8,675 research outputs found
Simulation-based assessment of the stationary tail distribution of a stochastic differential equation
A commonly used approach to analyzing stochastic differential equations (SDEs) relies on performing
Monte Carlo simulation with a discrete-time counterpart. In this paper we study the impact of such a
time-discretization when assessing the stationary tail distribution. For a family of semi-implicit Euler
discretization schemes with time-step h > 0, we quantify the relative error due to the discretization, as a
function of h and the exceedance level x. By studying the existence of certain (polynomial and exponential)
moments, using a sequence of prototypical examples, we demonstrate that this error may tend to 0 or ¥.
The results show that the original shape of the tail can be heavily affected by the discretization. The cases
studied indicate that one has to be very careful when estimating the stationary tail distribution using Euler
discretization schemes
How close are time series to power tail L\'evy diffusions?
This article presents a new and easily implementable method to quantify the
so-called coupling distance between the law of a time series and the law of a
differential equation driven by Markovian additive jump noise with heavy-tailed
jumps, such as -stable L\'evy flights. Coupling distances measure the
proximity of the empirical law of the tails of the jump increments and a given
power law distribution. In particular they yield an upper bound for the
distance of the respective laws on path space. We prove rates of convergence
comparable to the rates of the central limit theorem which are confirmed by
numerical simulations. Our method applied to a paleoclimate time series of
glacial climate variability confirms its heavy tail behavior. In addition this
approach gives evidence for heavy tails in data sets of precipitable water
vapor of the Western Tropical Pacific.Comment: 30 pages, 10 figure
A spliced Gamma-Generalized Pareto model for short-term extreme wind speed probabilistic forecasting
Renewable sources of energy such as wind power have become a sustainable
alternative to fossil fuel-based energy. However, the uncertainty and
fluctuation of the wind speed derived from its intermittent nature bring a
great threat to the wind power production stability, and to the wind turbines
themselves. Lately, much work has been done on developing models to forecast
average wind speed values, yet surprisingly little has focused on proposing
models to accurately forecast extreme wind speeds, which can damage the
turbines. In this work, we develop a flexible spliced Gamma-Generalized Pareto
model to forecast extreme and non-extreme wind speeds simultaneously. Our model
belongs to the class of latent Gaussian models, for which inference is
conveniently performed based on the integrated nested Laplace approximation
method. Considering a flexible additive regression structure, we propose two
models for the latent linear predictor to capture the spatio-temporal dynamics
of wind speeds. Our models are fast to fit and can describe both the bulk and
the tail of the wind speed distribution while producing short-term extreme and
non-extreme wind speed probabilistic forecasts.Comment: 25 page
Half-tapering strategy for conditional simulation with large datasets
Gaussian conditional realizations are routinely used for risk assessment and
planning in a variety of Earth sciences applications. Conditional realizations
can be obtained by first creating unconditional realizations that are then
post-conditioned by kriging. Many efficient algorithms are available for the
first step, so the bottleneck resides in the second step. Instead of doing the
conditional simulations with the desired covariance (F approach) or with a
tapered covariance (T approach), we propose to use the taper covariance only in
the conditioning step (Half-Taper or HT approach). This enables to speed up the
computations and to reduce memory requirements for the conditioning step but
also to keep the right short scale variations in the realizations. A criterion
based on mean square error of the simulation is derived to help anticipate the
similarity of HT to F. Moreover, an index is used to predict the sparsity of
the kriging matrix for the conditioning step. Some guides for the choice of the
taper function are discussed. The distributions of a series of 1D, 2D and 3D
scalar response functions are compared for F, T and HT approaches. The
distributions obtained indicate a much better similarity to F with HT than with
T.Comment: 39 pages, 2 Tables and 11 Figure
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