Giulio Cottone

On the use of fractional calculus for the probabilistic characterization of random variables

In this paper, the classical problem of the probabilistic characterization of a random variable is reexamined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of αα-stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of αα-stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.

Fractional Calculus Approach to the Statistical Characterization of Random Variables and Vectors

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

A novel exact representation of stationary colored Gaussian processes (fractional differential approach)

A novel representation of functions, called generalized Taylor form, is applied to the filtering of white noise processes. It is shown that every Gaussian colored noise can be expressed as the output of a set of linear fractional stochastic differential equations whose solution is a weighted sum of fractional Brownian motions. The exact form of the weighting coefficients is given and it is shown that it is related to the fractional moments of the target spectral density of the colored noise.

Fractional mechanical model for the dynamics of non-local continuum

In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both model shapes and natural frequency of the non-local systems are then studied.

Fractional spectral moments for digital simulation of multivariate wind velocity fields

Abstract In this paper, a new method for the digital simulation of multivariate wind velocity fields by fractional spectral moments function is proposed. Firstly, a digital linear filter whose coefficients are fractional spectral moments of the systems transfer function is constructed. Then, it is shown that by applying some basic concepts of fractional calculus, the samples of the target process can be simulated as superposition of Riesz fractional derivatives of a Gaussian white noise processes.

Composite laminates buckling optimization through Lévy based ant colony optimization

In this paper, the authors propose the use of the Levy probability distribution as leading mechanism for solutions differentiation in an efficient and bio-inspired optimization algorithm, ant colony optimization in continuous domains, ACOR. In the classical ACOR, new solutions are constructed starting from one solution, selected from an archive, where Gaussian distribution is used for parameter diversification. In the proposed approach, the Levy probability distributions are properly introduced in the solution construction step, in order to couple the ACOR algorithm with the exploration properties of the Levy distribution. The proposed approach has been tested on mathematical test functions and on a real world problem of structural engineering, the composite laminates buckling load maximization. In the latter case, as in many other cases in real world problems, the function to be optimized is multi-modal, and thus the exploration ability of the Levy perturbation operator allow the attainment of better results.

Path integral solution by fractional calculus

In this paper, the Path Integral solution is developed in terms of complex moments. The method is applied to nonlinear systems excited by normal white noise. Crucial point of the proposed procedure is the representation of the probability density of a random variable in terms of complex moments, recently proposed by the first two authors. Advantage of this procedure is that complex moments do not exhibit hierarchy. Extension of the proposed method to the study of multi degree of freedom systems is also discussed.

Ship Roll Motion under Stochastic Agencies Using Path Integral Method

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamical models and then applied for ship roll dynamics under random impulsive white noise excitation.

Statistics of nonlinear stochastic dynamical systems under Lévy noises by a convolution quadrature approach

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar nonlinear systems excited by Levy white noises. The proposed numerical procedure relies on the introduction of an integral transform of the Wiener–Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: (1) statistics of systems with several different drift terms can be handled in an efficient way, independently from the kind of white noise; (2) the particular form of Wiener–Hopf integral transform and its numerical evaluation, both introduced in this study, are generalizations of fractional integro-differential operators of potential type and Grunwald–Letnikov fractional derivatives, respectively.

ROC and Cost Curves for SHM Performance Characterization in a Multilevel Damage Classification Framework: Application to Impact Damage in Aircraft Composites Structures

In this paper, the optimization of maintenance schemes for aircraft structures is treated by a Bayesian updating approach. A built-in structural health monitoring system for multilevel damage detection based on ultrasonic guided waves is considered. Statistical evaluation of the SHM performances are dealt by means of multi-class ROC analysis, probability of detection and false alarm maps. Two methods, based on the minimization of the service life cost statistics are implemented. An application on a square panel of side 1.2m subjected to random impacts is considered. doi: 10.12783/SHM2015/18