Alpha stable distribution

Transient vibration component of mechanical systems, such as, amplitude of frequency modulation are not easily detected by using the conventional Fourier analysis. Therefore, it is necessary to use appropriate analysis techniques in order to extract the signal components in both time and frequency domains. Hence, the objective of this work is to apply time-frequency energy distributions, namely, the Wigner Distribution and the Choi-Williams Distribution in the processing of the mechanical systems response in the time domain. These time-frequency analysis techniques will be applied in the study of the vibration of the mechanical systems in two different situations. In the first case the objective is to extract amplitude modulation components of vibration signals generated during the metal machining, or turning process. In the second situation, the time-frequency distributions were applied in the analysis of the frequency modulation components caused by the acceleration of a rotating ...

Source separation aims at decomposing a vector into additive components. This is often done by first estimating source parameters before feeding them into a filtering method, often based on ratios of covariances. The whole pipeline is traditionally rooted in some probabilistic framework providing both the likelihood for parameter estimation and the separation method. While Gaussians are ubiquitous for this purpose, many studies showed the benefit of heavy-tailed models for estimation. However, there is no counterpart filtering method to date exploiting such formalism, so that related studies revert to covariance-based filtering after estimation is finished. Here, we introduce a new multivariate separation technique, that fully exploits the flexibility of α-stable heavy-tailed distributions. We show how a spatial representation can be exploited, which decomposes the observation as an infinite sum of contributions originating from all directions. Two methods for separation are derived. The first one is non-linear and similar to a beamforming technique, while the second one is linear, but minimizes a covariation criterion, which is the counterpart of the covariance for α-stable vectors. We evaluate the proposed techniques in a large number of challenging and adverse situations on synthetic experiments, demonstrating their performance for the extraction of signals from strong interferences.

This paper studies the base station (BS) spatial distributions across different scenarios in urban, rural, and coastal zones, based on real BS deployment data sets obtained from two European countries (i.e., Italy and Croatia). Basically, this paper takes into account different representative statistical distributions to characterize the probability density function of the BS spatial density, including Poisson, generalized Pareto, Weibull, lognormal, and α-Stable. Based on a thorough comparison with real data sets, our results clearly assess that the α-Stable distribution is the most accurate one among the other candidates in urban scenarios. This finding is confirmed across different sample area sizes, operators, and cellular technologies (GSM/UMTS/LTE). On the other hand, the lognormal and Weibull distributions tend to fit better the real ones in rural and coastal scenarios. We believe that the results of this paper can be exploited to derive fruitful guidelines for BS deployment in a cellular network design, providing various network performance metrics, such as coverage probability, transmission success probability, throughput, and delay.

Generalized Hecke group Hp,∞(λ) is generated by X(z) = −(z − λp) −1 and Y (z) = −(z + λ) −1 where λp = 2 cos π p , p ≥ 2 integer and λ ≥ 2. Extended generalized Hecke group Hp,∞(λ) is obtained by adding the reection R(z) = 1/z to the generators of generalized Hecke group Hp,∞(λ). In this paper, we study the commutator subgroups of extended generalized Hecke groups Hp,∞(λ). Also, we determine the power subgroups of generalized Hecke groups Hp,∞(λ) and extended generalized Hecke groups Hp,∞(λ).

Activités numériques I 1 Autour des nombres • L'ensemble d'entier naturel N = {...}. • L'ensemble d'entier relatif Z = {...}. • L'ensemble de nombre décimal D = {}. • L'ensemble de nombre rationnel Q = {}. • L'ensemble de nombre réel R est l'ensemble de nombre rationnel et de nombre irrationnel.

A new approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics computed from observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior; exact inference being feasible only if the statistics are sufficient. With no reduction to sufficiency being possible in the state space setting, we pursue summaries via the maximization of an auxiliary likelihood function. We derive conditions under which this auxiliary likelihood-based approach achieves Bayesian consistency and show that-in a precise limiting sense-results yielded by the auxiliary maximum likelihood estimator are replicated by the auxiliary score. Particular attention is given to a structure in which the state variable is driven by a continuous time process, with exact inference typically infeasible in this case due to intractable transitions. Two models for continuous time stochastic volatility are used for illustration, with auxiliary likelihoods constructed by applying computationally efficient filtering methods to discrete time approximations. The extent to which the conditions for consistency are satisfied is demonstrated in both cases, and the accuracy of the proposed technique when applied to a square root volatility model also demonstrated numerically. In multiple parameter settings a separate treatment of each parameter, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality associated with ABC methods.

In this paper, the concept of a neutrosophic stable random variable is introduced. Two definitions of a neutrosophic random variable are presented. We introduced both the neutrosophic probability distribution function and the neutrosophic probability density function, and the convolution with the neutrosophic concept. In addition, we proved some properties of a neutrosophic stable random variable, and three examples are discussed.

In this paper, the concept of a neutrosophic stable random variable is introduced. Two definitions of a neutrosophic random variable are presented. We introduced both the neutrosophic probability distribution function and the neutrosophic probability density function, and the convolution with the neutrosophic concept. In addition, we proved some properties of a neutrosophic stable random variable, and three examples are discussed.

We propose a novel approach to approximate Bayesian computation (ABC) that seeks to cater for possible misspecification of the assumed model. This new approach can be equally applied to rejection-based ABC and to popular regression adjustment ABC. We demonstrate that this new approach mitigates the poor performance of regression adjusted ABC that can eventuate when the model is misspecified. In addition, this new adjustment approach allows us to detect which features of the observed data can not be reliably reproduced by the assumed model. A series of simulated and empirical examples illustrate this new approach.

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach -including a computationally efficient version based on the auxiliary score -achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.

Dans ce travail, nous etudions de maniere approfondie les processus a memoire longue saisonniers avec variance infinie des innovations. Dans le premier chapitre, nous rappelons les differentes proprietes des lois -stables univariees (stabilite, calcul des moments, simulation, : : :). Nous introduisons ensuite deux modeles a variance infinie largement utilises dans la litterature statistique : les modeles ARMA -stables et les modeles ARFIMA -stables developpes respectivement par Mikosch et al. [57] et Kokoszka et Taqqu [45]. Constatant les limites de ces modeles, nous construisons dans le second chapitre de nouveaux modeles appeles processus ARFISMA symetriques -stables. Ces modeles nous permettent de prendre en compte dans une modelisation la presence eventuelle des trois elements caracteristiques suivants : memoire longue, saisonnalite et variance infinie, que l'on rencontre souvent en finance, en telecommunication ou en hydrologie. Apres avoir conclu le chapitre par l'etud...

This paper studies the base station (BS) spatial distributions across different scenarios in urban, rural, and coastal zones, based on real BS deployment data sets obtained from two European countries (i.e., Italy and Croatia). Basically, this paper takes into account different representative statistical distributions to characterize the probability density function of the BS spatial density, including Poisson, generalized Pareto, Weibull, lognormal, and α-Stable. Based on a thorough comparison with real data sets, our results clearly assess that the α-Stable distribution is the most accurate one among the other candidates in urban scenarios. This finding is confirmed across different sample area sizes, operators, and cellular technologies (GSM/UMTS/LTE). On the other hand, the lognormal and Weibull distributions tend to fit better the real ones in rural and coastal scenarios. We believe that the results of this paper can be exploited to derive fruitful guidelines for BS deployment in a cellular network design, providing various network performance metrics, such as coverage probability, transmission success probability, throughput, and delay.

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach-including a computationally efficient version based on the auxiliary score-achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.

The machinery fault signal is a typical non-Gaussian and non-stationary process. The fault signal can be described by SαS distribution model because of the presence of impulses. Time-frequency distribution is a useful tool to extract helpful information of the machinery fault signal. Various fractional lower order (FLO) time-frequency distribution methods have been proposed based on fractional lower order statistics, which include fractional lower order short time Fourier transform (FLO-STFT), fractional lower order Wigner-Ville distributions (FLO-WVDs), fractional lower order Cohen class time-frequency distributions (FLO-CDs), fractional lower order adaptive kernel time-frequency distributions (FLO-AKDs) and adaptive fractional lower order time-frequency auto-regressive moving average (FLO-TFARMA) model time-frequency representation method. The methods and the exiting methods based on second order statistics in SαS distribution environments are compared, simulation results show that the new methods have better performances than the existing methods. The advantages and disadvantages of the improved time-frequency methods have been summarized. Last, the new methods are applied to analyze the outer race fault signals, the results illustrate their good performances.