Heavy-tailed Probability Density Functions Research Articles

Performance of Detectors for Cooperative Spectrum Sensing Under Laplacian Noise

Electronic systems in general can be impaired by impulsive noise generated by a variety of sources. Spectrum sensors are of particular interest herein, since their probabilities of detection and false alarm can be severely degraded under this impairment. Several models for impulsive noise have been studied in the literature, all of them having the common characteristic of being well represented by heavy-tailed probability density functions, like Laplace and some Stable distributions. This article addresses the performances of state-of-the-art detectors for cooperative spectrum sensing when the received signal is impaired by Laplacian noise. This is made by means of estimating the probability of detection for a fixed false alarm rate, when important system parameters are varied. It is demonstrated that the robustness against impulsive noise varies significantly depending on the adopted detection strategy.

Geometrical attacks resilient statistical watermark decoder using polar harmonic Fourier moments

This paper presents a new robust multiplicative watermark detector. Due to the strong robustness against various attacks, polar harmonic Fourier moment (PHFM) magnitudes are used as the employed watermark carrier. The distribution of PHFM magnitudes is highly non-Gaussian and can be properly modeled by a heavy-tailed probability density function (PDF). In this paper, we proved that Weibull distribution can suitably fit the distribution of PHFM magnitudes, and based on this, we presented a statistics-based watermark decoder by using the Weibull as a prior for the PHFM magnitudes. In watermark embedding, a multiplicative manner was used to embed watermark information in PHFM magnitudes of the highest entropy blocks to achieve better robustness and imperceptibility. In watermark detection, we developed a Weibull distribution-based statistical watermark decoder, which uses the maximum likelihood (ML) decision rule. Compared with Bessel K form (BKF), Cauchy, and generalized Gaussian (GG)-based decoders, the Weibull-based decoder demonstrates stronger robustness. In addition, the proposed watermark decoder is more robust against geometrical and common image processing attacks than existing statistical watermark decoders.

Multimodal Neuroimaging Fusion in Nonsubsampled Shearlet Domain Using Location-Scale Distribution by Maximizing the High Frequency Subband Energy

A fusion of medical imaging data obtained from different modalities plays an important role in the current clinical practice. In this paper, we propose a novel multimodal fusion algorithm for brain imaging data based on the statistical properties of nonsubsampled shearlet transform (NSST) coefficients and a novel energy maximization fusion rule. The marginal distributions of the high-frequency NSST coefficients exhibit heavier tails than the Gaussian distribution. As a consequence, after studying its characteristics, we use a heavy-tailed probability density function, student’s $t$ location-scale distribution, to describe the highly non-Gaussian statistics of empirical NSST coefficients by learning the parameters using maximum likelihood estimation. Then, we employ this model to develop a maximum a posteriori estimator to obtain the noise-free coefficients. Then, for the first time, a novel fusion rule for obtaining the fused NSST coefficients based on maximizing the energy in the high-frequency subbands is proposed. Experiments are carried out on fusing two or more multimodal neuroimages taken from the BrainWeb, Alzheimer’s Disease Neuroimaging Initiative (ADNI), and Whole Brain Atlas databases. It is seen from the subjective and objective results that the proposed multimodal neuroimaging fusion method significantly outperforms the state-of-the-art methods including under noisy scenarios, and hence, it is more robust. It is also observed that the signal intensities in the fused images are better enhanced when a more number of source images are being fused. The proposed technique should benefit the medical professionals in diagnosing neurological disorders, such as Alzheimer, epilepsy, and multiple sclerosis.

Heavy-Tailed Response of Structural Systems Subjected to Stochastic Excitation Containing Extreme Forcing Events

We characterize the complex, heavy-tailed probability density functions (pdfs) describing the response and its local extrema for structural systems subject to random forcing that includes extreme events. Our approach is based on recent probabilistic decomposition-synthesis (PDS) technique (Mohamad, M. A., Cousins, W., and Sapsis, T. P., 2016, “A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities,” J. Comput. Phys., 322, pp. 288–308), where we decouple rare event regimes from background fluctuations. The result of the analysis has the form of a semi-analytical approximation formula for the pdf of the response (displacement, velocity, and acceleration) and the pdf of the local extrema. For special limiting cases (lightly damped or heavily damped systems), our analysis provides fully analytical approximations. We also demonstrate how the method can be applied to high dimensional structural systems through a two-degrees-of-freedom (TDOF) example system undergoing extreme events due to intermittent forcing. The derived formulas can be evaluated with very small computational cost and are shown to accurately capture the complicated heavy-tailed and asymmetrical features in the probability distribution many standard deviations away from the mean, through comparisons with expensive Monte Carlo simulations.

Analysis of Array Model Errors in GARCH-Based DOA Estimation

In this paper, we consider the theoretical performance analysis of a DOA (direction-of-arrival) estimation method using GARCH (generalized autoregressive conditional heteroscedasticity) noise model. GARCH models are feasible for heavy tailed probability density functions and time-varying variances of stochastic process and also have flexible forms. We addressed a GARCH(1,1) based noise model in the maximum likelihood approach for the estimation of DOAs of impinging sources. In this paper, by means of perturbation approach we consider error analysis of GARCH-based array processing method and derive the theoretical DOA estimation errors versus array model errors such as gain, phase and sensors positions errors. The validity of the proposed analysis is confirmed by using numerical results.

Myriad non‐linearity for GNSS robust signal processing

The robustness of standard correlation-based Global Navigation Satellite System (GNSS) signal processing can be significantly improved by pre-processing the input samples with a zero-memory non-linearity (ZMNL). A paradigm for the design of ZMNLs is provided by the M-estimator framework where heavy-tailed probability density functions (pdfs) are used to model the impairments affecting the input samples. The myriad non-linearity, obtained considering a Cauchy pdf, is analysed for the acquisition and tracking of GNSS signals in the presence of pulsed interference. The impact of the myriad non-linearity is theoretically characterised and Monte Carlo simulations are used to support theoretical findings. Finally, real GNSS signals collected in the presence of jamming are processed: the myriad non-linearity provides a significant performance improvement with respect to standard GNSS signal processing which is unable to acquire and track the samples affected by interference.

Robust EM kernel-based methods for linear system identification

Recent developments in system identification have brought attention to regularized kernel-based methods. This type of approach has been proven to compare favorably with classic parametric methods. However, current formulations are not robust with respect to outliers. In this paper, we introduce a novel method to robustify kernel-based system identification methods. To this end, we model the output measurement noise using random variables with heavy-tailed probability density functions (pdfs), focusing on the Laplacian and the Student’s t distributions. Exploiting the representation of these pdfs as scale mixtures of Gaussians, we cast our system identification problem into a Gaussian process regression framework, which requires estimating a number of hyperparameters of the data size order. To overcome this difficulty, we design a new maximum a posteriori (MAP) estimator of the hyperparameters, and solve the related optimization problem with a novel iterative scheme based on the Expectation–Maximization (EM) method. In the presence of outliers, tests on simulated data and on a real system show a substantial performance improvement compared to currently used kernel-based methods for linear system identification.

Statistical analysis of polychaete population density: dynamics of dominant species and scaling properties in relative abundance fluctuations

Abstract. We consider here the dynamics of two polychaete populations based on a 20 yr temporal benthic survey of two muddy fine sand communities in the Bay of Morlaix, Western English Channel. These populations display high temporal variability, which is analyzed here using scaling approaches. We find that population densities have heavy tailed probability density functions. We analyze the dynamics of relative species abundance in two different communities of polychaetes by estimating in a novel way a "mean square drift" coefficient which characterizes their fluctuations in relative abundance over time. We show the usefulness of using new tools to approach and model such highly variable population dynamics in marine ecosystems.

Heavy-Tailed Prediction Error: A Difficulty in Predicting Biomedical Signals of1/fNoise Type

A fractal signal x(t) in biomedical engineering may be characterized by 1/f noise, that is, the power spectrum density (PSD) divergences at f = 0. According the Taqqu's law, 1/f noise has the properties of long-range dependence and heavy-tailed probability density function (PDF). The contribution of this paper is to exhibit that the prediction error of a biomedical signal of 1/f noise type is long-range dependent (LRD). Thus, it is heavy-tailed and of 1/f noise. Consequently, the variance of the prediction error is usually large or may not exist, making predicting biomedical signals of 1/f noise type difficult.

Asymptotic front behavior in anA+B→2Areaction under subdiffusion

We discuss the front propagation in the $A+B\rightarrow 2A$ reaction under subdiffusion which is described by continuous time random walks with a heavy-tailed power law waiting time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front propagation: an intermediate asymptotic regime given by the front solution of the corresponding continuous equation, and the final asymptotics, which is fluctuation-dominated and therefore lays out of reach of the continuous scheme. We moreover show that the continuous reaction subdiffusion equation indeed possesses a front solution that decelerates and becomes narrow in the course of time. This continuous description breaks down for larger times when the front gets atomically sharp. We show that the velocity of such fronts decays in time faster than in the continuous regime.

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We show that, under suitable confinement conditions, the ordinary Fokker–Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed pdfs known in the literature can be obtained this way. For the underlying diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when an initially infinite number of pdf moments decreases to a few or none at all. The time dependence of the variance (if in existence), ∼tγ with 0<γ<2, may in principle be interpreted as a signature of subdiffusive, normal diffusive or superdiffusive behavior under confining conditions; the exponent γ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.

Robust Time-Frequency Distributions with Complex-Lag Argument

The robust time-frequency distributions with complex-lag argument are proposed. They can provide an accurate estimation of fast varying instantaneous frequency in the presence of noise with heavy-tailed probability density function. The L-estimate form of this distribution is defined and it includes the L-estimate form of Wigner distribution as a special case. A modification for multicomponent signal representation is proposed, as well. Theoretical considerations are illustrated by the examples.

Detection-Threshold Approximation for Non-Gaussian Backgrounds

The detection-threshold (DT) term in the sonar equation describes the signal-to-noise ratio (SNR) required to achieve a specified probability of detection (Pd) for a given probability of false alarm (Pfa). Direct evaluation of DT requires obtaining the detector threshold (ft.) as a function of Pfa and then using h while inverting the often complicated relationship between SNR and Pd. However, easily evaluated approximations to DT exist when the background additive noise or reverberation is Gaussian (i.e., has a Rayleigh-distributed envelope). While these approximations are extremely accurate for Gaussian backgrounds, they are erroneously low when the background has a heavy-tailed probability density function. In this paper, it is shown that by obtaining h appropriately from the non-Gaussian background while approximating Pd for a target in the non-Gaussian background by that for a Gaussian background, the easily evaluated approximations to DT extend to non-Gaussian backgrounds with minimal loss in accuracy. Both fluctuating targets (FTs) and nonfluctuating targets (NFTs) are considered in Weibulland K-distributed backgrounds. While the Pd approximation for FTs is very accurate, it is coarser for NFTs, necessitating a correction factor to the DT approximations.

Characterization of short time fluctuations in atmospheric wind speeds by LIDAR measurements

Wind velocity measurements were taken using a Leosphere Windcube LIDAR system, which operates as a pulsed laser Doppler anemometer. Here we report on first results, which show typical characteristics of atmospheric wind velocities. Especially pronounced intermittent (or heavy-tailed) probability density functions (PDF) of wind speed increments are present on all observed time scales. A transition for large time scales towards a more Gaussian shape is found if only ten minute intervals are selected which share the same mean wind speed. Both typical shapes of PDF are well described and modeled by two previously proposed models.

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad-tailed distribution is similar for shear flows and time-varying nonsheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long-time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We document that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of small-scale scalar variation within the weak shear regions. For cellular flows we document a lack of self-similarity in the PDFs when periodic time dependence is present, in contrast to the self-similar decay for time independent flow. Finally, we analyze the behavior of the PDFs for coherent initial fields and the parametric dependence of the variance decay rate on the Péclet number and the initial wavenumber of the scalar field.

Particle acceleration in turbulent flows: A class of nonlinear stochastic models for intermittency

The problem of modeling the velocity and acceleration of inertial particles in turbulent flows is discussed. Particular attention is focused on the modeling of the particle Lagrangian velocity increment, especially, but not exclusively, in the case in which only the low frequencies of the carrier turbulent flow field are available. The need for suitable models arises in the simulation of particle laden flows by the means of new computational techniques such as large-eddy simulation. For this, stochastic differential equations, sde, are often introduced, though there is a lack of clarity in how such models should deal with the experimental observed far from Gaussian statistics, intermittency, and heavy tailed probability density function for particle acceleration. It is well known that Langevin-type equations are not capable of reproducing such features. It is first shown how the stochastic model for the particle Lagrangian velocity increments is far from being a Langevin equation, and it is characterized by nonlinear drift and diffusion; the statistical characteristics of this first model are shown to be in qualitative agreement with experimental findings. These results suggest an improved model for the particle dynamics based upon a more general family of nonlinear sde; the family, which is generated by a single parameter, includes both the Langevin equation and the first model as special cases. An analysis of the statistical properties of the new sde shows that the model is capable of accurately reproducing the strong deviations from Gaussianity observed in recent experiments.

T-distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p( x) ∼ 1/ x μ+1 , and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.

Detectors based on stochastic resonance, Part 2: Convergence analysis and perturbative corrections

This paper considers convergence properties and perturbative corrections for stochastic resonant (SR) detectors operating in a realistic marine environment. The description of such a detector is reviewed in the weak signal limit. A deterministic algorithm is developed to globally optimize the performance of the SR detector. It is established that this algorithm converges with logarithmic complexity. Scaling arguments demonstrate an improvement over standard, deterministic algorithms in two important limiting cases: (i) increasing accuracy of the optimization procedure and (ii) increasingly heavy-tailed marine noise probability density functions (PDFs), corresponding to increasingly turbulent ocean conditions. Perturbative corrections due to small nonzero input SNRs, temporal drift in the marine noise PDF and the effect of uncertainty in signal frequency are derived. The correction due to frequency error is found to be of second order, and hence subdominant to the former two which are of first order. For a class of marine noise PDFs, these corrections are expressed in terms of standard mathematical functions which are easy to compute in real-time. Numerical simulations indicate that the SR detector is stable under the perturbative corrections considered and that finite input SNRs constitute the dominant perturbative effect for standard ocean acoustic scenarios. It is stressed that these results imply efficient and inexpensive upgrades to existing sonar hardware.

High quantile estimation for heavy-tailed distributions

Different estimators of high quantiles, such as x p c proposed in [N.M. Markovitch, U.R. Krieger, The estimation of heavy-tailed probability density functions, their mixtures and quantiles. Computer Networks 40 (3) (2002) 459–474], Weissman’s estimator x p w and the POT-method are considered. Regarding the estimators x p c and x p w the asymptotic normality of the logarithms of ratios of these estimators to the true value of the quantile is proved. These estimators are applied to real data of Web sessions and pages. Furthermore, bootstrap confidence intervals of x p c and x p w are constructed for modelled data of different heavy-tailed distributions as well as for Web-traffic data.