Oracle Inequalities for Efromovich–Pinsker Blockwise Estimates

Oracle inequality is a relatively new statistical tool for the analysis of nonparametric adaptive estimates. Oracle is a good pseudo-estimate that is based on both data and an underlying estimated curve. An oracle inequality shows how well an adaptive estimator mimics the oracle for a particular underlying curve. The most advanced oracle inequalities have been recently obtained by Cavalier and Tsybakov (2001) for Stein type blockwise estimates used in filtering a signal from a stationary white Gaussian process. The authors also conjecture that a similar result can be obtained for Efromovich–Pinsker (EP) type blockwise estimators where their approach, based on Stein's formula for risk calculation, does not work. This article proves the conjecture and extends it upon more general models which include not stationary and dependent processes. Other possible extensions, a discussion of practical implications and a numerical study are also presented.     

Novel scaling behavior of directed polymers: Disorder distribution with long tails

We study the problem of directed polymers (DP) on a square lattice. The distribution of disorder is assumed to be independent but non-Gaussian. We show that for distributions with a power-law tailP() 1/||¹⁺ , where>2, so that the mean and variance are well defined, the scaling exponentv of the DP model depends on in a continuous fashion.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Extinction Effects of Multiplicative Non-Gaussian L′evy Noise in a Tumor Growth System with Immunization

The extinction phenomenon induced by multiplicative non-Gaussian Levy noise in a tumor growth model with immune response is discussed. Under the influence of the stochastic immune rate, the model is analyzed in terms of a stochastic differential equation with multiplicative noise. By means of the theory of the infinitesimal generator of Hunt processes, the escape probability, which is used to measure the noise-induced extinction probability of tumor cells, is explicitly expressed as a function of initial tumor cell density, stability index and noise intensity. Based on the numerical calculations, it is found that for different initial densities of tumor cells, noise parameters play opposite roles on the escape probability. The optimally selected values of the multiplicative noise intensity and the stability index are found to maximize the escape probability.     

Limit Gaussian Behavior of the Solutions of the Multidimensional Burger'; Equation with Weak-Dependent Initial Conditions

Asymptotic normality of the solutions of the initial-value problems for multidimensional Burgers' equation with weakly dependent possible non-Gaussian initial condition is proved.     

Multiple resonances with time delays in scale-free networks of Hodgkin-Huxley neurons subjected to non-Gaussian noise

The spiking behavior with varying time delay in scale-free networks of Hodgkin-Huxley neurons with non-Gaussian noise has been studied,and the effect of non-Gaussian noise on the delay-induced spiking behavior is discussed. It was found that multiple spatio-temporal resonances occur when the delay lengths are integer multiples of the spiking periods,and the resonances may be strengthened when the non-Gaussian noise is appropriate. This result shows that time delays can optimize the spiking temporal regulari...     

Structural first failure times under non-Gaussian stochastic behavior

An analytical moment-based method for calculating structural first failure times under non-Gaussian stochastic behavior is proposed.In the method,a power series that constants can be obtained from response moments (skewness,kurtosis,etc.) is used firstly to map a non-Gaussian structural response into a standard Gaussian process,then mean up-crossing rates,mean clump size and the initial passage probability of a critical barrier level by the original structural response are estimated,and finally,the formula for calculating first failure times is established on the assumption that corrected up-crossing rates are independent.An analysis of a nonlinear single-degree-of-freedom dynamical system excited by a Gaussian model of load not only demonstrates the usage of the proposed method but also shows the accuracy and efficiency of the proposed method by comparisons between the present method and other methods such as Monte Carlo simulation and the traditional Ganssian model.     

Image denoising using trivariate prior model in nonsubsampled dual-tree complex contourlet transform domain and non-local means filter in spatial domain

Image denoising is always the basic problem of image processing, and the main challenge is how to effectively remove the noise and preserve the detailed information. This paper presents a new image denoising algorithm based on the combination of trivariate prior model in nonsubsampled dual-tree complex contourlet transformlet transform (NSDTCT) domain and non-local means filter (NLMF) in spatial domain. Firstly, NSDTCT is constructed by combining the dual-tree complex wavelet transform (DTCWT) and nonsubsampled directional filter banks (NSDFB). The noisy image is decomposed by using NSDTCT. Secondly, based on the correlation between the interscale and intrascale dependencies of NSDTCT coefficients, the distribution of the high frequency coefficients is modeled with the trivariate non-Gaussian distribution model. A nonlinear trivariate shrinkage function is derived in the framework of Bayesian theory, and then the denoised coefficients are obtained and inverse NSDTCT is performed to get the initial denoised image. Finally, NLMF is used to smooth the initial denoised image. Simulation experiment shows that our algorithm can obtain better performances than those outstanding denoising algorithms in terms of peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) as well as visual quality.