Α-stable Distribution Research Articles

Non-Gaussian amplitude PDF modeling of ultrasound images based on a novel generalized Cauchy-Rayleigh mixture

In this paper, a mixture of generalized Cauchy distribution and Rayleigh distribution that possesses a closed-form expression is proposed for modeling the heavy-tailed Rayleigh (HTR) distribution. This new approach is developed for analytically modeling the amplitude distribution of ultrasound images based on the HTR distribution. HTR as a non-Gaussian distribution is basically the amplitude probability density function (PDF) of the complex isotropic symmetric α-stable (S α S) distribution which appears in the envelope distribution of ultrasonic images. Analytic expression for HTR distribution is a momentous consideration in signal processing with stable random variables. Furthermore, we introduce a mixture ratio estimator based on the energy of amplitude PDF which contains both α and γ parameters. For a quantitative assessment, we compare the accuracy and computational complexity of the proposed mixture with other approximations of HTR distribution through several numeral simulations on synthetic random samples. Experimental results obtained from the Kolmogorov-Smirnov (K-S) distance and Kullback-Leibler (K-L) divergence as the goodness-of-fit tests on real ultrasound images reveal the favor of the new mixture model.

Application of tempered stable distribution for selection of optimal frequency band in gearbox local damage detection

Local damage detection methods based on the vibration signal analysis are widely discussed in the literature. One of the most popular approach is signal enhancement by filtering focused on extraction of informative content at so called informative frequency band (IFB). In such approach the vibration signal first is decomposed into time-frequency representation and then the measures of impulsivity are applied to the appropriate sub-signals. Till now, kurtosis was preferred as criterion for IFB search. However, for some cases more robust criteria should be used. Jablonski proposed to calculate kurtosis from envelope spectrum, Urbanek suggested MID (Modulation Intensity Distribution), set of alternative selectors as extension of spectral kurtosis (SK) have been defined by Obuchowski. Further extension was recently proposed by Zak, namely statistics related to α-stable distribution were used as IFB indicators. This distribution is especially important in modeling data with outliers so it was reasonable to apply this approach in the considered problem. As it was shown, the α-stable based methodology for some vibration signals more clearly indicates the IFB in contrast to the classical ones. However, in case of early stage of fault development or developed fault with high level of noise the mentioned criteria (including stability index as a measure of impulsivity) may be insufficient. It is related to the fact that cyclic impulses related to damage are often hidden in the noise (even after time-frequency decomposition). In this paper we propose to apply (instead of kurtosis or stability index) the parameters of tempered stable distribution. This distribution is an extension of the α-stable one, however, it possesses many properties of Gaussian systems. It is associated with two parameters which may indicate properly the IFB in case of poor signal-to-noise ratio. In this paper we remind the approach of IFB selection based on the kurtosis and α-stable distribution and describe how to localize the information of the fault by using tempered stable approach. The real vibration signal from gearbox is analyzed in the context of presented methodology. Finally, we examine two examples of vibration signals for which the proposed approach is superior with respect to the classical methodology.

Self-adjoint integral operator for bounded nonlocal transport.

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

PDF approximations to estimation and detection in time-correlated alpha-stable channels

This paper considers the estimation and detection problems for statistically dependent heavy-tailed signals with no closed-form probability density function (PDF). We propose two parametric PDF approximations for symmetric α-stable (SαS) distribution to be utilized in approaches based on the Maximum likelihood (ML) criterion. The nonlinear least square (LS) and curve fitting are used to compute parameters of the new formulations which are functions of the characteristic exponent. Moreover, we study binary signal detection in channels with time-dependent heavy-tailed noise modeled by SαS distribution and first order autoregressive (AR(1)) process. Using the novel PDF approximations in the ML estimator, an algorithm for model parameters estimation of the noise is initially developed. Then, new suboptimal receivers are designed through the use of the new PDF formulations and parameter estimates. Numerical results demonstrate the superiority of the proposed approximations over the existing formulations, and also good accuracy for the estimation algorithm. Additionally, it is shown that the proposed detectors operate near optimal receiver and also outperform the other suboptimal detectors, especially when α is small.

Robust digital non-linear self-interference cancellation in full duplex radios with maximum correntropy criterion

Full duplex radio increases the frequency efficiency but its performance is limited by the self-interference (SI). We first analyze the multiple noises in the full duplex radio system and model such noises as an α-stable distribution. Then we formulate a novel non-Gaussian SI problem. Under the maximum correntropy criterion (MCC), a robust digital non-linear self-interference cancellation algorithm is proposed for the SI channel estimation. A gradient descent based algorithm is derived to search the optimal solution. Simulation results show that the proposed algorithm can achieve a smaller estimation error and a higher pseudo signal to interference plus noise ratio (PSINR) than the well-known least mean square (LMS) algorithm and least square (LS) algorithm.

Series expansions for the all-time maximum of α-stable random walks

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

Delving into α-stable distribution in noise suppression for seizure detection from scalp EEG

Objective. There is serious noise in EEG caused by eye blink and muscle activities. The noise exhibits similar morphologies to epileptic seizure signals, leading to relatively high false alarms in most existing seizure detection methods. The objective in this paper is to develop an effective noise suppression method in seizure detection and explore the reason why it works. Approach. Based on a state-space model containing a non-linear observation function and multiple features as the observations, this paper delves deeply into the effect of the α-stable distribution in the noise suppression for seizure detection from scalp EEG. Compared with the Gaussian distribution, the α-stable distribution is asymmetric and has relatively heavy tails. These properties make it more powerful in modeling impulsive noise in EEG, which usually can not be handled by the Gaussian distribution. Specially, we give a detailed analysis in the state estimation process to show the reason why the α-stable distribution can suppress the impulsive noise. Main results. To justify each component in our model, we compare our method with 4 different models with different settings on a collected 331-hour epileptic EEG data. To show the superiority of our method, we compare it with the existing approaches on both our 331-hour data and 892-hour public data. The results demonstrate that our method is most effective in both the detection rate and the false alarm. Significance. This is the first attempt to incorporate the α-stable distribution to a state-space model for noise suppression in seizure detection and achieves the state-of-the-art performance.

A formal power series expansion–regularization approach for Lévy stable distributions: the symmetric case with (M positive integer)

In practice, the Lévy α-stable distribution is usually expressed in terms of the Fourier integral of its characteristic function. Indeed, known closed form expressions are relatively scarce given the huge parameters space: , and . Hence, systematic efforts have been made towards the development of proper methods for analytically solving the mentioned integral. As a further contribution in this direction, here we propose a new way to tackle the problem. We consider an approach in which one first solves the Fourier integral through a formal (thus not necessarily convergent) series representation. Then, one uses (if necessary) a pertinent sum-regularization procedure to the resulting divergent series, so as to obtain an exact formula for the distribution, which is amenable to direct numerical calculations. As a concrete study, we address the centered, symmetric, unshifted and unscaled distribution (, , ), with , . Conceivably, the present protocol could be applied to other sets of parameter values.

Characteristic function based parameter estimation of skewed alpha-stable distribution: An analytical approach

This paper introduces a new technique for analytical parameter estimation of skewed α-stable distribution with 1<α≤2. Stable distribution as a four-parameter non-Gaussian distribution is completely characterized by its characteristic function (CF). There are some serious limitations in parameter estimation of α-stable distribution due to the lack of closed-form expression for the general α-stable probability density function (PDF). The proposed estimator uses a hierarchical framework based on the skewed α-stable CF, and hence, allows a rapid estimation of parameters with high accuracy in real-time signal processing algorithms. In our scheme, only two values of α-stable CF, which has analytic formula, are utilized to estimate the parameters of α-stable density. In addition, the closed-form expression for estimating the required values of CF is derived. To provide a precise quantitative assessment, our proposed approach is compared with three other state-of-the-art estimators which have analytic formulas through a series of goodness-of-fit tests. Simulation results also demonstrate that the proposed method has a good accuracy both for the symmetric and non-symmetric (skewed) α-stable distributions. Furthermore, the advantage of the proposed CF based method becomes more evident through the experimental results obtained from the high-resolution SAR images.

PTAS distributions with application to risk management

The family of positive tempered α-stable (pTAS) or sometimes also tempered one-sided α-stable distributions dates back to Tweedie (1984) and Hougaard (1986) who discussed it in the context of frailty distribution in life table methods for heterogenous populations. The pTAS family generalizes the well-known gamma distribution and allows for heavier tails depending on the parameter α. Because of this property, pTAS distributions appear to be useful in the context of risk management. Against this background, the contribution of his work is three-fold: Firstly, we summarize the properties of the pTAS family. Secondly, we describe its numerical implementation and illustrate the functions by means of R examples in the Appendix. Thirdly, we empirically demonstrate that this family can be successfully applied in risk management. Concretely, applications to credit and operational risk are given.

Rate-distortion function for [formula omitted]-stable sources

In this paper, we develop a numerical approximation based on the Blahut–Arimoto algorithm to the rate distortion function of sources with α-stable distribution both for the symmetric and the skewed cases. The calculated rate-distortion function provides bounds for lossy source coding/data compression and the achievable rates for a given distortion.

Power-law cross-correlations estimation under heavy tails

We examine the performance of six estimators of the power-law cross-correlations—the detrended cross-correlation analysis, the detrending moving-average cross-correlation analysis, the height cross-correlation analysis, the averaged periodogram estimator, the cross-periodogram estimator and the local cross-Whittle estimator—under heavy-tailed distributions. The selection of estimators allows to separate these into the time and frequency domain estimators. By varying the characteristic exponent of the α-stable distributions which controls the tails behavior, we report several interesting findings. First, the frequency domain estimators are practically unaffected by heavy tails bias-wise. Second, the time domain estimators are upward biased for heavy tails but they have lower estimator variance than the other group for short series. Third, specific estimators are more appropriate depending on distributional properties and length of the analyzed series. In addition, we provide a discussion of implications of these results for empirical applications as well as theoretical explanations.

Robust adaptive algorithm for active control of impulsive noise

Active noise control (ANC) systems employing adaptive filters suffer from stability issues in the presence of impulsive noise. To overcome this limitation, new methods must be investigated. In this paper, we propose the filtered-x state-space recursive least square (FxSSRLS), an SSRLS-based practical and adaptive algorithm for ANC. Computer simulations are executed to verify the enhanced performance of the FxSSRLS algorithm. Symmetric α-stable (SαS) distributions are used to model impulsive noise. The results show that the proposed FxSSRLS algorithm is more robust in eliminating high-peaked impulses than the recently reported algorithms for ANC applications. Moreover, the suggested solution exhibits better stability and faster convergence, without jeopardizing the performance of the proposed solution in terms of residual noise suppression in the presence of impulses.

Data-driven vibration signal filtering procedure based on the α-stable distribution

A novel procedure for data-driven enhancement of informative signal is presented in this paper. The introduced methodology covers decomposition of the signal via time-frequency spectrogram into set of narrowband sub-signals. Furthermore, each of the sub-signals is considered as a sample of independent identically distributed random variables and we model the distribution of the sample, in contrast to the classical methodology where the simple statistics, for example kurtosis, for each sub-signal was calculated. This approach provides a new perspective in the signal processing techniques for local damage detection. Using our methodology one can eliminate potential risk related to high sensitivity towards single outlier. In the proposed procedure we model each sub-signal in time-frequency representation by α-stable distribution. This distribution is a generalization of standard Gaussian one and allows us for modeling sub-signals related to both informative and non-informative frequencies. As a result, we obtain distribution of stability parameter vs. frequencies that is analogy to spectral kurtosis approach well known in the literature. Such characteristic is basis for filter design used for raw signal enhancement. To evaluate efficiency of our method we compare raw and filtered signal in time, time-frequency and frequency (envelope spectrum) domains. Moreover, we present comparison to the spectral kurtosis approach. The presented methodology we applied to simulated signal and real vibration signal from two stage heavy duty gearbox used in mining industry.

Application of α-stable mutation in a hierarchic evolutionary inverse solver

The multi-deme Hierarchic Genetic Strategy (HGS) developed at the end of the 20th century already proved its capabilities of solving ill-conditioned multi-modal continuous global optimization problems, both benchmarks and real-world engineering inversions. As a standard it uses the mutation operator based on the normal probability distribution. It is a common choice in the continuous evolutionary optimization, but in practice it exhibits some properties that significantly reduce its exploratory abilities, which are crucial in the search for multiple solutions. Those drawbacks can be largely overcome if we replace the normal distribution with a special α-stable distribution for α<2. In this paper, we study the application of such distribution in the HGS mutation operator. First, we execute standard multi-modal benchmarks to show the impact of particular values of the stable distribution parameters. Then, using selected values of those parameters we employ the HGS with the α-stable mutation in solving an advanced ill-conditioned inverse parametric problem connected to the oil and gas resource investigation. The obtained results show that the α-stable mutation delivers more solutions than the classical normal mutation within a slightly better time budget. Another important conclusion is that the number of solutions is significantly more predictable in the case of α-stable mutation, which is a very advantageous feature from the point of view of the application of a stochastic strategy in the inverse problem solution.

A discussion on the innovation distribution of the Markov regime-switching GARCH model

The Markov Regime-Switching Generalized autoregressive conditional heteroskedastic (MRS-GARCH) model is a widely used approach to model the financial volatility with potential structural breaks. The original innovation of the MRS-GARCH model is assumed to follow the Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Many existing studies point out that this problem can lead to inconsistent estimates. To overcome it, the Student's t-distribution and General Error Distribution (GED) are the two most popular alternatives. However, a recent study points out that the Student's t-distribution lacks stability. Also, it incorporates the α-stable distribution in the GARCH-type model. The issue of the α-stable distribution is that its second moment does not exist. To solve this problem, the tempered stable distribution, which retains most characteristics of the α-stable distribution and has defined moments, is a natural candidate. In this paper, we conduct a series of simulation studies to demonstrate that MRS-GARCH model with tempered stable distribution consistently outperform that with Student's t-distribution and GED. Our empirical study on the S&P 500 daily return volatility also generates robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modeling the financial volatility in general contexts with a MRS-GARCH-type specification.

Finite Length Analysis of Low-Density Parity-Check Codes on Impulsive Noise Channels

Low-density parity-check (LDPC) codes with very long block lengths are well known for their powerful error correction, but it is not always desirable to employ long codes in communication systems, where latency is a serious issue, such as voice and video communication between multiple users. Finite length analyses of LDPC codes have already been presented in the literature for the additive white Gaussian noise channel, but in this paper, we consider the finite length analysis of LDPC codes for channels that exhibit impulsive noise. First, an exact uncoded bit error probability (BEP) of an impulsive noise channel, modeled as a symmetric α-stable (SαS) distribution, is derived. Then, to obtain the LDPC-coded performance, density evolution is applied to evaluate the asymptotic performance of LDPC codes on SαS channels and determine the threshold signal-to-noise ratio. Finally, we derive closed-form expressions for the BEP and block error probability of short LDPC codes on these channels, which are shown to match closely with simulated results on channels with different levels of impulsiveness, even for block lengths as low as 1000 b.

Has the 2008 financial crisis affected stock market efficiency? The case of Eurozone

In this paper, the impact of the 2008 financial crisis on the weak-form efficiency of twelve Eurozone stock markets is investigated empirically. Efficiency is tested via the Generalized Hurst Exponent method, while dynamic Hurst exponents are estimated by means of the rolling window technique. To account for biases associated with the finite sample size and the leptokurtosis of the financial data, the statistical significance of the Hurst exponent estimates is assessed through a series of Monte-Carlo simulations drawn from the class of α-stable distributions. According to our results, the 2008 crisis has adversely affected stock price efficiency in most of the Eurozone capital markets, leading to the emergence of significant mean-reverting patterns in stock price movements.

Entanglement Properties and Quantum Phases for a Fermionic Disordered One-Dimensional Wire with Attractive Interactions.

A fermionic disordered one-dimensional wire in the presence of attractive interactions is known to have two distinct phases, a localized and superconducting, depending on the strength of interaction and disorder. The localized region may also exhibit a metallic behavior if the system size is shorter than the localization length. Here we show that the superconducting phase has a distribution of the entanglement entropy distinct from the metallic regime. The entanglement entropy distribution is strongly asymmetric with a Lévy α-stable distribution (compared to the Gaussian metallic distribution), as is seen also for the second Rényi entropy distribution. Thus, entanglement properties may reveal properties which cannot be detected by other methods.

The St. Petersburg paradox: An experimental solution

The St. Petersburg paradox refers to a gamble of infinite expected value, where people are likely to spend only a small entrance fee for it. There is a huge volume of literature that mostly concentrates on the psychophysics of the game; experiments are scant. Here, rather than focusing on the psychophysics, we offer an experimental, “physical” solution as if robots played the game. After examining the time series formed by one billion plays, we: confirm that there is no characteristic scale for this game; explicitly formulate the implied power law; and identify the type of α-stable distribution associated with the game. We find an α=1 and, thus, the underlying distribution of the game is a Cauchy flight, as hinted by Paul Samuelson.