Fractional Moments Research Articles

Efficient seismic reliability analysis of large-scale coupled systems including epistemic and aleatory uncertainties

Reliability analysis of large coupled systems is a challenging task in dam safety. Not only the number of random variables is increased, but also a complex numerical model makes the simulations computationally expensive. To overcome this problem, an efficient analytical model should be developed which does not affect the accuracy of reliability analysis. In this paper, a gravity dam-foundation-reservoir coupled system is taken as a vehicle for the numerical simulations. Epistemic (i.e. material) uncertainties, as well as the ground motion record-to-record variability (i.e. aleatory) are incorporated in the simulations. A new reliability technique is proposed and its efficiency is compared to the Latin Hypercube Sampling (LHS). In this technique, a continuous probability density function is replaced by a finite number of fractional moments. Furthermore, an improved bivariate dimension reduction method is developed for dynamic systems. Several parametric simulations are performed to investigate the impact of the reservoir water level and the dam size. Results of the proposed method are in good agreement with the LHS. Finally, a response surface meta-model is proposed to estimate the dam behavior as a function of ground motion intensity measures.

A model-free, non-parametric method for density determination, with application to asset returns

The distribution of the returns of an asset is still an open problem despite the variety of models and methods devised to deal with it. In this note we propose a model-free, nonparametric method for the estimation of asset returns, which is applicable also to a wider set of similar problems. The Laplace transform of a shifted rate of return or other percent change of a positive random variables can be transformed into a fractional moment problem which can then be solved by the method of maximum entropy. This is a very robust method and requires only a few values of the Laplace transform to provide good reconstructions. This method can be applied in multiple fields where model-free density estimation of functionals like percent changes of random variables is of interest.

An improved maximum entropy method via fractional moments with Laplace transform for reliability analysis

The fractional moment-based maximum entropy method (FM-MEM) attracts more and more attention in reliability analysis recently, comparing with the common integer moment-based maximum entropy method. In the present paper, a new approach for reliability analysis is proposed from the improvement of the fractional moment-based maximum entropy method via the Laplace transformation and dimension reduction method (DRM). Different with the traditional FM-MEM with a double-loop multivariate optimization formulation, the proposed method introduces a single-loop univariate optimization algorithm through solving a set of linear equations. Firstly, the single-loop algorithm is proposed for bounded positive random variable and then is extended to arbitrary positive random variable with the help of Laplace transform. Then, a univariate strategy is given to simplify the calculation for the unknown parameters in the single-loop algorithm. Through the improvement, not only the proposed method can predict failure probability accurately but also the computational cost is considerably reduced. Due to the univariate optimization strategy, the proposed method is more robust than common reliability analysis methods, which are sensitive to the initial points, such as the traditional FM-MEM and the first-order reliability method. Several numerical examples are studied to illustrate the efficiency, robustness, and accuracy of the proposed method for the prediction of the failure probability in comparison with other methods.

Glitch Behavior of Pulsars and Contribution from Neutron Star Crust

Pulsars are highly magnetized rotating neutron stars with a very stable rotation speed. Irrespective of their stable rotation rate, many pulsars have been observed with the sudden jump in the rotation rate, which is known as pulsar glitch. The glitch phenomena are considered to be an exhibit of superfluidity of neutron matter inside the neutron star's crustal region. The magnitude of such rapid change in rotation rate relative to their stable rotation frequency can quantify the moment of inertia of the crustal region to the total moment of inertia of the star called as the fractional moment of inertia (FMI). In this paper, we have calculated FMI for different masses of the star using six different representative unified equations of state (EoS) constructed under Relativistic Mean Field (RMF) framework. We have performed an event-wise comparison of FMI obtained from data with that of theoretically calculated values with and without considering the entrainment effect. It is found that larger glitches can't be explained by crustal FMI alone, even without the entrainment.

New integral inequalities for (r,alpha)-fractional moments of continuous random variables

New integral inequalities for (r,alpha)-fractional moments of continuous random variables

Adaptive scaled unscented transformation for highly efficient structural reliability analysis by maximum entropy method

The approximation of the probability density function (PDF) of the performance function, especially for the tail distribution, is of paramount importance in structural reliability analysis. In this paper, a new method is proposed to derive the PDF of the performance function with accuracy and high efficiency. The derivation is based on the maximum entropy method(MEM), where the fractional moments are adopted as constraints. Since the MEM dose not involve deterministic structural analysis, the efficiency and accuracy of the proposed method is dependent on the evaluation of fractional moments. In this regard, an adaptive scaled unscented transformation (ASUT) is developed to obtain the fractional moments with only a few of sample evaluations. The proposed ASUT is applicable to problems with correlated/uncorrelated random variables. Besides, it can circumvent the so-called “curse of dimensionality” to some extent. Thus, the proposed method could be taken as a general tool for highly efficient structural reliability analysis. Numerical examples involving explicit and implicit performance functions are used to illustrate the implementation of the proposed method, which shows the great efficacy of the proposed method, particularly for a high-dimensional problem. The problems to be further investigated are also pointed out.

Performance analysis of spectrum sensing schemes based on fractional lower order moments for cognitive radios in symmetric α-stable noise environments

Abstract Spectrum sensing schemes based on fractional lower order moment (FLOM) are often used in impulsive noise environments in which traditional energy detectors are not applicable. The performance of FLOM-based detectors operating in an α-stable noise environment is difficult to evaluate. This is because α-stable random variables can usually only be modeled by the characteristic function since closed-form expressions are not available, except for the special values of the characteristic exponent that correspond to the Cauchy and Gaussian noise distributions. In this paper, we derive closed-form expressions for the probability density function (PDF) and corresponding complementary cumulative distribution function (CDF) for a symmetric α-stable random variable with an arbitrary characteristic exponent α (0

Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in the natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag–Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.

Flutter reliability analysis of suspension bridges based on multiplicative dimensional reduction method

A reliability analysis method is proposed in this paper based on the maximum entropy (MaxEnt) principle in which constraints are specified in terms of the fractional moments instead of integer moments. Then a multiplicative dimensional reduction method (M-DRM) is introduced to compute the fractional moments. The method is applicable for both explicit and implicit limit state functions of complex structures. After two examples illustrate the accuracy and efficiency of this method in comparison to the Monte Carlo simulation (MCS), the method is used to analyze the flutter reliability of suspension bridge. The results show that the empirical formula method in which the limit state function is explicitly represented as a function of variables is only a too conservative estimate for flutter reliability analysis but is not accurate adequately. So it is not suitable for reliability analysis of bridge flutter. The actual flutter reliability analysis should be conducted based on a finite element method in which limit state function is implicitly represented as a function of variables. The proposed M-DRM provide an alternate and efficient way to analyze a much more complicated flutter reliability of long span suspension bridge.

Solution method of fractional moments involved in probability density estimation of structural output response

Solution method of fractional moments involved in probability density estimation of structural output response

A vine copula–based method for analyzing the moment-independent importance measure of the multivariate output

The moment-independent importance measure technique for exploring how uncertainty allocates from output to inputs has been widely used to help engineers estimate the degree of confidence of decision results and assess risks. Solving the Borgonovo moment-independent importance measure in the presence of the multivariate output is still a challenging problem due to “curse of dimensionality,” and it is investigated in this contribution. For easily estimating the moment-independent importance measure, a novel method based on the vine copula is proposed. In the proposed method for estimating moment-independent importance measure, three steps are included. First, the moment-independent importance measure is expressed as a product of bivariate copula density functions through the vine copula trees. Second, the marginal probability density functions are obtained by the maximum entropy under the constraint of the fractional moments. Finally, the post-processed is executed to directly estimate the moment-independent importance measure by estimated copula density functions. The proposed method can handle multivariate output easily. The results of several examples indicate the validity and benefits of the proposed method.

An improved complex fractional moment-based approach for the probabilistic characterization of random variables

Recently, the concept of complex fractional moments has been introduced as an extension to ordinary moments. It has been shown that complex fractional moments never diverge and they are able to represent both probability density function (PDF) and characteristic function (CF). In this paper, we develop an improved complex fractional moments-based approach to recover the PDF and the CF, in which the PDF is re-expressed in two parts such that the complex fractional moment coincide with Mellin transform in each part. The proposed PDF decomposition scheme is a most natural way to connect complex fractional moments and Mellin transform while it represent the PDF (or CF) in a more straightforward and more distinct way. By virtue of inverse Mellin transform theorem, the PDF (or CF) can be exactly recovered in terms of complex fractional moments. In addition, the developed approach is further extended to estimate the PDF (or CF) from a finite number of samples of variables. Three illustrative distributions are used to demonstrate the proposed complex fractional moment-based approach.

A joint optimization of energy harvesting and spectrum sensing for wireless sensor networks under Middleton Class A noise

With the growing popularity of wireless sensor networks, the environment in which the network is located becomes more undesirable. In addition, the problems of spectrum scarcity and the short sensor lifetime have become increasingly prominent. In this article, we incorporate the two technologies of cognitive radio and energy harvesting to solve the above problems of wireless sensor networks under impulsive noise. First, we use a Middleton Class A noise model to imitate the practical environment and the fractional lower order moments detector is employed to perform spectrum sensing for the sensors of wireless sensor networks, which are performing as the second users. Second, a new time-slots structure is proposed for the self-powered second user and the analytical expression of the second user’s average throughput is derived. Finally, we maximize the second user’s average throughput by a joint optimization of the sensing duration and data transmission duration while giving the primary user sufficient protection. Simulation shows that a much better performance can be achieved by fractional lower order moment detector than the traditional energy detector. Moreover, our optimization of the time-slots allocation is feasible and the maximum second user’s average throughput can be obtained.

An efficient method for moment-independent global sensitivity analysis by dimensional reduction technique and principle of maximum entropy

Probability density function (PDF)-based and failure probability (FP)-based moment-independent global sensitivity indices can commendably reflect the influence of model input on the whole distribution and partial distribution (or called FP) of model output respectively, yet how to efficiently and accurately estimate these two indices for guiding the engineering practice still remains an essential and challenging problem. In this paper, a novel PDF estimation based method is proposed, which equivalently transforms the computation of these two indices into that of the unconditional and conditional fractional moments of model output. To estimate them, an efficient and simple way is introduced based on a multiplicative version of the dimensional reduction method. The proposed method remarkably reduces the computational cost and can obtain these two indices simultaneously by reusing the information in the integration grid. Results of three case studies demonstrate the effectiveness of the proposed method and its good engineering application.

Truncated fractional moments of stable laws

Expressions are given for the truncated fractional moments E(X−a)+p of a general stable law. These involve families of special functions that arose out of the study of multivariate stable densities and probabilities. This main result yields expressions for E(X−a)+ when the index of stability α>1, and expressions for fractional absolute and signed moments for any stable law.

Dynamic threshold-setting for RF-powered cognitive radio networks in non-Gaussian noise

The increasing demand for spectral and energy efficient communication networks has drawn an upsurge of interest in frequency (RF) energy harvesting (EH) cognitive radio networks (CRNs), and the spectrum analyzer is the brain to provide different spectral access and energy harvesting opportunities in wireless-powered cognitive radio networks (WP-CRNs). The performance of detecting the existence of primary signals will be more crucial for the EH node. However, to the best of our knowledge, no dynamic threshold-setting method to achieve target detection performance, particularly in the practical non-Gaussian noise, is reported in the literature. Traditional assumption of additive white Gaussian noise (AWGN) fails to model the behavior of certain types of noise found in practice.In this paper we give a dynamic threshold-setting method to detect the existence of primary signals for the energy harvesting node in non-Gaussian noise background. Combined with the fractional lower order statistics (FLOS), we propose a weighting factor for dynamic threshold-setting to exploit the existence of the PU to harvest RF energy, namely weighting factor based fractional lower order moment (WF-FLOM) detector. Numerical simulation results verify that the proposed method is effective for the EH node, when the signal-to-noise ratio is low or the back ground noise varies.

High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluationof the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

Nonparametric Blind Signal Detection Based on Logarithmic Moments in Very Impulsive Noise

The detection problem in impulsive noise modeled by the symmetric alpha stable SαS distribution is studied. The traditional detectors based on the second or higher order moments fail in SαS noise, and the method based on the fractional lower order moments (FLOMs) performs poorly when the noise distribution has small values of characteristic exponent. In this paper, a detector based on the logarithmic moments is investigated. The analytical expressions of the false alarm and detection probabilities are derived in nonfading channels as well as Rayleigh fading channels. The effect of noise uncertainty on the performance is discussed. Simulation results show that the logarithmic detector performs better than the FLOM and Cauchy detectors in very impulsive noise. In addition, the logarithmic detector is a nonparametric method and avoids estimating the parameter of the noise distribution, which makes the logarithmic detector easier to implement than the FLOM detector.

Fractional Quaternion Zernike Moments for Robust Color Image Copy-Move Forgery Detection

In this paper, fractional Zernike moments (FrZMs) for complex signals are generalized to fractional quaternion Zernike moments (FrQZMs) for quaternion signal processing in a holistic manner by the quaternion algebra. We first present the definition of FrQZMs and an efficient implementation algorithm for speeding up the computation of FrQZMs through FrZMs of each component of the quaternion signal. The performance of the proposed FrQZMs is evaluated by considering robust color image copy–move forgery detection. The proposed robust copy–move forgery-detection algorithm considers the FrQZMs as a feature and a modified PatchMatch algorithm as a feature matching algorithm. Experimental results on two publicly available data sets (FAU and GRIP data set) have demonstrated that the proposed FrQZM-based algorithm can achieve an overall better performance than the state-of-the-art algorithms, especially in some additional operation cases.

An Analysis of a Nonlinear System Excited by Combined Gaussian and Poisson White Noises Using Complex Fractional Moments

An Analysis of a Nonlinear System Excited by Combined Gaussian and Poisson White Noises Using Complex Fractional Moments