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| 2. |
- Faouzi, Tarik, et al.
(författare)
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A deep look into the dagum family of isotropic covariance functions
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The Dagum family of isotropic covariance functions has two parameters thatallow for decoupling of the fractal dimension and Hurst effect for Gaussianrandom fields that are stationary and isotropic over Euclidean spaces.Sufficient conditions that allow for positive definiteness in Rd of the Dagumfamily have been proposed on the basis of the fact that the Dagum familyallows for complete monotonicity under some parameter restrictions.The spectral properties of the Dagum family have been inspected to a verylimited extent only, and this paper gives insight into this direction. Specifically,we study finite and asymptotic properties of the isotropic spectral density(intended as the Hankel transform) of the Dagum model. Also, we establishsome closed forms expressions for the Dagum spectral density in terms of theFox–Wright functions. Finally, we provide asymptotic properties for such aclass of spectral densities.
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| 3. |
- Faouzi, Tarik, et al.
(författare)
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A deep look into the Dagum family of isotropic covariance functions
- 2022
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Ingår i: Journal of Applied Probability. - Cambridge : Cambridge University Press. - 0021-9002 .- 1475-6072. ; 59:4, s. 1026-1041
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Tidskriftsartikel (refereegranskat)abstract
- The Dagum family of isotropic covariance functions has two parameters that allow fordecoupling of the fractal dimension and the Hurst effect for Gaussian random fields thatare stationary and isotropic over Euclidean spaces. Sufficient conditions that allow forpositive definiteness in $R^d$ of the Dagum family have been proposed on the basis ofthe fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a verylimited extent only, and this paper gives insight into this direction. Specifically, we studyfinite and asymptotic properties of the isotropic spectral density (intended as the Hankeltransform) of the Dagum model. Also, we establish some closed-form expressions forthe Dagum spectral density in terms of the Fox–Wright functions. Finally, we provideasymptotic properties for such a class of spectral densities.
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| 4. |
- Malyarenko, Anatoliy, 1957-, et al.
(författare)
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Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces
- 2024
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Ingår i: Journal of theoretical probability. - 0894-9840 .- 1572-9230.
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Tidskriftsartikel (refereegranskat)abstract
- Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).
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| 5. |
- Zhang, Xian, et al.
(författare)
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Elastodynamic problem on tensor random fields with fractal and Hurst effects
- 2022
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Ingår i: Meccanica (Milano. Print). - : Springer Science+Business Media B.V.. - 0025-6455 .- 1572-9648. ; 57:4, s. 957-970
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Tidskriftsartikel (refereegranskat)abstract
- This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.
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