Abstract
Non-Gaussian stochastic fields are introduced by means of integrals with respect to independently scattered stochastic measures distributed according to generalized Laplace laws. In particular, we discuss stationary second order random fields that, as opposed to their Gaussian counterpart, have a possibility of accounting for asymmetry and heavier tails. Additionally to this greater flexibility the models discussed continue to share most spectral properties with Gaussian processes. Their statistical distributions at crossing levels are computed numerically via the generalized Rice formula. The potential for stochastic modeling of real life phenomena that deviate from the Gaussian paradigm is exemplified by a stochastic field model with Matérn covariances.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Åberg, S.: Wave intensities and slopes in Lagrangian seas. Adv. Appl. Probab. 39, 1018–1035 (2007)
Åberg, S., Podgórski, K., Rychlik, I.: Fatigue damage assessment for a spectral model of non-Gaussian random loads. Probab. Eng. Mech. 24, 608–617 (2009)
Barndorff-Nielsen, O.E.: Superposition of Ornstein-Uhlenbeck type processes. Theory Probab. Appl. 45, 175–194 (2000)
Barndorff-Nielsen, O.E., Pérez-Abreu, V.: Stationary and self-similar processes driven by Lévy processes. Stoch. Proc. Appl. 84, 357–369 (1999)
Bengtsson, A., Bogsjö, K., Rychlik, I.: Uncertainty of estimated rainflow damage for random loads. Mar. Struct. 22(2), 261–274 (2008)
Bondesson, L.: Generalized gamma convolutions and related classes of distributions and densities. Lecture Notes in Statistics, vol. 76. Springer Verlag (1992)
Cambanis, S., Podgórski, K., Weron, A.: Chaotic behavior of infinitely divisible processes. Stud. Math. 115, 109–127 (1995)
Cramér, H., Leadbetter, M.R.: Stationary and related stochastic processes. Sample Function Properties and Their Applications. Wiley, New York. Reprinted by Dover Publications, Inc., Mineola, New York (1967)
Galtier, T.: Note on the estimation of crossing intensity for Laplace moving averages. Extremes (2010, in print)
Gihman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes. Springer Verlag, Berlin (1974)
Goda, Y.: Random waves and spectra. In: Handbook of coastal and ocean engineering, wave and coastal structures, vol. 1, pp. 175–212 (1990)
Gurley, K.R., Kareem, A., Tognarelli, M.A.: Simulation of a class of non-normal random processes. Int. J. Non-Linear Mech. 31, 601–617 (1996)
Hardin, C.D.: On the spectral representation of symmetric stable processes. J. Multivar. Anal. 12, 385–401 (1982)
Kotz, S., Kozubowski, T.J., Podgórski, K.: The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhaüser, Boston (2001)
Lagaros, N.D., Stefanou, G., Papadrakakis, M.: An enhanced hybrid method for the simulation of highly skewed non-Gaussian stochastic fields. Comput. Methods Appl. Mech. Eng. 194, 4822–4844 (2005)
Matérn, B.: Spatial variation. Lecture Notes in Statistics, 2nd edn., vol. 36. Springer-Verlag, New York (1986)
Molz, F.J., Kozubowski, T.J., Podgórski, K., Castle, J.W.: A generalization of the fractal/facies model. Hydrogeol. J. 15, 809–816 (2006)
Palacios, M.B., Steel, M.F.J.: Non-Gaussian Bayesian geostatistical modeling. JASA 101, 604–618 (2006)
Rajput, B.S., Rosiński, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82, 451–487 (1989)
Røislien, J., Omre, H.: T-distributed random fields: a parametric model for heavy-tailed well-log data. Math. Geol. 38, 821–849 (2006)
Wright, A.L.: An ergodic theorem for the square of a wide-sense stationary process. Ann. Probab. 4, 829–835 (1976)
Zähle, U.: A general Rice formula, Palm measures, and horizontal-window conditioning for random fields. Stoch. Proc. Appl. 17, 265–283 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of both researchers was supported in part by the Gothenburg Stochastic Center and the Swedish foundation for Strategic Research through GMMC, Gothenburg Mathematical Modelling Center. Research of the second author partially supported by the Swedish Research Council Grant 2008-5382.
Rights and permissions
About this article
Cite this article
Åberg, S., Podgórski, K. A class of non-Gaussian second order random fields. Extremes 14, 187–222 (2011). https://doi.org/10.1007/s10687-010-0119-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-010-0119-1
Keywords
- Laplace distribution
- Spectral density
- Covariance function
- Stationary second order processes
- Rice formula