Α-stable Model Research Articles

Simple Lévy α-Stable Model Analysis of Elastic pp and pp¯ Low-|t| Data from SPS to LHC Energies

A simple Lévy α-stable (SL) model is used to describe the data on elastic pp and pp¯ scattering at low-|t| from SPS energies up to LHC energies. The SL model is demonstrated to describe the data with a strong non-exponential feature in a statistically acceptable manner. The energy dependence of the parameters of the model is determined and analyzed. The Lévy α parameter of the model has an energy-independent value of 1.959 ± 0.002 following from the strong non-exponential behavior of the data. We strengthen the conclusion that the discrepancy between TOTEM and ATLAS elastic pp differential cross section measurements arises only in the normalization and not in the shape of the distribution of the data as a function of t. We find that the slope parameter has different values for pp and pp¯ elastic scattering at LHC energies. This may be the effect of the odderon exchange or the jump in the energy dependence of the slope parameter in the energy interval 3 GeV ≲s≲ 4 GeV.

Lévy α-Stable Model for the Non-Exponential Low-|t| Proton–Proton Differential Cross-Section

It is known that the Real Extended Bialas–Bzdak (ReBB) model describes the proton–proton (pp) and proton–antiproton (pp¯) differential cross-section data in a statistically non-excludible way, i.e., with a confidence level greater than or equal to 0.1% in the center of mass energy range 546 GeV ≤s≤8 TeV and in the squared four-momentum transfer range 0.37 GeV2 ≤ −t ≤ 1.2 GeV2. Considering, instead of Gaussian, a more general Lévy α-stable shape for the parton distributions of the constituent quark and diquark inside the proton and for the relative separation between them, a generalized description of data is obtained, where the ReBB model corresponds to the α=2 special case. Extending the model to α<2, we conjecture that the validity of the model can be extended to a wider kinematic range, in particular, to lower values of the four-momentum transfer −t. We present the formal Lévy α-stable generalization of the Bialas–Bzdak model and show that a simplified version of this model can be successfully fitted, with α<2, to the non-exponential, low −t differential cross-section data of elastic proton–proton scattering at s=8 TeV.

Flexible ECG signal modeling and compression using alpha stable functions

In this paper, we propose a flexible modeling technique for the ECG signals. Modeling is achieved by considering the weighted summation of elementary functions representing the waveforms that describe each component of the cardiac cycle. We thus evaluate the benefit brought by α-stable functions with respect to Gaussian functions in terms of modeling precision. Seven records from the MIT-BIH arrhythmia database have been used to assess the performances of the proposed modeling method, including Normal beat, Premature Ventricular Contraction beat, Right Bundle Block Branch beat and Paced beat. When applied on the chosen records, it turns out that α-stable modeling always outperforms Gaussian modeling. Since each waveform is related to a particular physiological event in the ECG cardiac cycle, we also exploit flexibility of choosing α-stable modeling instead of Gaussian one for some event of medical interest in order to solve compression purpose efficiency-quality compromise. The comparison of the α-stable model applied in compression with other techniques proves the efficiency of the proposed method, mainly in term of quality score.

Information theoretic evaluation of Lorentzian, Gaussian, Voigt, and symmetric alpha-stable models of reversible transverse relaxation in cervical cancer in vivo at 3T.

To better characterize cervical cancer at 3T. MRI transverse relaxation patterns hold valuable biophysical information about cellular scale microstructure. Lorentzian modeling is typically used to represent intravoxel frequency distributions, resulting in mono-exponential decay of reversible transverse relaxation. However, deviations from mono-exponential decay are expected theoretically and observed experimentally. We compared the information content of four models of signal attenuation with reversible transverse relaxation. Biological phantoms and six women with cervical squamous cell carcinoma were imaged using a gradient-echo sampling of the spin-echo (GESSE) sequence. Lorentzian, Gaussian, Voigt, and Symmetric α-Stable (SAS) models were ranked using Akaike's Information Criterion (AIC), and the model retaining the highest information content was identified at each voxel as the best model. The Lorentzian model resulted in information loss in large fractions of the phantoms and cervix. Gaussian and SAS models frequently had higher information content than the Lorentzian in much of the areas of interest. The Voigt model rarely surpassed the three other models in terms of information content. Gaussian and SAS models provide better fitting of data in much of the human cervix at 3T. Minimizing information loss through improved tissue modeling may have important implications for identifying reliable biomarkers of tumor hypoxia and iron deposition.

Expectation–Maximization algorithm for finite mixture of [formula omitted]-stable distributions

A Gaussian Mixture Model (GMM) is a parametric probability density function built as a weighted sum of Gaussian distributions. Gaussian mixtures are used for modelling the probability distribution in many fields of research nowadays. Nevertheless, in many real applications, the components are skewed or heavy tailed. For that reason, it is useful to model the mixtures as components with α-stable distribution.In this work, we present a mixture of skewed α-stable model where the parameters are estimated using the Expectation–Maximization algorithm. As the Gaussian distribution is a particular limiting case of α-stable distribution, the proposed model is a generalization of the widely used GMM.The proposed algorithm is much faster than the parameter estimation of the α-stable mixture model using a Bayesian approach and Markov chain Monte Carlo methods. Therefore, it is more suitable to be used for large vector observations.

Merging the Bernoulli–Gaussian and symmetric [formula omitted]-stable models for impulsive noises in narrowband power line channels

To model impulsive noise in power line channels, both the Bernoulli–Gaussian model and the symmetric α-stable model are usually applied. Towards a merge of existing noise measurement databases and a simplification of communication system design, the compatibility between the two models is of interest. In this paper, we show that they can be approximately converted to each other under certain constrains, although never generally unified. Based on this, we propose a fast model conversion.

Sign Function Based Sparse Adaptive Filtering Algorithms for Robust Channel Estimation under Non-Gaussian Noise Environments

Robust channel estimation is required for coherent demodulation in multipath fading wireless communication systems which are often deteriorated by non-Gaussian noises. Our research is motivated by the fact that classical sparse least mean square error (LMS) algorithms are very sensitive to impulsive noise while standard SLMS algorithm does not take into account the inherent sparsity information of wireless channels. This paper proposes a sign function based sparse adaptive filtering algorithm for developing robust channel estimation techniques. Specifically, sign function based least mean square error (SLMS) algorithms to remove the non-Gaussian noise that is described by a symmetric α-stable noise model. By exploiting channel sparsity, sparse SLMS algorithms are proposed by introducing several effective sparse-promoting functions into the standard SLMS algorithm. The convergence analysis of the proposed sparse SLMS algorithms indicates that they outperform the standard SLMS algorithm for robust sparse channel estimation, which can be also verified by simulation results.

On the Use of Alpha-Stable Distributions in Noise Modeling for PLC

Power-line communication (PLC) deals with the challenge of transmitting data over a channel impaired by background and impulsive noises. In this paper, we provide evidence that the marginal distribution of the noise found in PLC systems exhibits some statistical properties that can be well captured by using α-stable distribution. Motivated by this result, we present a practical application of the α-stable model to synthesize noise in the power line. Furthermore, we also show that this family of distributions is a suitable alternative in order to model the PLC background noise that is commonly assumed as Gaussian. Finally, we study the performance of an orthogonal frequency-division multiplexing communication system under both kinds of background noise (i.e., Gaussian and α-stable). From these experiments, we conclude that if highly impulsive background traffic is mistakenly assumed as Gaussian in evaluation studies, the system performance is largely overestimated.

Understanding the Traffic Nature of Mobile Instantaneous Messaging in Cellular Networks: A Revisiting to <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-Stable Models

Mobile instantaneous messaging (MIM) services significantly facilitate personal and business communications, inevitably consume substantial network resources, and potentially affect the network stability. In this paper, we aim to understand the traffic nature of MIM in cellular networks. Specifically, in order to reach credible conclusions, our research takes account of practical measurement records of MIM services from China Mobile at two different levels. First, a data set of individual message level (IML) traffic is exploited and reveals power-law distributed message length and lognormal distributed interarrival time, the heavy-tailness of which completely diverts from the geometric model and the exponential model recommended by the 3rd generation partnership project (3GPP). Second, another data set considers the statistical pattern of aggregated traffic within one whole base station, and demonstrates the accuracy of α-stable models for the aggregated traffic. Furthermore, it verifies that the α-stable models are suitable for characterizing the traffic in both the conventional fixed core networks and the cellular access networks. At last, with the aid of the generalized central limit theorem, we build up a theoretical relation between the distributions of IML traffic and aggregated traffic.

Dynamic alpha-stable method for CDO pricing

The goal of this paper is to develop the dynamic alpha (α)-stable method for collateralized debt obligation (CDO) pricing based on the α-stable distributions, which will resolve the two issues caused by using traditional static factor copula method in the practice, which means when pricing CDOs, the traditional static factor Copula method does not only exhibit the correlation smile phenomenon which is not inconsistent with the model's assumption, but also cannot be used in pricing CDOs or credit portfolio derivatives for the underlying portfolio with different maturities. As the applications, we present calibration and empirical numerical results for iTraxx Europe Tranches quotes from the market data on March 30, 2007. Thus, our new method under the framework of the dynamic α-stable model is the way for CDO pricing in the practice, and should be useful for the risk management in the practice too.

Joint estimation of state and noise parameters in a linear dynamic system with impulsive measurement noise: Application to OFDM systems

Interference mitigation is one of the main challenges in wireless communication, especially in ad hoc networks. In such context, the Multiple Access Interference (MAI) is known to be of an impulsive nature. Therefore, the conventional Gaussian assumption is inadequate to model this type of interference. Nevertheless, it can be accurately modeled by stable distributions. In fact, it was shown in literature that the α-stable distribution is a useful tool to model impulsive data. In this paper, we tackle the problem of noise compensation in ad hoc networks. More precisely, this issue is addressed within an Orthogonal Frequency Division Multiplexing (OFDM) transmission link assuming a symmetric α-stable model for the signal distortion due to MAI. Based on Bayesian estimation, the proposed approach estimates the transmitted OFDM symbols in the time domain using the Sequential Monte Carlo (SMC) methods. Unlike existing schemes, we consider the more realistic case where the impulsive noise parameters are assumed to be unknown at the receiver. Consequently, our approach deals also with the difficult task of noise parameters estimation which can be very useful for other purposes such as target tracking in wireless sensor networks or channel estimation. Simulations results, provided in terms of Mean Square Error (MSE) and Bit Error Rate (BER), illustrate the efficiency and the robustness of this scheme.

Image quality/distortion metric based on α-stable model similarity in wavelet domain

Quality assessment is of central importance in numerous image processing tasks. State-of-the-art objective image quality assessment (IQA) algorithms are generally devised for specific distortion types or based on training procedure of large databases. In this work, we propose a general-purpose full-reference/no-reference (FR/NR) IQA framework for image distortions, nominated by Image Quality/Distortion Metric (IQDM). The leptokurtic and heavy-tailed behaviors of image wavelet coefficients are characterized by symmetric α-stable (SαS) density, and the statistical studies indicate that the model parameters may be altered because of the presence of distortion. This important priori knowledge of original image’s distribution is then used to gauge the distortion between degraded and reference SαS models in multi-scale wavelet sub-bands. We investigate the relationship between original and degraded parameters over scales, accordingly infer the original parameters from the degraded ones. A characteristic probability density function for SαS and its closed-form Kullback–Leibler distance are derived for FR/NR-IQDM using the model parameters. Extensive experiments and comparisons demonstrate that the proposed FR/NR-IQDM scheme is efficacious to most common types of distortion, and leads to a highly comparable performance to the benchmarks and prevalent competitors in consistency with subjective judgements.

PERFORMANCE EVALUATION OF ADAPTIVE RECEIVERS FOR UWB COMMUNICATION

UWB (Ultra Wide Band) is a radio technology used at very low energy levels for short-range, high-bandwidth communications using a large portion of the radio spectrum. But MAI (Multiple Access Interference) degrades the performance of UWB systems. To suppress this, an α-stable model along with Fama/Roll-McCulloch parameter estimation technique is used. This model adapts to the noise and interference in the environment and hence it forms a robust technique for adaptive receiver designs. But this technique shows BER performance clearly only in high SNR region. Hence matched filter techniques are proposed here. From simulation results, it is verified that the proposed method provides better BER than the linear and myriad filter detectors.

Novel Adaptive Nonlinear Receivers for UWB Multiple Access Communications

Ultra-wide bandwidth communication systems are known to suffer from non-Gaussian interference. In general, wireless ad-hoc networks, due to the lack of perfect power control and the random placement of nodes, are very likely to experience non-Gaussian multiple access interference. In systems with additive non-Gaussian noise, non-Gaussian signal detection and nonlinear signal processing can play important roles, especially in suppressing the multiple access interference. Practical ultra-wide bandwidth receiver designs based on the α-stable non-Gaussian additive impairment model are proposed and evaluated. It is demonstrated that the α-stable model together with a fractile based parameter estimation technique constitute a robust technique for adaptive receiver designs. The proposed receivers based on the α-stable model suppress multiple access interference, adapt to the noise and the interference in the environment, and are robust to estimation errors in the receiver parameters. The proposed receivers provide much better bit error rates than linear detectors and previously known nonlinear detectors.

Features modeling with an α-stable distribution: Application to pattern recognition based on continuous belief functions

The aim of this paper is to show the interest in fitting features with an α-stable distribution to classify imperfect data. The supervised pattern recognition is thus based on the theory of continuous belief functions, which is a way to consider imprecision and uncertainty of data. The distributions of features are supposed to be unimodal and estimated by a single Gaussian and α-stable model. Experimental results are first obtained from synthetic data by combining two features of one dimension and by considering a vector of two features. Mass functions are calculated from plausibility functions by using the generalized Bayes theorem. The same study is applied to the automatic classification of three types of sea floor (rock, silt and sand) with features acquired by a mono-beam echo-sounder. We evaluate the quality of the α-stable model and the Gaussian model by analyzing qualitative results, using a Kolmogorov–Smirnov test (K–S test), and quantitative results with classification rates. The performances of the belief classifier are compared with a Bayesian approach.

Likelihood-free Bayesian inference for [formula omitted]-stable models

α-stable distributions are utilized as models for heavy-tailed noise in many areas of statistics, finance and signal processing engineering. However, in general, neither univariate nor multivariate α-stable models admit closed form densities which can be evaluated pointwise. This complicates the inferential procedure. As a result, α-stable models are practically limited to the univariate setting under the Bayesian paradigm, and to bivariate models under the classical framework. A novel Bayesian approach to modelling univariate and multivariate α-stable distributions is introduced, based on recent advances in “likelihood-free” inference. The performance of this procedure is evaluated in 1, 2 and 3 dimensions, and through an analysis of real daily currency exchange rate data. The proposed approach provides a feasible inferential methodology at a moderate computational cost.

α-stable multiple access interference modelling for amplify-and-forward multihop ad hoc networks

A multihop amplify-and-forward relaying network that considers multiple access interference is investigated. Owing to the ad hoc configuration, the interference distribution is not Gaussian. A symmetric α-stable model is proposed that adapts to the impulsive behaviour owing to its heavy tailed distribution. Based on the stability property, analytical work is developed for multihop interference modelling and the theoretical expressions of the error probability that depend on two parameters are then derived.

α-stable interference modeling and cauchy receiver for an IR-UWB Ad Hoc network

In this paper we investigate ad hoc networks based on impulse radio ultra wideband. Due to multiple access, the interference distribution is not Gaussian. One important reason for errors is the presence of close interferers generating pulse collision. However such events are rare and we propose an α-stable model compatible with this fact due to its heavy tailed distribution. We derive the analytical expression of the two significant parameters. They depend on the attenuation coefficient, the users' density, the pulse collision probability and the pulse shape. We finally propose receiver strategies (Cauchy receiver and p-norm) that outperforms the classical Gaussian receiver.

Characterizing impulsive network traffic using truncated α-stable processes

It has been recently recognized that aggregated traffic in a variety of networks exhibits a similar impulsiveness over a wide range of aggregation levels, but approaches a Gaussian distribution in the limit as the aggregation level grows. Although several traffic models have been proposed in the past decade, their accuracy in simultaneously characterizing the above properties still needs to be further improved. In this letter, we propose a truncated α-stable process model which is able to capture the impulsiveness of observed network traffic as well as its tendency toward the Gaussian distribution with aggregation. An inherent physical mechanism is also proposed to give insight into the underlying meaning of the proposed model. Simulation results show that the proposed process achieves close agreement with real traffic and outperforms previous models.

On alpha stable distribution of wind driven water surface wave slope

We propose a new formulation of the probability distribution function of wind driven water surface slope with an α-stable distribution probability. The mathematical formulation of the probability distribution function is given under an integral formulation. Application to represent the probability of time slope data from laboratory experiments is carried out with satisfactory results. We compare also the α-stable model of the water surface slopes with the Gram–Charlier development and the non-Gaussian model of Liu et al. [J. Phys. Oceanogr. 27, 782 (1997)]. Discussions and conclusions are conducted on the basis of the data fit results and the model analysis comparison.