various geophysical indices (Dst, Kp, F10.7). In
this paper extreme value theory (EVT) is used to investigate the probability of
extreme solar flares. Previous work has assumed that the distribution of solar
flares follows a power law. However such an approach can lead to a poor
estimation of the return times of such events due to uncertainties in the tails
of the probability distribution function. Using EVT and GOES X-ray flux data it
is shown that the expected 150-year return level is approximately an X60 flare
whilst a Carrington-like flare is a one in a 100-year event. It is also shown
that the EVT results are consistent with flare data from the Kepler space
telescope mission.Comment: 13 pages, 4 figures; updated content following reviewer feedbac
Non-stationarity in peaks-over-threshold river flows:a regional random effects model
Under the influence of local- and large-scale climatological processes, extreme river flow events often show long-term trends, seasonality, inter-year variability and other characteristics of temporal non-stationarity. Properly accounting for this non-stationarity is vital for making accurate predictions of future floods. In this paper, a regional model based on the generalised Pareto distribution is developed for peaks-over-threshold river flow data sets when the event sizes are non-stationary. If observations are non-stationary and covariates are available then extreme value (semi-)parametric regression models may be used. Unfortunately the necessary covariates are rarely observed and, if they are, it is often not clear which process, or combination of processes, to include in the model. Within the statistical literature, latent process (or random effects) models are often used in such scenarios. We develop a regional time-varying random effects model which allows identification of temporal non-stationarity in event sizes by pooling information across all sites in a spatially homogeneous region. The proposed model, which is an instance of a Bayesian hierarchical model, can be used to predict both unconditional extreme events such as the m-year maximum, as well as extreme events that condition on being in a given year. The estimated random effects may also tell us about likely candidates for the climatological processes which cause non-stationarity in the flood process. The model is applied to UK flood data from 817 stations spread across 81 hydrometric regions
Monte Carlo-based tail exponent estimator
In this paper we propose a new approach to estimation of the tail exponent in
financial stock markets. We begin the study with the finite sample behavior of
the Hill estimator under {\alpha}-stable distributions. Using large Monte Carlo
simulations, we show that the Hill estimator overestimates the true tail
exponent and can hardly be used on samples with small length. Utilizing our
results, we introduce a Monte Carlo-based method of estimation for the tail
exponent. Our proposed method is not sensitive to the choice of tail size and
works well also on small data samples. The new estimator also gives unbiased
results with symmetrical confidence intervals. Finally, we demonstrate the
power of our estimator on the international world stock market indices. On the
two separate periods of 2002-2005 and 2006-2009, we estimate the tail exponent
Generalized Extreme Value distribution parameters as dynamical indicators of Stability
We introduce a new dynamical indicator of stability based on the Extreme
Value statistics showing that it provides an insight on the local stability
properties of dynamical systems. The indicator perform faster than other based
on the iteration of the tangent map since it requires only the evolution of the
original systems and, in the chaotic regions, gives further information about
the information dimension of the attractor. A numerical validation of the
method is presented through the analysis of the motions in a Standard map
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