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Random Fields for Spatial Data Modeling pp 393–432Cite as

Spartan Random Fields and Langevin Equations

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Part of the book series: Advances in Geographic Information Science ((AGIS))

Abstract

In this chapter we look at Spartan random fields from a different perspective. Our first goal is to show that solutions of linear stochastic partial differential (Langevin) equations are random fields with rational spectral densities [694]. In addition, the respective covariance function is the Green’s function of a suitable (i.e., derivable from Langevin equation) partial differential equation. Finally, the joint dependence of random fields that satisfy Langevin equations driven by a Gaussian white noise process can be expressed in terms of an exponential Gibbs-Boltzmann pdf; the latter has a quadratic energy function that involves local (i.e., based on low-order field derivatives) terms.

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Richard Feynman

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Notes

  1. 1.

    To simplify the notation and avoid confusing ω with the angular frequency w, we replace X(t; ω) with x(t), where the latter represents the trajectory for a specific realization.

  2. 2.

    w corresponds to the wavenumber k which represents the “spatial” frequency in reciprocal space.

  3. 3.

    The horizontal axis corresponds to time, while the vertical axis denotes the position.

  4. 4.

    For the sake of space economy we use the abbreviation \(\sum \) for \(\sum _{i=1}^{d}\sum _{j=1}^{d}\).

  5. 5.

    Note that \(\mathcal {L}\) is the operator acting on the covariance, while \(\mathcal {L}^{\ast }\) is the operator acting on the random field.

  6. 6.

    In contrast with Sect. 9.3, herein we absorb the coefficient σ 0 in and consider forcing with unit-variance white noise.

  7. 7.

    The Green’s function equation is sometimes expressed as \(\mathcal {L} [G(\mathbf {s} - {\mathbf {s}}')] = -\delta (\mathbf {s} - \mathbf {s}')\), i.e., with a negative sign in front of the delta function.

  8. 8.

    In the time series literature the autocovariance function is usually denoted by the symbol γ, which we have reserved for the variogram as is common in spatial statistics.

  9. 9.

    The positivity of the coefficients is ensured by the fact that α 1, α 3 > 0 and α 2 > 1.

  10. 10.

    The equation for the realizations x t holds for Xt(ω) as well.

References

  1. Anderson, E.R., Duvall Jr., T.L., Jefferies, S.M.: Modeling of solar oscillation power spectra. Astrophys. J. Part 1 364, 699–705 (1990)

    Article  ADS  Google Scholar 

  2. Besag, J., Mondal, D.: First-order intrinsic autoregressions and the de Wijs process. Biometrika 92(4), 909–920 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bolin, D., Lindgren, F.: Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping. Ann. Appl. Stat. 5(1), 523–550 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  5. Brillinger, D.: Modeling spatial trajectories. In: Gelfand, A.E., Diggle, P., Guttorp, P., Fuentes, M. (eds.) Handbook of Spatial Statistics, pp. 463–474. CRC Press, Boca Raton, FL, USA (2010)

    Chapter  Google Scholar 

  6. Carlson, J.M., Langer, J.S., Shaw, B.E., Tang, C.: Intrinsic properties of a Burridge-Knopoff model of an earthquake fault. Phys. Rev. A 44(2), 884–897 (1991)

    Article  ADS  Google Scholar 

  7. Caughey, T.K., Stumpf, H.J.: Transient response of a dynamic system under random excitation. J. Appl. Mech. 28(4), 563–566 (1961)

    Article  ADS  MATH  Google Scholar 

  8. Chilès, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty, 2nd edn. John Wiley & Sons, New York, NY, USA (2012)

    Book  MATH  Google Scholar 

  9. Crawford, F.S.: Berkeley Physics Course: Waves, vol. 3. McGraw-Hill, New York, NY, USA (1968)

    Google Scholar 

  10. Cressie, N.: Spatial Statistics. John Wiley & Sons, New York, NY, USA (1993)

    MATH  Google Scholar 

  11. Dolph, C.L., Woodbury, M.A.: On the relation between Green’s functions and covariances of certain stochastic processes and its application to unbiased linear prediction. Trans. Am. Math. Soc. 72(3), 519–550 (1952)

    MathSciNet  MATH  Google Scholar 

  12. Einstein, A.: Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. Annalen der Physik 322(8), 549–560 (1905). https://doi.org/10.1002/andp.19053220806

    Article  ADS  MATH  Google Scholar 

  13. Elogne, S.N., Hristopulos, D.T., Varouchakis, E.: An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities. Stochastic Environ. Res. Risk Assess. 22(5), 633–646 (2008)

    Article  MathSciNet  Google Scholar 

  14. Farmer, C.L.: Bayesian field theory applied to scattered data interpolation and inverse problems. In: Iske, A., Levesley, J. (eds.) Algorithms for Approximation, pp. 147–166. Springer, Heidelberg, Germany (2007)

    Chapter  Google Scholar 

  15. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 1. Basic Books, New York, NY, USA (2010)

    MATH  Google Scholar 

  16. Foreman-Mackey, D., Agol, E., Ambikasaran, S., Angus, R.: Fast and scalable Gaussian process modeling with applications to astronomical time series. Astron. J. 154(6), 220–240 (2017)

    Article  ADS  Google Scholar 

  17. Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, New York, NY, USA (1987)

    Book  MATH  Google Scholar 

  18. Goldstein, H., Poole, C.P., Safko, J.S.: Classical Mechanics, 3rd edn. Addison-Wesley, San Francisco (2000)

    MATH  Google Scholar 

  19. Hamilton, J.D.: Time Series Analysis, vol. 2. Princeton University Press, Princeton, NJ, USA (1994)

    MATH  Google Scholar 

  20. Hauer, B., Maciejko, J., Davis, J.: Nonlinear power spectral densities for the harmonic oscillator. Ann. Phys. 361, 148–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Higham, N.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Hochberg, D., et al.: Effective action for stochastic partial differential equations. Phys. Rev. E 60(6), 6343–6360 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Hristopulos, D.T.: Permissibility of fractal exponents and models of band-limited two-point functions for fGn and fBm random fields. Stoch. Environ. Res. Risk Assess. 17(3), 191–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hristopulos, D.T.: Spartan Gibbs random field models for geostatistical applications. SIAM J. Sci. Comput. 24(6), 2125–2162 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hristopulos, D.T.: Approximate methods for explicit calculations of non-Gaussian moments. Stoch. Environ. Res. Risk Assess. 20(4), 278–290 (2006)

    Article  MathSciNet  Google Scholar 

  26. Hristopulos, D.T.: Covariance functions motivated by spatial random field models with local interactions. Stoch. Environ. Res. Risk Assess. 29(3), 739–754 (2015)

    Article  MathSciNet  Google Scholar 

  27. Hristopulos, D.T., Christakos, G.: Practical calculation of non-Gaussian multivariate moments in spatiotemporal Bayesian maximum entropy analysis. Math. Geol. 33(5), 543–568 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hristopulos, D.T., Elogne, S.: Analytic properties and covariance functions of a new class of generalized Gibbs random fields. IEEE Trans. Inf. Theory 53(12), 4667–4679 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jacobs, K.: Stochastic Processes for Physicists: Understanding Noisy Systems. Cambridge University Press, Cambridge, UK (2010)

    Book  MATH  Google Scholar 

  30. Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)

    Article  ADS  MATH  Google Scholar 

  31. Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II: Nonequilibrium Statistical Mechanics, 2nd edn. Springer Science & Business Media, Berlin, Germany (1991)

    Book  MATH  Google Scholar 

  32. Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lang, A.: Simulation of stochastic partial differential equations and stochastic active contours. Ph.D. thesis, Universität Mannheim, Germany (2007). https://ub-madoc.bib.uni-mannheim.de/1770/

  34. Lim, S., Teo, L.: Generalized Whittle-Matérn random field as a model of correlated fluctuations. J. Phys. A Math. Theor. 42(10), 105202 (2009)

    Article  ADS  MATH  Google Scholar 

  35. Lindgren, F., Rue, H., Lindström, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the SPDE approach. J. R. Stat. Soc. Ser. B 73(4), 423–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M.M., Ainsworth, M., Karniadakis, G.E.: What is the fractional laplacian? arXiv preprint arXiv:1801.09767 (2018)

    Google Scholar 

  37. Marcotte, D., Allard, D.: Gibbs sampling on large lattice with GMRF. Comput. Geosci. 111, 190–199 (2018)

    Article  ADS  Google Scholar 

  38. Martin, P.C., Siggia, E.D., Rose, H.A.: Interpolation schemes for three dimensional velocity fields from scattered data using Taylor expansions. Phys. Rev. A 8(1), 423–437 (1973)

    Article  ADS  Google Scholar 

  39. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Mondal, D.: Applying Dynkin’s isomorphism: an alternative approach to understand the Markov property of the de Wijs process. Bernoulli 21(3), 1289–1303 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Montgomery, D.C., Jennings, C.L., Kulahci, M.: Introduction to Time Series Analysis and Forecasting. John Wiley & Sons, Hoboken, NJ, USA (2008)

    MATH  Google Scholar 

  42. Nørrelykke, S.F., Flyvbjerg, H.: Harmonic oscillator in heat bath: exact simulation of time-lapse-recorded data and exact analytical benchmark statistics. Phys. Rev. E 83(4), 041103 (2011)

    Article  ADS  Google Scholar 

  43. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin, Germany (2003)

    Book  MATH  Google Scholar 

  44. Ortigueira, M.D., Machado, J.A.T.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Ripley, B.D.: Spatial Statistics, vol. 575. John Wiley & Sons, Hoboken, NJ, USA (2005)

    MATH  Google Scholar 

  46. Rozanov, J.A.: Markov random fields and stochastic partial differential equations. Math. USSR-Sbornik 32(4), 515 (1977). http://stacks.iop.org/0025-5734/32/i=4/a=A08

    Article  MATH  Google Scholar 

  47. Rue, H., Martino, S., Chopin, N.: Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B (Stat Methodol.) 71(2), 319–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam, Netherlands (1993)

    MATH  Google Scholar 

  49. Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3), 521–541 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Shumway, R.H., Stoffer, D.S.: Time Series Analysis and its Applications. Springer Science & Business Media, New York, NY, USA (2000)

    Book  MATH  Google Scholar 

  51. Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319–328 (1977)

    Article  ADS  Google Scholar 

  52. Swift, J.B., Hohenberg, P.C.: Swift-Hohenberg equation. Scholarpedia 3(9), 6395 (2008). https://doi.org/10.4249/scholarpedia.6395. Revision #91841

    Article  ADS  Google Scholar 

  53. Vanmarcke, E.: Random Fields: Analysis and Synthesis. World Scientific, Hackensack, NJ, USA (2010)

    Book  MATH  Google Scholar 

  54. Vergara, R.C.: Development of geostatistical models using stochastic partial differential equations. Ph.D. thesis, MINES, Paris Tech (2018). http://cg.ensmp.fr/bibliotheque/public/CARRIZO_These_02513.pdf

  55. Vergara, R.C., Allard, D., Desassis, N.: A general framework for SPDE-based stationary random fields. arXiv preprint arXiv:1806.04999 (2018)

    Google Scholar 

  56. Whittle, P.: On stationary processes in the plane. Biometrika 41(3/4), 434–449 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  57. Whittle, P.: Stochastic processes in several dimensions. Bull. Int. Stat. Inst. 40(2), 974–994 (1963)

    MathSciNet  MATH  Google Scholar 

  58. Yaremchuk, M., Smith, S.: On the correlation functions associated with polynomials of the diffusion operator. Q. J. R. Meteorol. Soc. 137(660), 1927–1932 (2011)

    Article  ADS  Google Scholar 

  59. Zhang, Z., Karniadakis, G.: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol. 196. Springer, Cham, Switzerland (2017)

    Chapter  MATH  Google Scholar 

  60. Žukovič, M., Hristopulos, D.T.: Environmental time series interpolation based on Spartan random processes. Atmos. Environ. 42(33), 7669–7678 (2008)

    Article  ADS  Google Scholar 

  61. Žukovič, M., Hristopulos, D.T.: Spartan random processes in time series modeling. Physica A: Stat. Mech. Appl. 387(15), 3995–4001 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  62. Žukovič, M., Hristopulos, D.T.: The method of normalized correlations: a fast parameter estimation method for random processes and isotropic random fields that focuses on short-range dependence. Technometrics 51(2), 173–185 (2009)

    Article  MathSciNet  Google Scholar 

  63. Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, New York, NY, USA (2001)

    MATH  Google Scholar 

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  1. Technical University of Crete, Chania, Greece

    Dionissios T. Hristopulos

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Hristopulos, D.T. (2020). Spartan Random Fields and Langevin Equations. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_9

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