Anomalous transport and diffusion versus extreme value theory

In the present work we match the biased hierarchical continuous-time random flight (HCTRF) on a regular lattice (based on hierarchical waiting-time distribution) and the extreme event theory (EVT). This approach extends the understanding of the anomalous transport and diffusion (for example, found in some amorphous, vitreous solids as well as in conducting and light-emitting organic polymers). Both independent approaches were developed in terms of random-trap or valley model where the disorder of energy landscape is exponentially distributed while the corresponding mean residence times in traps obey the power-law. This type of disorder characterizes several amorphous (even used commercially) materials which makes it possible to apply the HCTRF formalism. By using the EVT we additionally show that the rare (stochastic) events are indeed responsible for the transport and diffusion in these materials.     

Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph.     

Adaptive walks with noisy fitness measurements

Summary Adaptive walks constitute an optimization technique for searching a space of possible solutions, for example, a space of different molecules. The goal is to find a point in space (a molecule) that is optimal or near-optimal in some property, generally referred to as the fitness, such as its ability to bind to a given receptor. Adaptive walking, an analog of natural selection, is a powerful technique for searching landscapes. However, errors in the measurements will cause errors in the adaptive walks. Mutant molecules of higher fitness may be ignored or mutants of lower fitness may be accepted. To examine the effect of measurement error on adaptive walks, we simulate single-agent hill-climbing walks on NK landscapes of varying ruggedness where Gaussian noise is added to the fitness values to model measurement error. We consider both constant measurement noise and noise whose variance decays exponentially with fitness. We show that fitness-independent noise can cause walks to melt off the peaks in a landscape, wandering in larger regions as the noise increases. However, we also show that a small amount of noise actually helps the walk perform better than with no noise. For walks in which noise decreases exponentially with fitness, the most characteristic behavior is that the walk meanders throughout the landscape until it stumbles across a point of relatively high fitness, then it climbs the landscape towards the nearest peak. Finally, we characterize the balance between selection pressure and noise and show that there are several classes of walk dynamic behavior.     

Value-at-risk estimation with wavelet-based extreme value theory: Evidence from emerging markets

This paper introduces wavelet-based extreme value theory (EVT) for univariate value-at-risk estimation. Wavelets and EVT are combined for volatility forecasting to estimate a hybrid model. In the first stage, wavelets are used as a threshold in generalized Pareto distribution, and in the second stage, EVT is applied with a wavelet-based threshold. This new model is applied to two major emerging stock markets: the Istanbul Stock Exchange (ISE) and the Budapest Stock Exchange (BUX). The relative performance of wavelet-based EVT is benchmarked against the Riskmetrics-EWMA, ARMA-GARCH, generalized Pareto distribution, and conditional generalized Pareto distribution models. The empirical results show that the wavelet-based extreme value theory increases predictive performance of financial forecasting according to number of violations and tail-loss tests. The superior forecasting performance of the wavelet-based EVT model is also consistent with Basel II requirements, and this new model can be used by financial institutions as well.     

Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios

Recently it has been found that composite Brownian walk searches are more efficient than any Lévy walk when searching is non-destructive and when the Lévy walks are not responsive to conditions found in the search. Here a new class of adaptive Lévy walk searches is presented that encompasses composite Brownian walks as a special case. In these new models, bouts of Lévy walk searching alternate with bouts of more intensive Brownian walk searching. Switching from extensive to intensive searching is prompted by the detection of a target. And here, switching back to extensive searching arises if a target is not located after travelling a distance equal to the ‘giving-up distance’. It is found that adaptive Lévy walks outperform composite Brownian walks when searching for sparsely distributed resources. Consequently there is stronger selection pressures for Lévy processes when resources are sparsely distributed within unpredictable environments. The findings reconcile Lévy walk search theory with the ubiquity of two modes of searching by predators and with their switching search mode immediately after finding a prey.     

Constrained random walks and vortex filaments in turbulence theory

We consider a simplified model of vorticity configurations in the inertial range of turbulent flow, in which vortex filaments are viewed as random walks in thermal equilibrium subjected to the constraints of helicity and energy conservation. The model is simple enough so that its properties can be investigated by a relatively straightforward Monte-Carlo method: a pivot algorithm with Metropolis weighting. Reasonable values are obtained for the intermittency dimensionD, a Kolmogorov-like exponent , and higher moments of the velocity derivatives. Qualitative conclusions are drawn regarding the origin of non-gaussian velocity statistics and regarding analogies with polymers and with systems near a critical point.This work was supported in part by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, US Department of Energy, under Contract Number DE-AC03-76SF000098     

Adaptive random walks on the class of Web graphs

We study random walk with adaptive move strategies on a class of directed graphs with variable wiring diagram. The graphs are grown from the evolution rules compatible with the dynamics of the world-wide Web [B. Tadić, Physica A 293, 273 (2001)], and are characterized by a pair of power-law distributions of out- and in-degree for each value of the parameter β, which measures the degree of rewiring in the graph. The walker adapts its move strategy according to locally available information both on out-degree of the visited node and in-degree of target node. A standard random walk, on the other hand, uses the out-degree only. We compute the distribution of connected subgraphs visited by an ensemble of walkers, the average access time and survival probability of the walks. We discuss these properties of the walk dynamics relative to the changes in the global graph structure when the control parameter β is varied. For β≥ 3, corresponding to the world-wide Web, the access time of the walk to a given level of hierarchy on the graph is much shorter compared to the standard random walk on the same graph. By reducing the amount of rewiring towards rigidity limit β↦β_c≲ 0.1, corresponding to the range of naturally occurring biochemical networks, the survival probability of adaptive and standard random walk become increasingly similar. The adaptive random walk can be used as an efficient message-passing algorithm on this class of graphs for large degree of rewiring.     

Time-dependent random walks and the theory of complex adaptive systems

Motivated by novel results in the theory of complex adaptive systems, we analyze the dynamics of random walks in which the jumping probabilities are time dependent. We determine the survival probability in the presence of an absorbing boundary. For an unbiased walk, the survival probability is maximized in the case of large temporal oscillations in the jumping probabilities. On the other hand, a random walker who is drifted towards the absorbing boundary performs best with a constant jumping probability. We use the results to reveal the underlying dynamics responsible for the phenomenon of self-segregation and clustering observed in the evolutionary minority game.     

Universal properties of self-avoiding walks from two-dimensional field theory

We use the recently conjectured exact S-matrix of the massive O(n) model to derive its form factors and ground state energy. This information is then used in the limit n → 0 to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio C/D which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over m-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to m 2 gives results which are accurate to at least one part in 10³. This remarkable fact is traced to a softening of the m > 2 branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body S-matrix, S(O) = −1. Since this is a general property of interacting 2D field theories, it suggests that similar approximation may well hold for other models. However, we also study the moments of the area of self-avoiding loops, and show that, in this case, the two-particle approximation is not valid.     

The theory of ordered spans of unrestricted random walks

The spans of ann-step random walk on a simple cubic lattice are the sides of the smallest rectangular box, with sides parallel to the coordinate axes, that contains the random walk. Daniels first developed the theory in outline and derived results for the simple random walk on a line. We show that the development of a more general asymptotic theory is facilitated by introducing the spectral representation of step probabilities. This allows us to consider the probability density for spans of random walks in which all moments of single steps may be infinite. The theory can also be extended to continuous-time random walks. We also show that the use of Abelian summation simplifies calculation of the moments. In particular we derive expressions for the span distributions of random walks (in one dimension) with single step transition probabilities of the formP(j) 1/j ¹⁺, where 0<<2. We also derive results for continuous-time random walks in which the expected time between steps may be infinite.     

Measuring daily Value-at-Risk of SSEC index: A new approach based on multifractal analysis and extreme value theory

Recent studies in the econophysics literature reveal that price variability has fractal and multifractal characteristics not only in developed financial markets, but also in emerging markets. Taking high-frequency intraday quotes of the Shanghai Stock Exchange Component (SSEC) Index as example, this paper proposes a new method to measure daily Value-at-Risk (VaR) by combining the newly introduced multifractal volatility (MFV) model and the extreme value theory (EVT) method. Two VaR backtesting techniques are then employed to compare the performance of the model with that of a group of linear and nonlinear generalized autoregressive conditional heteroskedasticity (GARCH) models. The empirical results show the multifractal nature of price volatility in Chinese stock market. VaR measures based on the multifractal volatility model and EVT method outperform many GARCH-type models at high-risk levels.     

Quantum walks with an anisotropic coin II: scattering theory

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest.

     

Universal power laws in the threshold network model: A theoretical analysis based on extreme value theory

We theoretically and numerically investigated the threshold network model with a generic weight function where there were a large number of nodes and a high threshold. Our analysis was based on extreme value theory, which gave us a theoretical understanding of the distribution of independent and identically distributed random variables within a sufficiently high range. Specifically, the distribution could be generally expressed by a generalized Pareto distribution, which enabled us to formulate the generic weight distribution function. By using the theorem, we obtained the exact expressions of degree distribution and clustering coefficient which behaved as universal power laws within certain ranges of degrees. We also compared the theoretical predictions with numerical results and found that they were extremely consistent.     

Gusts within plant canopies are extreme value processes

Plant canopy turbulence is characterised by energetic high-velocity downdrafts (sweeps) and updrafts (ejections) that punctuate otherwise quiescent flow and thereby make the dominant contribution to turbulent transport. It is suggested that this turbulence provides a natural setting for the emergence of extreme value processes. The suggestion is supported by simulation data obtained from Lagrangian probability density function models. Predicted wind speed frequency distributions within the lower portions of plant canopies are consistent with observations and are very well represented by Gumbel (extreme value) distributions. This is interpreted as a signature of strong (maximal) intermittent gusts making the dominant contribution to the turbulence. The findings complement the apparent ubiquity of generalised Gumbel distributions of global quantities in boundary-layer turbulence, equilibrium phase transitions and non-equilibrium models with self-organised criticality. In these examples, however, there are no known direct relationships between the global quantities and extreme value processes. The new findings are interesting from a practical perspective because correct parametrisation of the wind speed frequency distribution is required for the accurate quantification of seed abscission and the removal of pollens and passively released pathogenic spores from plant surfaces.     

Some extreme value statistics of laser speckle patterns

Extreme value statistics of the maximum intensity at a fixed point in the laser speckle pattern are evaluated. The results are employed to set up criteria to determine when the intensity fluctuations exceed the dynamic range of the recording medium.     

Derivation of critical exponents from extreme value distributions

It is shown that all critical exponents, with the exception of the heat capacity exponent other than for the mean field theory, can be derived from the characteristic exponent of an extreme value distribution for the smallest value and the dimensionality of the space. The relation between the characteristic exponent and the dimensionalityd of the space imposes the conditiond4. This is borne out by direct evaluation of the spatial correlation function.     

Level density of a bose gas and extreme value statistics

We establish a connection between the level density of a gas of noninteracting bosons and the theory of extreme value statistics. Depending on the exponent that characterizes the growth of the underlying single-particle spectrum, we show that at a given excitation energy the limiting distribution function for the number of excited particles follows the three universal distribution laws of extreme value statistics, namely, the Gumbel, Weibull, and Fréchet distributions. Implications of this result, as well as general properties of the level density at different energies, are discussed.     

极值温差红外图像的自适应双局部增强算法

针对目前红外焦平面成像系统在观察目标、特别是极值温差目标时,各温度段灰度描述不均匀和细节不够的问题,提出了一种自适应红外图像双局部增强算法。详细介绍了通过空间分布和灰度统计特性两个方向实现对极值温差图像自适应增强的方法,该方法首先从红外图像的空间分布特性出发,将图像切割成多个局部图像,然后再从直方图灰度分布出发,将局部图像的直方图进行聚类分段,并对分段直方图均衡增强,最后对生成的每个局部图像增强结果进行线性插值拼接完成增强算法。通过在红外焦平面系统中实验证明了极值温差自适应的红外图像双局部增强算法的可行性,并获得了很好的效果,成像质量有明显提高。     

极值温差红外图像的自适应双局部增强算法

针对目前红外焦平面成像系统在观察目标、特别是极值温差目标时,各温度段灰度描述不均匀和细节不够的问题,提出了一种自适应红外图像双局部增强算法。详细介绍了通过空间分布和灰度统计特性两个方向实现对极值温差图像自适应增强的方法,该方法首先从红外图像的空间分布特性出发,将图像切割成多个局部图像,然后再从直方图灰度分布出发,将局部图像的直方图进行聚类分段,并对分段直方图均衡增强,最后对生成的每个局部图像增强结果进行线性插值拼接完成增强算法。通过在红外焦平面系统中实验证明了极值温差自适应的红外图像双局部增强算法的可行性,并获得了很好的效果,成像质量有明显提高。