Symmetric Stable Distribution Research Articles

Mixture of Symmetric Stable Distributions for Macular Pathology Detection in Optical Coherence Tomography Scans.

Optical coherence tomography (OCT) is widely used to detect retinal disorders. In this study a new methodology is proposed for automatic detection of macular pathologies in the OCT images. Our approach is based on modeling the normal and abnormal OCT images with α-stable mixture model represented by stochastic differential equations (SDE). Parameters of the model are used to detect abnormal OCT images. The α-stable mixture model is created after applying a fractional Laplacian operator to the image and Expectation-Maximization (EM) algorithm is applied to estimate its parameters. The classification of an OCT image as normal or abnormal would be done by training SVM classifier based on estimated parameters of the mixture model. This method is examined for macular abnormality detection such as AMD, DME, and MH and achieve maximum accuracy of 97.8%. Clinical Relevance - This study establishes automatic method for anomaly detection on OCT images and provides fast and accurate OCT interpretation in clinical application.

Estimation of parameters of multiple chirp signal in presence of additive alpha-stable errors

In this paper, we consider the estimation of the unknown parameters of the multiple chirp signal model in presence of additive error. The chirp signals are quite common in many areas of science and engineering, specially sonar, radar, audio signals etc. The observed signals are usually corrupted by noise. In different signal processing applications it is observed that the errors may be heavy tailed. In this paper it is assumed that the additive errors have mean zero but may not have finite variance and are independent and identically distributed. We consider the least squares estimators and the approximate least squares estimators which maximize a periodogram like function. It has been observed that both the estimators are strongly consistent. The asymptotic distribution of the least squares estimators is obtained under the assumption that the additive errors are from a symmetric stable distribution. The approximate least squares estimators have the same asymptotic distribution as the least squares estimators. We perform some numerical simulations to see how the proposed estimators work. It is observed that the least squares estimators perform slightly better than the approximate least squares estimators in terms of the biases and mean absolute deviation, although the theoretical properties of the approximate least squares estimators can be established in a slightly weaker conditions.

The ECF-WS estimator for univariate symmetric stable distributions with application in seismic trace signals

The parameter that is of most interest in applications of stable distributions is the stability index α, which determines how “heavy” the tail is. For applications when the main interest is on the tail, we develop an estimator of α for univariate symmetric stable distributions, based on the characteristic function and on the principle of wavelet shrinkage. We analyze the proposed estimator by the Monte Carlo simulations and prove that it is a consistent estimator. The presence of human-made explosions in a seismic catalog leads to errors in the statistical analysis of seismicity. Here, we also proposed the CWS coefficient, based on the proposed estimator, like an additional classification tool to discriminate between natural tectonic earthquakes and human-made explosions.

An empirical study of the behaviour of the sample kurtosis in samples from symmetric stable distributions

An empirical study of the behaviour of the sample kurtosis in samples from symmetric stable distributions

Applications of Fractional Lower Order Synchrosqueezing Transform Time Frequency Technology to Machine Fault Diagnosis

Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signal analysis. The short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based synchrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. The fault signals belong to a non-Gaussian and nonstationary alpha (α) stable distribution with 1<α<2 and even the noises being also α stable distribution. The conventional FSST and SSST methods degenerate and even fail under α stable distribution noisy environment. Motivated by the fact that fractional low order STFT and fractional low order S-transform work better than the traditional STFT and S-transform methods under α stable distribution noise environment, we propose in this paper the fractional lower order FSST (FLOFSST) and the fractional lower order SSST (FLOSSST). In addition, we derive the corresponding inverse FLOSST and inverse FLOSSST. The simulation results show that both FLOFSST and FLOSSST perform better than the conventional FSSST and SSST under α stable distribution noise in instantaneous frequency estimation and signal reconstruction. Finally, FLOFSST and FLOSSST are applied to analyze the time frequency distribution of the outer race fault signal. Our results show that FLOFSST and FLOSSST extract the fault features well under symmetric stable (SαS) distribution noise.

A formula for hidden regular variation behavior for symmetric stable distributions

We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric.
 Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

The role of the asymmetric Ekman dissipation term on the energetics of the two-layer quasi-geostrophic model

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We present theoretical results that show that in either the standard or the extrapolated configurations, the Ekman term dissipates energy at large scales, but does not dissipate potential enstrophy. It also creates an approximately symmetric stable distribution of potential enstrophy between the two layers. The behavior of the Ekman term changes fundamentally at small scales. Under the standard formulation, the Ekman term will unconditionally dissipate energy and also dissipate, under very minor conditions, potential enstrophy at small scales. However, under the extrapolated formulation, there exist small “negative regions”, which are defined over a two-dimensional phase space, capturing the distribution of energy per wavenumber between baroclinic energy and barotropic energy, and the distribution of potential enstrophy per wavenumber between the upper layer and the lower layer, where the Ekman term may inject energy or potential enstrophy.

An Universal, Simple, Circular Statistics-Based Estimator of α for Symmetric Stable Family

The aim of this article is to obtain a simple and efficient estimator of the index parameter of symmetric stable distribution that holds universally, i.e., over the entire range of the parameter. We appeal to directional statistics on the classical result on wrapping of a distribution in obtaining the wrapped stable family of distributions. The performance of the estimator obtained is better than the existing estimators in the literature in terms of both consistency and efficiency. The estimator is applied to model some real life financial datasets. A mixture of normal and Cauchy distributions is compared with the stable family of distributions when the estimate of the parameter α lies between 1 and 2. A similar approach can be adopted when α (or its estimate) belongs to (0.5,1). In this case, one may compare with a mixture of Laplace and Cauchy distributions. A new measure of goodness of fit is proposed for the above family of distributions.

Means as Improper Integrals

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

Approximate analytical time-domain Green’s functions for space-fractional wave equations

Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the power law exponent and attenuation constant for breast and for liver. The results for both breast and liver converge very close to the source, and similar performance is observed in time-domain Green’s functions computed for linear with frequency attenuation. Despite minor differences in the arguments, the approximate analytical time-domain Green’s functions derived for dispersive time-fractional and space-fractional wave equations are also quite similar. [Work supported in part by NIH Grants EB023051 and EB012079.] Approximate and exact time-domain Green’s functions are available for time-fractional wave equations that describe power law attenuation in soft tissue, where each expression contains a stable probability distribution function. Previous work has also demonstrated that the exact time-domain Green’s functions for time-fractional and space-fractional wave equations that describe power law attenuation are similar. Approximate analytical time-domain Green’s functions have recently been derived for the Chen-Holm and Treeby-Cox space-fractional wave equations, where the approximate time-domain Green’s function for the Chen-Holm wave equation contains a symmetric stable probability distribution function and the approximate time-domain Green’s function for the Treeby-Cox wave equation contains a maximally skewed stable probability distribution function. Comparisons between the exact numerical and approximate analytical expressions for these time-domain Green’s functions are evaluated for published values of the po...

Upper bound estimation of positioning error for ground-based augmentation system

Ground-based augmentation systems (GBAS) calculate protection levels (PL), which are bounds of the GBAS positioning errors associated with given integrity levels. In practice, PLs calculated using a Gaussian overbound method tend to be overestimated when the actual (non-Gaussian-distributed) GBAS ranging errors exhibit heavy tails. We propose a stable distribution-based method to overcome this problem. The heavy-tailed stable distribution gives a more appropriate representation of the GBAS ranging error. Based on a symmetric stable distribution, the overbound of the GBAS ranging error is estimated using numerical computations. The stable overbound can tightly bound both the core and the tails of the GBAS ranging error. The PL calculated using this stable overbound is less conservative than that calculated using the Gaussian overbound, although both methods have similar computational complexity. A performance evaluation based on simulated measurements collected from a GBAS prototype shows that the proposed approach increases the availability of GBAS.

Lévy flight with absorption: A model for diffusing diffusivity with long tails.

We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of "diffusing diffusivity" [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)JPCBFK1520-610610.1021/acs.jpcb.6b01527], it was shown that the mean square displacement of particle remains Fickian, i.e., 〈x^{2}(T)〉∝T at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of D in that model has a steady state limit which is exponential, i.e., π_{e}(D)∼e^{-D/D_{0}}. In the present study, we model the diffusivity of a particle as a Lévy flight process so that D has a power-law tailed distribution, viz., π_{e}(D)∼D^{-1-α} with 0<α<1. We find that in the short time limit, the width of displacement distribution is proportional to sqrt[T], implying that the diffusion is Fickian. But for long times, the width is proportional to T^{1/2α} which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index 2α which preserves its shape at all times.

Asymmetric stable Paretian distribution testing

Two new tests for the symmetric stable Paretian distribution with tail index 1 < α < 2 are proposed. The test statistics and their associated approximate p-values are instantly computed and do not require use of the stable density or distribution or maximum likelihood estimation. They exhibit high power against a variety of alternatives, and much higher power than the existing test based on the empirical characteristic function. The two tests are combined to yield a test that has substantially higher power. A fourth test based on likelihood ratio is also studied. Extensions are proposed to address the asymmetric case and are shown to have reasonable actual size properties and high power against several viable alternatives.

SIMPLE ESTIMATOR AND CONSISTENT STRONGLY OF STABLE DISTRIBUTIONS

Stable distributions are extensively used to analyze earnings of financial assets, such as exchange rates and stock prices assets. In this paper we propose a simple and strongly consistent estimator for the scale parameter of a symmetric stable Levy distribution. The advantage of this estimator is that your computational time is minimum thus it can be used to initialize intensive computational procedure such as maximum likelihood. With random samples of sized n we tested the efficacy of these estimators by Monte Carlo method. We also included applications for three data sets.

Parameter-estimation methods for symmetric stable distributions: Application to small samples of spatial fluctuations of rainfall

The estimation of parameters of stable distributions is hampered by the lack of closed-form expressions for the density and distribution functions. Moreover, the accuracy of estimation methods is affected noticeably by small samples that fail to provide sufficiently descriptive information about the empirical distribution. This study reviewed the methods most commonly used and examined the possibility of improving the estimation of parameters of symmetric stable distributions using mathematical properties derived from the absolute value of symmetric stable variables. On one hand, considering the absolute value might be beneficial to the adjustment of small samples because insufficient data in one tail of the empirical distribution could be complemented with data from the other. Thus, the performance of techniques that rely on distribution quantiles might be improved. On the other hand, the evaluation of the distribution function at fractional absolute moments was found to be a good estimator of the characteristic exponent of stable distributions. This technique is faster than regression-type methods and returns remarkably similar results. The performances of these parameter-estimation methods were evaluated by analyzing the results of an application to small samples from spatial fluctuations of rainfall in which extreme variations are frequently found.

A Characterization of Symmetric Stable Distributions

Characterization problems in probability are studied here. Using the characteristic function of an additive convolution we generalize some known characterizations of the normal distribution to stable distributions. More precisely, if a distribution of a linear form depends only on the sum of powers of the certain parameters, then we obtain symmetric stable distributions.

Suboptimum Detectors for AF Relaying With Gaussian Noise and S &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\alpha$&lt;/tex-math&gt;&lt;/inline-formula&gt;S Interference

The conventional Gaussian detector based on the sample mean has been widely used in amplify-and-forward (AF) relaying by considering the Gaussian noise only. In a practical wireless network, co-channel interference may exist due to multiple access or frequency reuse. Such interference often follows a symmetric alpha stable (S $\alpha$ S) distribution that may not have a finite mean. As a result, the conventional Gaussian detector performs very poorly when both impulsive interference and Gaussian noise are considered. In this paper, novel suboptimum detectors for signals corrupted by S $\alpha$ S interference and Gaussian noise are proposed for AF relaying. Numerical results are presented to show that the new suboptimum detectors have performance gains on the order of tens of decibels in effective signal-to-noise ratio (SNR) over the conventional Gaussian detector under the same conditions.

Measure of location-based estimators in simple linear regression

In this note we consider certain measure of location-based estimators (MLBEs) for the slope parameter in a linear regression model with a single stochastic regressor. The median-unbiased MLBEs are interesting as they can be robust to heavy-tailed samples and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the MLBEs. In the first case, the regressor and error is assumed to follow a symmetric stable distribution. In the second, other types of regressions, with potentially contaminated errors, are considered. For both cases the consistency and exact finite-sample distributions of the MLBEs are established. Some results for the corresponding limiting distributions are also provided. In addition, we illustrate how our results can be extended to include certain heteroskedastic and multiple regressions. Finite-sample properties of the MLBEs in comparison to the LSE are investigated in a simulation study.

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

.For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J(r) ∼ r−σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings JL∝ Lθμ(σ) is found to be whenever the long-ranged couplings are relevant θμ(σ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θμ(σ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(−u)1+μ for u → −∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.