Skip to main content

Introduction

  • Chapter
  • First Online:
Random Fields for Spatial Data Modeling

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

  • 1825 Accesses

Abstract

This chapter introduces various definitions and concepts that are useful in spatial data modeling: random fields, trends, fluctuations, spatial domain types, different spatial models, disorder and heterogeneity, noise and errors, inductive and empirical modeling, sampling and prediction, are among the topics discussed herein. There are also brief discussions of the connections between statistical mechanics and random fields as well as on stochastic versus nonlinear systems approaches.

As you set out for Ithaca hope the voyage is a long one, full of adventure, full of discovery.

Ithaca, by C. P. Cavafy

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    I do not like using “and/or”; herein, “A or B” shall mean that A, or B, or both A and B are valid.

  2. 2.

    In technical jargon, different colors may be used for different types of noise. For example, pink noise, also known as flicker, has a power-law spectral density , where ∥k∥ is the wavenumber and 0 < α < 2. The spectral density of colored fluctuations does not necessarily follow a power law.

  3. 3.

    The maximum integral range should be the reference scale, if the random field is anisotropic.

  4. 4.

    Note that X(s; ω) may represent a linear superposition of random fields with different properties.

  5. 5.

    It is not presumed that readers will be able to solve the groundwater flow problem without training. This example serves more as an illustration and motivation for further reading.

  6. 6.

    It will be useful to allow for dynamic changes of the noise; hence, in addition to space we include a time label for the following discussion.

  7. 7.

    To be more precise, for spatial noise case we should refer to wavenumbers instead of frequencies.

  8. 8.

    We use the letter “W” to denote the Wiener process as is commonly done in the literature.

  9. 9.

    For a precise definition of this statement see Sect. 5.1.

  10. 10.

    See the glossary for the definition of lattice used in this book.

References

  1. Abarbanel, H.: Analysis of Observed Chaotic Data. Springer, New York, NY, USA (1996)

    Book  MATH  Google Scholar 

  2. Abrahamsen, P.: A Review of Gaussian Random Fields and Correlation Functions. Tech. Rep. TR 917, Norwegian Computing Center, Box 114, Blindern, N-0314, Oslo, Norway (1997)

    Google Scholar 

  3. Acker, J.G., Leptoukh, G.: Online analysis enhances use of NASA earth science data. EOS Trans. Am. Geophys. Union 88(2), 14–17 (2007)

    Article  ADS  Google Scholar 

  4. Adler, P.M.: Porous Media, Geometry and Transports. Butterworth and Heinemann, Stoneham, UK (1992)

    Google Scholar 

  5. Adler, R.J.: The Geometry of Random Fields. John Wiley & Sons, New York, NY, USA (1981)

    MATH  Google Scholar 

  6. Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer Science & Business Media, New York, NY, USA (2009)

    MATH  Google Scholar 

  7. Amit, D.J.: Field Theory, the Renormalization Group, and Critical Phenomena, 2nd edn. World Scientific, New York, NY, USA (1984)

    Google Scholar 

  8. Anderson, P.W.: Basic Notions of Condensed Matter Physics. Benjamin-Cummings, New York, NY, USA (1984)

    Google Scholar 

  9. Armstrong, M.: Basic Linear Geostatistics. Springer, Berlin, Germany (1998)

    Book  MATH  Google Scholar 

  10. Baddeley, A., Gregori, P., Mahiques, J.M., Stoica, R., Stoyan, D.: Case Studies in Spatial Point Process Modeling. Lecture Notes in Statistics. Springer, New York, NY, USA (2005)

    Google Scholar 

  11. Besag, J.: Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Methodol. 36(2), 192–236 (1974)

    MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Brillinger, D.R.: Trend analysis: time series and point process problems. Environmetrics 5(1), 1–19 (1994)

    Article  Google Scholar 

  14. Brockmann, D., Helbing, D.: The hidden geometry of complex, network-driven contagion phenomena. Science 342(6164), 1337–1342 (2013)

    Article  ADS  Google Scholar 

  15. Brown, R.: A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philos. Mag. 4(21), 161–173 (1828)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Christakos, G.: Random Field Models in Earth Sciences. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  18. Christakos, G., Hristopulos, D.T.: Spatiotemporal Environmental Health Modelling. Kluwer, Boston (1998)

    MATH  Google Scholar 

  19. Cramér, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes. John Wiley & Sons, New York, NY, USA (1967)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  21. Cressie, N., Wikle, C.L.: Statistics for Spatio-temporal Data. John Wiley & Sons, New York, NY, USA (2011)

    MATH  Google Scholar 

  22. Cushman, J.H.: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles: Theory and Applications of Transport in Porous Media, 1st edn. Kluwer, Dordrecht, The Netherlands (1997)

    Book  Google Scholar 

  23. Dagan, G.: Flow and Transport in Porous Formations. Springer, Berlin, Germany (1989)

    Book  Google Scholar 

  24. Dagan, G., Neuman, S.P.: Subsurface Flow and Transport: A Stochastic Approach. Cambridge University Press, Cambridge, UK (2005)

    Google Scholar 

  25. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Probability and its Applications (New York), vol. I, 2nd edn. Springer, New York, NY, USA (2003)

    Google Scholar 

  26. Davison, A.C., Huser, R., Thibaud, E.: Geostatistics of dependent and asymptotically independent extremes. Math. Geosci. 45(5), 511–529 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Davison, A.C., Padoan, S.A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27(2), 161–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Deutsch, C.V.: Geostatistical Reservoir Modeling. Oxford, UK, New York, NY, USA (2002)

    Google Scholar 

  29. Diggle, P., Ribeiro, P.J.: Model-based Geostatistics. Springer Science & Business Media, New York, NY, USA (2007)

    Book  MATH  Google Scholar 

  30. Donsker, M.D.: Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Stat. 23(2), 277–281 (1952)

    Article  MATH  Google Scholar 

  31. E, W.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge, UK (2011)

    Google Scholar 

  32. 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 

  33. Feigelson, E.D., Babu, G.J.: Modern Statistical Methods for Astronomy. Cambridge University Press Textbooks, Cambridge, UK (2012)

    Book  Google Scholar 

  34. Franzke, C.L.E., O’Kane, T.J., Berner, J., Williams, P.D., Lucarini, V.: Stochastic climate theory and modeling. Wiley Interdiscip. Rev. Clim. Chang. 6(1), 63–78 (2015)

    Article  Google Scholar 

  35. Friedrich, R., Peinke, J., Sahimi, M., Tabar, M.R.R.: Approaching complexity by stochastic methods: from biological systems to turbulence. Phys. Rep. 506(5), 87–162 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  36. Gelfand, A.E., Diggle, P., Guttorp, P., Fuentes, M.: Handbook of Spatial Statistics. CRC Press, Boca Raton, FL, USA (2010)

    Book  MATH  Google Scholar 

  37. Gelhar, L.W.: Stochastic Subsurface Hydrology. Prentice Hall, Englewood Cliffs, NJ (1993)

    Google Scholar 

  38. Gelhar, L.W., Axness, C.L.: Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19(1), 161–180 (1983)

    Article  ADS  Google Scholar 

  39. Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Dover, Mineola, NY, USA (2003)

    MATH  Google Scholar 

  40. Ghil, M., Allen, M.R., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., Yiou, P.: Advanced spectral methods for climatic time series. Rev. Geophys. 40(1), 3.1–3.41 (2002)

    Article  Google Scholar 

  41. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, Reading, MA (1992)

    MATH  Google Scholar 

  42. Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford University Press, Oxford (1997)

    Google Scholar 

  43. Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D 9(1–2), 189–208 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Grigoriu, M.: Stochastic Calculus: Applications in Science and Engineering. Springer Science & Business Media, New York, NY, USA (2013)

    MATH  Google Scholar 

  45. Guyon, X.: Random Fields on a Network: Modeling, Statistics and Applications. Springer, New York, NY, USA (1995)

    MATH  Google Scholar 

  46. Hastie, T., Tibshirani, R.J., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York, NY, USA (2008)

    MATH  Google Scholar 

  47. Helbing, D.: Globally networked risks and how to respond. Nature 497(7447), 51–59 (2013)

    Article  ADS  Google Scholar 

  48. Hodge, V.J., Austin, J.: A survey of outlier detection methodologies. Artif. Intell. Rev. 22(2), 85–126 (2004)

    Article  MATH  Google Scholar 

  49. Hohn, M.E.: Geostastistics and Petroleum Geology. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  50. Hristopulos, D.T., Mouslopoulou, V.: Strength statistics and the distribution of earthquake interevent times. Physica A 392(3), 485–496 (2013)

    Article  ADS  Google Scholar 

  51. Hristopulos, D.T., Tsantili, I.C.: Space-time models based on random fields with local interactions. Int. J. Mod. Phys. B 29, 1541007 (2015)

    MathSciNet  MATH  Google Scholar 

  52. Itzykson, C., Drouffe, J.M.: Statistical Field Theory, vol. 2. Cambridge University Press, Cambridge, UK (1991)

    MATH  Google Scholar 

  53. Jackson, J.D.: Classical Electrodynamics, 3rd edn. John Wiley & Sons, New York, NY, USA (1998)

    MATH  Google Scholar 

  54. Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press, London, UK (1978)

    Google Scholar 

  55. Kamath, C.: Scientific Data Mining: A Practical Perspective. Society of Industrial and Applied Mathematics, Philadelphia, PA, USA (2009)

    Book  MATH  Google Scholar 

  56. Kanevski, M., Maignan, M.: Analysis and Modelling of Spatial Environmental Data. EPFL Press, Lausanne, Switzerland (2004)

    MATH  Google Scholar 

  57. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, vol. 7. Cambridge University Press, Cambridge, UK (2004)

    MATH  Google Scholar 

  58. Kardar, M.: Statistical Physics of Fields. Cambridge University Press, Cambridge, UK (2007)

    Book  MATH  Google Scholar 

  59. Kelkar, M., Perez, G.: Applied Geostatistics for Reservoir Characterization. Society of Petroleum Engineers, Richardson, TX, USA (2002)

    Google Scholar 

  60. Kitanidis, P.K.: Introduction to Geostatistics: Applications to Hydrogeology. Cambridge University Press, Cambridge, UK (1997)

    Book  Google Scholar 

  61. Kitanidis, P.K.: Persistent questions of heterogeneity, uncertainty, and scale in subsurface flow and transport. Water Resour. Res. 51(8), 5888–5904 (2015)

    Article  ADS  Google Scholar 

  62. Lacasa, L., Toral, R.: Description of stochastic and chaotic series using visibility graphs. Phys. Rev. E 82(3), 036120 (2010)

    Article  ADS  Google Scholar 

  63. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics: Statistical Physics, vol. 5, 3rd edn. Butterworth-Heinemann, Oxford, UK (1980)

    Google Scholar 

  64. Lantuéjoul, C.: Geostatistical Simulation: Models and Algorithms. Springer, Berlin, Germany (2002)

    Book  MATH  Google Scholar 

  65. Lemm, J.C.: Bayesian Field Theory. Johns Hopkins University Press, Baltimore, MD, USA (2005)

    MATH  Google Scholar 

  66. Longley, P., Goodchild, M.F., Maguire, D.J., Rhind, D.W.: Geographic Information Systems and Science. John Wiley & Sons, Hoboken, NJ, USA (2005)

    Google Scholar 

  67. Longtin, A.: Stochastic dynamical systems. Scholarpedia 5(4), 1619 (2010)

    Article  ADS  Google Scholar 

  68. Lovejoy, S., Schertzer, D.: The Weather and Climate: Emergent Laws and Multifractal Cascades. Cambridge University Press, Cambridge, UK (2013)

    Book  MATH  Google Scholar 

  69. Mariethoz, G., Caers, J.: Multiple-point Geostatistics: Stochastic Modeling with Training Images. John Wiley & Sons, Chichester, West Sussex, UK (2015)

    Google Scholar 

  70. McComb, W.D.: The Physics of Fluid Turbulence. Oxford University Press, Oxford, UK (1990)

    MATH  Google Scholar 

  71. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence. MIT Press, Cambridge, MA, USA (1971)

    Google Scholar 

  72. Moro, E.: Network analysis. In: Higham, N.J., Dennis, M.R., Glendinning, P., Martin, P.A., Santosa, F., Tanner, J. (eds.) The Princeton Companion to Applied Mathematics, pp. 360–374. Princeton University Press, Princeton, NJ, USA (2015)

    Google Scholar 

  73. Mussardo, G.: Statistical Field Theory. Oxford University Press, Oxford, UK (2010)

    MATH  Google Scholar 

  74. Newman, M.E.J.: Networks: an Introduction. Oxford University Press, Oxford, UK (2010)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  76. Olea, R.A.: Geostatistics for Engineers and Earth Scientists. Springer Science & Business Media, New York, NY, USA (2012)

    Google Scholar 

  77. Ortega, A., Frossard, P., Kovačević, J., Moura, J.M.F., Vandergheynst, P.: Graph signal processing: overview, challenges, and applications. Proc. IEEE 106(5), 808–828 (2018)

    Article  Google Scholar 

  78. Osborne, A., Provenzale, A.: Finite correlation dimension for stochastic systems with power-law spectra. Physica D: Nonlinear Phenom. 35(3), 357–381 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes, The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge, UK (1997)

    MATH  Google Scholar 

  80. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA, USA (2006). www.GaussianProcess.org/gpml. [Online; accessed on 31 Oct 2018]

  81. Rubin, Y.: Applied Stochastic Hydrogeology. Oxford, UK, New York, NY, USA (2003)

    Google Scholar 

  82. Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. Chapman and Hall/CRC, Boca Raton, FL, USA (2005)

    Book  MATH  Google Scholar 

  83. Ruelle, D.: Chance and Chaos. Princeton University Press, Princeton, NJ, USA (1991)

    Google Scholar 

  84. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall/CRC (2000)

    Google Scholar 

  85. Sangoyomi, T.B., Lall, U., Abarbanel, H.D.I.: Nonlinear dynamics of the great salt lake: dimension estimation. Water Resour. Res. 32(1) (1996)

    Article  ADS  Google Scholar 

  86. Schabenberger, O., Gotway, C.A.: Statistical Methods for Spatial Data Analysis. CRC Press, Boca Raton, FL, USA (2004)

    MATH  Google Scholar 

  87. Sherman, M.: Spatial Statistics and Spatio-temporal Data: Covariance Functions and Directional Properties. John Wiley & Sons, Chichester, West Sussex, UK (2011)

    MATH  Google Scholar 

  88. Shuman, D.I., Narang, S.K., Frossard, P., Ortega, A., Vandergheynst, P.: The emerging field of signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag. 30(3), 83–98 (2013)

    Article  ADS  Google Scholar 

  89. 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 

  90. Skøien, J.O., Baume, O.P., Pebesma, E.J., Heuvelink, G.B.: Identifying and removing heterogeneities between monitoring networks. Environmetrics 21(1), 66–84 (2010)

    MathSciNet  Google Scholar 

  91. Smith, R.C.: Uncertainty Quantification: Theory, Implementation, and Applications, vol. 12. SIAM, Philadelphia, PA, USA (2013)

    Google Scholar 

  92. Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York, NY, USA (1999)

    Book  MATH  Google Scholar 

  93. Süli, E.: Numerical solution of partial differential equations. In: Higham, N.J., Dennis, M.R., Glendinning, P., Martin, P.A., Santosa, F., Tanner, J. (eds.) The Princeton Companion to Applied Mathematics, pp. 306–318. Princeton University Press, Princeton, NJ, USA (2015)

    Google Scholar 

  94. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York, NY, USA (2002)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  96. Vapnik, V.N.: The Nature of Statistical Learning, 2nd edn. Springer, New York, NY, USA (2000)

    Book  MATH  Google Scholar 

  97. Wackernagel, H.: Multivariate Geostatistics. Springer, Berlin, Germany (2003)

    Book  MATH  Google Scholar 

  98. Wang, J.F., Stein, A., Gao, B.B., Ge, Y.: A review of spatial sampling. Spat. Stat. 2(1), 1–14 (2012)

    Article  Google Scholar 

  99. Ward, L.M., Greenwood, P.E.: 1/f noise. Scholarpedia 2(12), 1537 (2007). revision #90924

    Article  ADS  Google Scholar 

  100. Webster, R., Oliver, M.A.: Geostatistics for Environmental Scientists. John Wiley & Sons, Hoboken, NJ, USA (2007)

    Book  MATH  Google Scholar 

  101. Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods: A Mathematical Introduction. Springer, New York, NY, USA (1995)

    MATH  Google Scholar 

  102. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions, vol. I. Springer, New York, NY, USA (1987)

    Book  MATH  Google Scholar 

  103. Zhang, J., Atkinson, P., Goodchild, M.F.: Scale in Spatial Information and Analysis. CRC Press, Boca Raton, FL, USA (2014)

    Book  Google Scholar 

  104. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena, 4th edn. Oxford University Press, Oxford, UK (2004)

    MATH  Google Scholar 

  105. Žukovič, M., Hristopulos, D.T.: Reconstruction of missing data in remote sensing images using conditional stochastic optimization with global geometric constraints. Stoch. Environ. Res. Risk Assess. 27(4), 785–806 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature B.V.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hristopulos, D.T. (2020). Introduction. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_1

Download citation

Publish with us

Policies and ethics