Search results for "Fractional moment"

showing 10 items of 15 documents

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

2009

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. 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 $\alpha$--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 o…

Characteristic function (probability theory)FOS: Physical sciencesAerospace EngineeringMathematics - Statistics TheoryOcean EngineeringProbability density functionComplex order momentStatistics Theory (math.ST)Fractional calculusymbols.namesakeIngenieurwissenschaftenFOS: MathematicsTaylor seriesApplied mathematicsCharacteristic function serieMathematical PhysicsCivil and Structural EngineeringMathematicsGeneralized Taylor serieMechanical EngineeringStatistical and Nonlinear PhysicsProbability and statisticsMathematical Physics (math-ph)Condensed Matter PhysicsFractional calculusFourier transformNuclear Energy and EngineeringPhysics - Data Analysis Statistics and ProbabilitysymbolsFractional calculus; Generalized Taylor series; Complex order moments; Fractional moments; Characteristic function series; Probability density function seriesddc:620Series expansionFractional momentProbability density function seriesSettore ICAR/08 - Scienza Delle CostruzioniRandom variableData Analysis Statistics and Probability (physics.data-an)
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The moment equation closure method revisited through the use of complex fractional moments

2015

In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones the probability density function response of nonlinear systems may be written in discretized form in terms of complex fractional moment not requiring a closure scheme.

Fokker PlanckMoment equationsFokker Planck equationCalculationMathematical transformationFractional calculuFractional momentMellin transformProbability
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CROSS-POWER SPECTRAL DENSITY AND CROSS-CORRELATION REPRESENTATION BY USING FRACTIONAL SPECTRAL MOMENTS

2012

Fractional calculus Mellin transform Complex order moments Fractional moments Fractional spectral moments Cross-correlation function Cross-power spectral density function
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Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments

2014

In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem…

Mellin transformApplied MathematicsMechanical EngineeringMonte Carlo methodMathematical analysisProbabilistic logicAerospace EngineeringOcean EngineeringProbability density functionWhite noiseComplex fractional moment Kolmogorov-Feller Mellin transform Poisson white noise Probability density functionNonlinear systemLinear differential equationControl and Systems EngineeringMellin inversion theoremElectrical and Electronic EngineeringMathematics
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Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables

2012

Abstract The aim of this paper is the probabilistic representation of the probability density function (PDF) or the characteristic function (CF) in terms of fractional moments of complex order. It is shown that such complex moments are related to Riesz and complementary Riesz integrals at the origin. By invoking the inverse Mellin transform theorem, the PDF or the CF is exactly evaluated in integral form in terms of complex fractional moments. Discretization leads to the conclusion that with few fractional moments the whole PDF or CF may be restored. Application to the pathological case of an α -stable random variable is discussed in detail, showing the impressive capability to characterize…

Mellin transformFractional spectral momentDiscretizationCharacteristic function (probability theory)Mechanical EngineeringCharacteristic functionMathematical analysisAerospace EngineeringComplex order momentOcean EngineeringStatistical and Nonlinear PhysicsProbability density functionFractional calculuCondensed Matter PhysicsFractional calculusNuclear Energy and EngineeringProbability density functionApplied mathematicsFractional momentRandom variableCumulantMellin transformCivil and Structural EngineeringMathematicsTaylor expansions for the moments of functions of random variablesProbabilistic Engineering Mechanics
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Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises

2019

In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based …

Mellin transformGeneralizationInverseFokker-Planck equationProbability density functionWhite noiseComplex fractional momentNonlinear systemIntegerProbability density functionApplied mathematicsFokker–Planck equationSettore ICAR/08 - Scienza Delle CostruzioniMathematics
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Constructing transient response probability density of non-linear system through complex fractional moments

2014

Abstract The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Ito stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments an…

Mellin transformLaplace transformApplied MathematicsMechanical EngineeringMathematical analysisProbability density functionComplex fractional momentStochastic differential equationNonlinear systemTransient responseMellin transform.Mechanics of MaterialsOrdinary differential equationProbability density functionStochastic averagingMellin inversion theoremTwo-sided Laplace transformNon-linear stochastic systemMathematics
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Fokker Planck equation solved in terms of complex fractional moments

2014

Abstract In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones, the solution of the FPK equation in terms of a finite number of complex moments may be easily found. It is shown that the probability density function (PDF) solution of the FPK equation is restored in the whole domain, including the trend at infinity with the exception of the value of the PDF in zero.

Mellin transformMechanical Engineeringmedia_common.quotation_subjectFokker Planck equationMathematical analysisZero (complex analysis)Aerospace EngineeringOcean EngineeringStatistical and Nonlinear PhysicsProbability density functionComplex fractional momentCondensed Matter PhysicsInfinityDomain (mathematical analysis)Fractional calculusNuclear Energy and EngineeringFokker–Planck equationFinite setMellin transformRiesz fractional integrals.Civil and Structural EngineeringMathematicsmedia_commonProbabilistic Engineering Mechanics
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Poisson white noise parametric input and response by using complex fractional moments

2014

Abstract In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.

Mellin transformParametric Poisson white noiseGeneralizationMechanical EngineeringMathematical analysisAerospace EngineeringOcean EngineeringStatistical and Nonlinear PhysicsProbability density functionWhite noiseComplex fractional momentCondensed Matter PhysicsPoisson distributionsymbols.namesakeNonlinear systemModified Kolmogorov–Feller equationNuclear Energy and EngineeringProbability density functionsymbolsFractional Poisson processMellin transformCivil and Structural EngineeringParametric statisticsMathematicsProbabilistic Engineering Mechanics
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Probabilistic characterization of nonlinear systems under Poisson white noise parametric input via complex fractional moments

2014

In this paper the probabilistic characterization of a nonlinear system enforced by parametric Poissonian white noise in terms of complex fractional moments is presented. In fact the initial system driven by a parametric input could be transformed into a system with an external type of excitation through an invertible nonlinear transformation. It is shown that by using Mellin transform theorem and related concepts, the solution of the Kolmogorov-Feller equation for the system with external input may be obtained in a very easy way.

Mellin transformPoisson white noiseDifferential equationMathematical analysisLinear systemProbabilistic logicWhite noiseComplex fractional momentlaw.inventionNonlinear systemInvertible matrixlawparametric systemsParametric statisticsMathematicsICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014
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