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Geostatistics and the Role of Variogram in Time Series Analysis: A Critical Review

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Statistical Methods for Spatial Planning and Monitoring

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Abstract

Exploratory data analysis and prediction in time series modeling are not typically based on geostatistical techniques, although in several cases applying these techniques might be convenient.

This paper aims to illustrate the usefulness of using Geostatistics and its basic tool, such as the variogram, in time series, especially when an explicit model for the process is not an important goal of the analysis. Moreover, the main differences between the time-domain approach and Geostatistics are highlighted throughout the paper. In order to underline the role of the variogram for modeling and prediction purposes, several theoretical aspects, such as interpolation of missing values, temporal prediction, nonparametric estimation, and their computational problems, are faced through an extensive case study regarding an environmental time series. A modified version of GSLib routine for kriging is suitably developed in order to define appropriate temporal search neighborhoods for missing values treatment and prediction.

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References

  1. Alabert, F.: The practice of fast conditional simulations through the LU decomposition of the covariance matrix. Math. Geol. 19(5), 369–386 (1987)

    Article  Google Scholar 

  2. Banea, O., Solow, A.R., Stone, L.: On fitting a model to a population time series with missing values. Isr. J. Ecol. Evol. 52(1), 1–10 (2006)

    Article  Google Scholar 

  3. Bisgaard, S., Khachatryan, D.: Asymptotic confidence intervals for variograms of stationary time series. Qual. Reliab. Eng. Int. (2007). doi:10.1002/qre.1052

  4. Bloomfield, P.: Fourier Analysis of Time Series: An Introduction, 2nd Edition J. Wiley & Sons, Inc., USA (2000)

    Google Scholar 

  5. Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden Day, San Francisco (1976)

    MATH  Google Scholar 

  6. Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods. Springer, New York (1987)

    MATH  Google Scholar 

  7. Chilés, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  8. Christakos, G.: Modern Spatiotemporal Geostatistics. Oxford University Press, New York (2000)

    Google Scholar 

  9. Cressie, N.: A graphical procedure for determining nonstationarity in time series. J. Am. Stat. Assoc. 83(404), 1108–1116 (1988)

    Article  MathSciNet  Google Scholar 

  10. Cressie, N.: Statistics for Spatial Data, Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1993)

    Google Scholar 

  11. Cressie, N., Grondona, M.O.: A comparison of variogram estimation with covariogram estimation. In: Mardia, K.V. (ed.) The Art of Statistical Science. Wiley, Chichester (1992)

    Google Scholar 

  12. Cressie, N., Wikle, C.K.: Statistics for Spatio-Temporal Data. J. Wiley & Sons, Inc., Hoboken, New Jersey (USA) (2011)

    MATH  Google Scholar 

  13. Deutsch, C.V., Journel, A.G.: GSLib: Geostatistical Software Library and User’s Guide. Oxford University Press, New York (1998)

    Google Scholar 

  14. De Cesare, L., Myers, D.E., Posa, D.: FORTRAN 77 programs for space-time modeling. Comput. Geosci. 28(2), 205–212 (2002)

    Article  Google Scholar 

  15. De Iaco, S., Myers, D.E., Posa, D.: On strict positive definiteness of product and product-sum covariance models. J. Stat. Plan. Inf. 141(3), 1132–1140 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. De Iaco, S., Myers, D.E., Posa, D.: Strict positive definiteness of a product of covariance functions. Commun. Stat. Theory Methods 40(24), 4400–4408 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Iaco, S., Posa, D.: Predicting spatio-temporal random fields: some computational aspects. Comput. Geosci. 41, 12–24 (2012)

    Article  Google Scholar 

  18. De Iaco, S., Palma, M.: Convergence of realization-based statistics to model-based statistics for the LU unconditional simulation algorithm. Some numerical tests. Stoch. Environ. Res. Risk Assess. Springer 16(5), 333–341 (2002)

    Article  MATH  Google Scholar 

  19. Gandin, L.S.: Objective Analysis of Meteorological Fields. Gidrometeorologicheskoe Izdatelstvo, Leningrad (1963)

    Google Scholar 

  20. Gevers, M.: On the use of variograms for the prediction of time series. Syst. Control Lett. 6(1), 15–21 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gomez-Hernandez, J.J., Cassiraga, E.F.: Theory and practice of sequential simulation. In: Armstrong, M., Dowd, P.A. (eds) Geostat. Simul., pp. 111–124. Kluwer Academic, Norwell, Massachusetts (1994)

    Google Scholar 

  22. Harris, B.: Introduction to the Theory of Spectral Analysis of Time Series in Spectral Analysis of Time Series. Wiley, New York (1967)

    Google Scholar 

  23. Haslett, J.: On the sample variogram and sample autocovariance for non-stationary time series. Statistician 46(4), 475–485 (1997)

    Google Scholar 

  24. Hull, J., White, A.: The pricing of assets on options with stochastic volatilities. J. Finance 42(2), 281–300 (1987)

    Article  Google Scholar 

  25. Janis, M.J., Robeson, S.M.: Determining the spatial representativeness of air-temperature records using variogram-nugget time series. Phys. Geogr. 25(6), 513–530 (2004)

    Article  Google Scholar 

  26. Jenkins, G.M., Watts, D.G.: Spectral Analysis and Its Applications. Holden-Day, New York (1968)

    MATH  Google Scholar 

  27. Jones, R.H., Ackerson, L.M.: Serial correlation in unequally spaced longitudinal data. Biometrika 77(4), 721–732 (1990)

    Article  MathSciNet  Google Scholar 

  28. Journel, A.G.: Nonparametric estimation of spatial distributions. Mat. Geol. 15(3), 445–468 (1983)

    Article  MathSciNet  Google Scholar 

  29. Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic, London (1981)

    Google Scholar 

  30. Journel, A.G., Rossi, E.M.: When do we need a trend model in kriging? Mat. Geol. 21(7), 715–739 (1989)

    Article  Google Scholar 

  31. Khachatryan, D., Bisgaard, S.: Some results on the variogram in time series analysis. Qual. Reliab. Eng. Int. (2009). doi:10.1002/qre.1013

  32. Kolmogorov, A.N.: The local structure of turbulence in an incomprehensible fluid at very large Reynolds numbers. Dokl. Acad. Nauk. SSSR 30(4), 229–303 (1941)

    Google Scholar 

  33. Kyriakidis, P.C., Journel, A.G.: Geostatistical space-time models: a review. Mat. Geol. 31(6), 651–684 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Krige, D.G.: A statistical approach to some basic mine valuation problems on the Witwatersrand. J. Chem. Metall. Min. Soc. S. Afr. 52(6), 119–139 (1951)

    Google Scholar 

  35. Little, R.J.A., Rubin, D.B.: Statistical Analysis with Missing Data. Wiley, New York (2002)

    MATH  Google Scholar 

  36. Luceño, A.: Estimation of missing values in possibly partially nonstationary vector time series. Biometrika 84(2), 495–499 (1997)

    Article  MATH  Google Scholar 

  37. Ma, C.: The use of the variogram in construction of stationary time series models. J. Appl. Probab. 41(4), 1093–1103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ma, C.: Stochastic processes with a particular type of variograms. Res. Lett. Signal Process (2007). doi:10.1155/2007/61579

  39. Ma, C.: Recent developments on the construction of spatio-temporal covariance models. Stoch. Environ. Res. Risk Assess. 22(S1), 39–47 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Matheron, G.: Principles of Geostatistics. Econ Geol 58(8), 1246–1266 (1963)

    Article  Google Scholar 

  41. Matheron, G.: Les variables régionalisées et leur estimation. Masson, Paris (1965)

    Google Scholar 

  42. Matheron, G.: The intrinsic random functions and their applications. Adv. Appl. Probab. 5(3), 439–468 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  43. Micchelli, C.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Myers, D.E.: Interpolation with positive definite functions. Sci. Terre 28, 251–265 (1988)

    Google Scholar 

  45. Myers, D.E.: Kriging, cokriging, radial basis functions and the role of positive definiteness. Comput. Math. Appl. 24(12), 139–148 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. Myers, D.E., De Iaco, S., Posa, D., De Cesare, L.: Space-time radial basis functions. Comput. Math. Appl. 43(3–5), 539–549 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Posa, D.: Limiting stochastic operations for stationary spatial processes. Math. Geol. 23(5), 695–701 (1991)

    Article  MathSciNet  Google Scholar 

  48. Posa, D.: The indicator formalism in spatial data analysis. J. Appl. Stat. 19(1), 83–101 (1992)

    Article  Google Scholar 

  49. Posa, D., Rossi, M.: Applying stationary and non-stationary kriging. Metron XLVII(1–4), 295–312 (1989)

    Google Scholar 

  50. Posa, D., De Iaco, S.: Geostatistica: teoria e applicazioni. Giappichelli editore, Torino (2009)

    Google Scholar 

  51. Priestley, M.B.: Spectral Analysis and Time Series. Academic, London (1981)

    MATH  Google Scholar 

  52. Schoellhamer, D.H.: Singular spectrum analysis for time series with missing data. Geophys Res Lett 28(16), 3187–3190 (2001)

    Article  Google Scholar 

  53. Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Mat. Soc. 44(3), 522–536 (1938)

    Article  MathSciNet  Google Scholar 

  54. Solow, A.R.: The analysis of second-order stationary processes: times series analysis, spectral analysis, harmonic analysis, and geostatistics. In: Verly, G., et al. (eds.) Geostatistics for Natural Resources Characterization, Part I, pp. 573–585. D. Reidel Publishing Co, Dordrecht (1984)

    Google Scholar 

  55. Stein, M.L.: The screening effect in kriging. Ann. Stat. 30(1), 298–323 (2002)

    Article  MATH  Google Scholar 

  56. Tong, H.: A personal journey through time series in Biometrika. Biometrika 88(1), 195–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. Uysal, M.: Reconstruction of time series data with missing values. J. Appl. Sci. 7(6), 922–925 (2007)

    Article  Google Scholar 

  58. Weerasinghe, S.: A missing values imputation method for time series data: an efficient method to investigate the health effects of sulphur dioxide levels. Environmetrics (2009). doi:10.1002/env.990

  59. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Springer, New York (1987)

    Google Scholar 

  60. Yaglom, A.M.: An Introduction to the Theory of Stationary Random Functions. Dover, New York (2004)

    Google Scholar 

Download references

Acknowledgements

The research activities involved in this paper were partially supported by the Project “5 per mille per la ricerca” entitled “Modelli di Interpolazione Stocastica per il Monitoraggio Ambientale: Sviluppi Teorici e Applicativi” given by University of Salento for the period 2011–2012 (scientific coordinator prof. S. De Iaco).

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De Iaco, S., Palma, M., Posa, D. (2013). Geostatistics and the Role of Variogram in Time Series Analysis: A Critical Review. In: Montrone, S., Perchinunno, P. (eds) Statistical Methods for Spatial Planning and Monitoring. Contributions to Statistics. Springer, Milano. https://doi.org/10.1007/978-88-470-2751-0_3

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