Modeling of super-extreme events: An application to the hierarchical Weierstrass-Mandelbrot Continuous-time Random Walk

We analytically demonstrate and numerically simulate two utmost cases of dragon-kings' impact on the (unnormalized) velocity autocorrelation function (VACF) of a complex time series generated by stochastic random walker. The first type of dragon-kings corresponds to a sustained drift whose duration time is much longer than that of any other event. The second type of dragon-kings takes the form of an abrupt shock whose amplitude velocity is much larger than those corresponding to any other event. The stochastic process in which the dragon-kings occur corresponds to an enhanced diffusion generated within the hierarchical Weierstrass-Mandelbrot Continuous-time Random Walk (WM-CTRW) formalism. Our analytical formulae enable a detailed study of the impact of the two super-extreme events on the VACF calculated for a given random walk realization on the form of upward deviations from the background power law decay present in the absence of dragon-kings. This allows us to provide a unambiguous distinction between the super-extreme dragon-kings and ‘normal' extreme "black swans”. The results illustrate diagnostic that could be useful for the analysis of extreme and super-extreme events in real empirical time serie

Extremal-point Densities of Interface Fluctuations

We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of non-equilibrium surface fluctuations. We give a number of exact, analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface growth models. We show that in spite of the non-universal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic to the macroscopic observables of the system. The quantities investigated here present us with tools that give an entirely un-orthodox approach to the dynamics of surface morphologies: a statistical analysis from the short wavelength end of the Fourier decomposition spectrum. In addition to surface growth applications, our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete event simulations, which are extensively used in Monte-Carlo type simulations on parallel architectures.Comment: 30 pages, 5 ps figure

Estimation of the Hurst Exponent for the Burdekin River using the Hurst-Mandelbrot Rescaled Range Statistic

The issue of the impact of agricultural run-off in north Queensland on nutrient accumulation in the Great Barrier Reef lagoon has recently been the subject of increased scientific, media and policy debate. Models of river flow and of nutrient plumes at the mouth of Queensland rivers are an essential prerequisite for studying the impact of agriculture on the Great Barrier Reef ecosystem. Hurst found long-term dependency in fluctuations at the mouth of the Nile River the theoretical basis of which was later found by Mandelbrot to involve fractional Brownian motion. Approximately one quarter of the inflow into the Barrier Reef lagoon comes from the Burdekin and Fitzroy region. Using a data set collected by Isdale et al. that covers the period 1644-1980, that is based on reconstructing river flow data for the Burdekin river by analysing luminescence lines in coral reefs at the river mouth, we have estimated the Hurst exponent of the Burdekin river for the purpose of developing a river flow simulation model based on fractional Brownian motion

High-dimensional order-free multivariate spatial disease mapping

Despite the amount of research on disease mapping in recent years, the use of multivariate models for areal spatial data remains limited due to difficulties in implementation and computational burden. These problems are exacerbated when the number of small areas is very large. In this paper, we introduce an order-free multivariate scalable Bayesian modelling approach to smooth mortality (or incidence) risks of several diseases simultaneously. The proposal partitions the spatial domain into smaller subregions, fits multivariate models in each subdivision and obtains the posterior distribution of the relative risks across the entire spatial domain. The approach also provides posterior correlations among the spatial patterns of the diseases in each partition that are combined through a consensus Monte Carlo algorithm to obtain correlations for the whole study region. We implement the proposal using integrated nested Laplace approximations (INLA) in the R package bigDM and use it to jointly analyse colorectal, lung, and stomach cancer mortality data in Spanish municipalities. The new proposal permits the analysis of big data sets and provides better results than fitting a single multivariate model

Dragon-kings: mechanisms, statistical methods and empirical evidence

This introductory article presents the special Discussion and Debate volume "From black swans to dragon-kings, is there life beyond power laws?" published in Eur. Phys. J. Special Topics in May 2012. We summarize and put in perspective the contributions into three main themes: (i) mechanisms for dragon-kings, (ii) detection of dragon-kings and statistical tests and (iii) empirical evidence in a large variety of natural and social systems. Overall, we are pleased to witness significant advances both in the introduction and clarification of underlying mechanisms and in the development of novel efficient tests that demonstrate clear evidence for the presence of dragon-kings in many systems. However, this positive view should be balanced by the fact that this remains a very delicate and difficult field, if only due to the scarcity of data as well as the extraordinary important implications with respect to hazard assessment, risk control and predictability.Comment: 20 page