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Rare events statistics for random walks on networks

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Docs 2 Watch. FICX 56 - Llendes. Use the HTML below. You must be a registered user to use the IMDb rating plugin. Photos Add Image Add an image Do you have any images for this title? Learn more More Like This. What Means Something. A Spell to Ward Off the Darkness Documentary Music. Trees Down Here Documentary Short. Two Years at Sea After working at sea, a man realizes his dream of moving to the middle of the forest. Biography Comedy Drama. Good Luck He Who Eats Children For this type of system, paths to a rare event concentrate around a most likely fluctuation path, which can be calculated as a minimiser of an action functional on the paths.

I have made use of this characteristic of the distribution of paths to construct a suitable objective function. The objective function is constructed such that it selects particles moving along the most likely path, which are more likely to end up in the rare event set.


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  4. The middle dashes lines show the estimated value. For dynamical systems more general than the weak noise diffusions I have furthermore developed a computational method to iteratively improve the objective function. With these methods I was able to vastly improve the accuracy of small probability estimates, or, equivalently, to significantly reduce the numerical cost of rare event sampling, since shorter GPA runs can attain in this way the same accuracy as longer brute force or even a naively constructed GPA simulation.

    The method initially uses a rough estimate of the path leading to an intermediate threshold, obtained from a short brute force simulation. This path can then be used to perform a first GPA simulation, allowing to estimate the path to a new, higher threshold, after which the GPA algorithm can be performed up to this new threshold. These first results on European heat waves confirm the connection between blocking high pressure patterns and heat wave conditions that has been proposed in the meteorology literature and demonstrates a new teleconnection pattern during heat wave conditions.

    Figure 3: Rare event simulation on the Planet Simulator climate model, using a genealogical particle analysis algorithm. Bottom: this allows for a very accurate estimation of for example the atmospheric flow pattern related to very rare heat waves. Extreme value theory is a field of statistics describing the distributions of extrema of random variables.

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    The theory is well developed for independent and weakly dependent random variables; application of this theory to deterministic dynamical systems is however relatively new. These relations depend on the type of observable and the partial dimensions of the attractor on the stable and unstable manifolds.

    This connection can therefore be used to improve estimation of extreme value distributions if estimates of the attractor dimensions are available from standard algorithms such as the Grassberger-Procaccia algorithm. Our method removes the ambiguity of selecting an arbitrary threshold as required in the standard autocorrelation or moment based methods Kuehn, We have exploited the fact that upon transitioning from sustained to transient turbulence the system makes large excursions to low-energy states, giving the energy distribution a fatter lower tail.

    In the sustained turbulence state on the other hand the energy distribution is lower bounded and therefore the minima are Weibull distributed, with a negative shape parameter. The transition can then be readily detected as a simple change in sign of the extreme value distribution shape parameter. Our method has the distinct advantage of having a clear threshold for the parameter where a transition is likely to occur, namely zero.

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    This contrasts to methods based on an increase in variance or skewness that do not have such a threshold. This advantage of the extremes based approach makes it more readily applicable and it has since been succesfully applied to estimate transition thresholds in an experiment of a dynamo magnetic field Faranda et al. Rare event simulation The cost of numerically calculating probabilities of rare events rapidly becomes prohibitive as the event of interest becomes rare.

    Extreme value theory Extreme value theory is a field of statistics describing the distributions of extrema of random variables. Rubino, G. Rare event simulation using Monte Carlo methods. Chichester, U. K: Wiley. Del Moral, P. Leadbetter, M. Extremes and related properties of random sequences and processes.

    New York: Springer-Verlag. Extreme value theory: an introduction. New York ; London: Springer. Embrechts, P. Modelling Extremal Events for Insurance and Finance.