Abstract
We present theoretical tools from statistical physics for the calculation of non-Gaussian moments. In particular, we focus on the variational approximation and the cumulant expansion. These methods enable approximate but explicit moment calculations with non-Gaussian probability density functions (pdfs). Their use is illustrated by calculating the variance and the excess kurtosis for a univariate non-Gaussian pdf. We comment on the potential application of these methods in estimating the parameters of the recently proposed Spartan spatial random fields.
Similar content being viewed by others
References
Abrahamsen P (1997) A review of Gaussian random fields and correlation functions. Technical Report 917, 2nd edn. Norwegian Computing Center, Oslo
Adler RJ (1981) The geometry of random fields. Wiley, New York
Barthelemy M, Orland H, Zerah G (1995) Propagation in random media: calculation of the effective dispersive permittivity by use of the replica method. Phys Rev E 52(1):1123–1127
Bochner S (1959) Lectures on fourier integrals. Princeton University Press, Princeton
Chaikin PM, Lubensky TC (1995) Principles of condensed matter physics. Cambridge University Press, Cambridge
Christakos G (1992) Random field models in earth sciences. Academic, San Diego
Creswick RJ, Farach HA, Poole CP (1991) Introduction to renormalization group methods in physics. Wiley, New York
Feynman RP (1982) Statistical mechanics. Benjamin and Cummings, Reading
Goldenfeld N (1993) Lectures on phase transitions and the renormalization group. Addison-Wesley, New York
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York
Hristopulos DT (2003a) Spartan Gibbs random field models for geostatistical applications. SIAM J Sci Comput 24:2125–2162
Hristopulos DT (2003b) Simulations of Spartan random fields. In: Simos TE (ed) Proceedings of the international conference of computational methods in sciences and engineering 2003. World Scientific, London, pp 242–247
Hristopulos DT (2003c) Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling. Adv Water Resour 26(12):1279–1308
Hristopulos DT, Christakos G (1997) A variational calculation of the effective fluid permeability of heterogeneous media. Phys Rev E 55(6):7288–7298
Hristopulos DT, Christakos G (2001) Practical calculation of non-Gaussian multivariate moments in spatiotemporal Bayesian maximum entropy analysis. Math Geol 33(5):543–568
Itzykson C, Drouffe J-M (1989) Statistical field theory, vol 1. Cambridge University Press, New York
Meurice Y (2002) Simple method to make asymptotic series of Feynman diagrams converge. Phys Rev Lett 88(14):1601–1604
Press WH, Teukolsky SA, Vettering WT, Flannery BP (1992) Numerical recipes in Fortran, vol 1. Cambridge University Press, New York
Rubin Y (2003) Applied stochastic hydrogeology. Oxford University Press, New York
Sornette D (2003) Critical phenomena. Springer, Berlin Heidelberg New York
Wackernagel H (2003) Multivariate geostatistics. Springer, Berlin Heidelberg New York
Yaglom AM (1987) Correlation theory of stationary and related random functions I: basic results. Springer, Berlin Heidelberg New York
Acknowledgments
This work is supported in part by the Greek Ministry of Education, Operational Programme for Education and Initial Vocational Training: Environment—Pithagoras II.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1
Here we derive Eqs. 52 and 53. First, note that the following is true:
In Eq. 82 summation is implied over the repeated indexes i l =1,...,m. The external field variables \(J_{i_{l}},\) l=1,...,m are moved outside the angled brackets, since they do not participate in the averaging. The derivatives act only on the external field, not the quantity inside the angled brackets. Let us assume that we are interested in the m-order partial derivative \(\partial^{m} ({\mathbf{J}} \cdot {\mathbf{X}})^{m}/\partial J_{i^{*}_{1}} \cdots J_{i^{*}_{m}}.\) The locations i *1 ··· i * m correspond to fixed locations (unlike i 1 ··· i m which are variable). It is not required that all i * k be different, in fact they can all denote the same location. Then, we obtain
We repeat the procedure m−1 times to obtain
The same can be shown by induction, i.e., by showing (1) that Eq. 52 holds for m=1, and (2) provided that Eq. 52 is valid for m−1, then it is also valid for m. Equation 53 is also proved by this approach.
Appendix 2
The fourth order cumulant is given by Eq. 50, i.e., v 4=〈V 4〉0 − 4〈V 3〉0〈V 〉0 −3〈V 2〉 20 +12〈V 2〉0〈V〉 20 − 6〈V〉 40 . Since 〈V〉0=〈H pert〉0, terms that are proportional to 〈V〉 m0 , where m>1 are nonlinear in H pert and do not contribute. Hence, only the terms 〈V 4〉0 and −3〈V 2〉 20 contribute to the active cumulant, given by Eq. 64. The latter is evaluated as follows.
Given that V=−H pert+i J·X, the first term in the active cumulant v′4 is given by
Of the terms in Eq. 85, the first three are of order O(H 2pert ), while the third vanishes after repeated differentiation. Hence, only the term 〈(J·X)4 〉0 contributes to Eq. 64.
The second term in v′4 is
Only the last term in Eq. 86, which is equal to 5 〈(J·X)4〉0 〈H pert〉0, contributes to linear order in H pert and is included in Eq. 66.
Next, we obtain the active fifth cumulant given by Eq. 66. The contributing terms of the fifth cumulant are given by v′5=〈V 5〉0 − 5〈V 4〉0 〈V〉0−10〈V 3〉0 〈V 2〉0+30〈V 2〉 20 〈V〉0. The first term in the active cumulant v′5 is given by
Of the terms in Eq. 87, the first four are of order O(H 2pert ), while the last one vanishes after differentiation at the limit J=0. Hence, only the term −5〈H pert(J·X)4〉0 contributes to Eq. 66.
The second term in v′5 is
Only the last term in Eq. 88, which is equal to 5〈(J·X)4 〉0 〈H pert 〉0, contributes to linear order in H pert and is included in Eq. 66.
The third term in v′5 is
Of the eight terms in Eq. 89, only the second to last term, which is equal to 30 〈H pert (J·X)2 〉0 〈(J·X)2〉0 contributes to Eq. 66.
The fourth term in v′5 is
Of the three terms in Eq. 90 only the second term, which is equal to 30〈(J·X)2〉 20 〈H pert〉0, contributes to Eq. 66.
Appendix 3
The numerical evaluation of the moment integrals in Sect. 7 of the general form
involves an infinite integration domain x∈[0,∞). The integrated function decays very rapidly at large values of x, and is practically different than zero in a narrow range, which depends strongly on the value of the non-Gaussian coefficient β. To overcome numerical errors due to this dependence, we transform the integral I n using the change of variable y=1/x as follows:
The integrand of the second term is zero at y=0 (because the numerator tends to zero exponentially while the denominator only as power law) and rises smoothly. As a result, it is possible to replace the lower limit of integration with a very small number ε (e.g., 10−16), to avoid the numerically undetermined division with zero.
Rights and permissions
About this article
Cite this article
Hristopulos, D.T. Approximate methods for explicit calculations of non-Gaussian moments. Stoch Environ Res Ris Assess 20, 278–290 (2006). https://doi.org/10.1007/s00477-005-0023-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-005-0023-4