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Recent advances to model anisotropic space–time data

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Abstract

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.

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References

  • Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New York

    Google Scholar 

  • Berg C, Christensen PR, Ressel P (1984) Harmonic analysis on semigroups. Springer, New York

    MATH  Google Scholar 

  • Berg C, Forst G (1975) Potential theory on locally compact abelian groups. Springer, New York

    MATH  Google Scholar 

  • Bevilacqua M, Gaetan C, Porcu E, Mateu J (2006) Composite likelihood methods for space–time data. In: International workshop on spatio-temporal modelling METMA3

  • Bochner S (1933) Monotone funktionen, Stiltjes integrale und harmonische analyse. Mathematische Annalen 108:378–410

    Article  MathSciNet  Google Scholar 

  • Christakos G (2000) Modern spatiotemporal geostatistics. Oxford University Press, Oxford

    Google Scholar 

  • Christakos G (2002) On a deductive logic-based spatiotemporal random field theory. Prob Theory Math Stat 66:54–65

    Google Scholar 

  • Cressie NAC, Huang C (1999) Classes of nonseparable, spatiotemporal stationary covariance functions. J Am Stat Assoc 94:1330–1340

    Article  MathSciNet  MATH  Google Scholar 

  • Curriero FC, Lele S (1999) A composite likelihood approach to semivariogram estimation. J Agric Biol Environ Stat 4(1):9–28

    Article  MathSciNet  Google Scholar 

  • De Cesare L, Myers DE, Posa D (2000) Product-sum covariance for space–time modeling: an environmental application. Environmetrics 12:11–23

    Article  Google Scholar 

  • Dimitrakopoulos R, Lou X (1994) Spatiotemporal modeling: covariances and ordinary kriging system. In: Dimitrakopoulos R (eds). Geostatistics for the next century. Kluwer, Dordrecht, pp 88–93

    Google Scholar 

  • Feller W (1966) An Introduction to probability theory and its applications, vol II. Wiley, New York

    MATH  Google Scholar 

  • Fernández-Casal R, González-Manteiga W, Febrero-Bande M (2003) Flexible spatio-temporal stationary variogram models. Stat Comput 13:127–136

    Article  MathSciNet  Google Scholar 

  • Fuentes M (2002) Spectral methods for nonstationary spatial processes. Biometrika 89:197–210

    Article  MathSciNet  MATH  Google Scholar 

  • Fuentes M, Smith RL (2001) A new class of nonstationary spatial models (unpublished preprint)

  • Gneiting T (2002a) Stationary covariance functions for space–time data. J Am Stat Assoc 97:590–600

    Article  MathSciNet  MATH  Google Scholar 

  • Gneiting T (2002b) Compactly supported correlation functions. J Multivar Anal 83:493–508

    Article  MathSciNet  MATH  Google Scholar 

  • Gneiting T, Genton MG, Guttorp P (2005) Geostatistical space–time models, stationarity, separability, and full symmetry. In: University of Washington Technical Report no. 475

  • Gneiting T, Schlather M (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev 46:269–282

    Article  MathSciNet  MATH  Google Scholar 

  • Gregori P, Porcu E, Mateu J, Sasvári Z (2007) On potentially negative space time covariances obtained as sum of products of marginal ones. Ann Inst Stat Math. doi:10.1007/s10463-007-0122-8

  • Guttorp P, Sampson PD, Newman K (1992) Non-parametric estimation of spatial covariance with application to monitoring network evaluation. Statistics in the environmental and Earth sciences. Edward Arnold, London

    Google Scholar 

  • Hart JF (1954) Central tendency in areal distributions. Econ Geogr 30:48–59

    Article  Google Scholar 

  • Janauer GA (2001) Is what has been measured of any direct relevance to the success of the macrophyte in its particular environment? Ravera O (ed) Scientific and legal aspects of biological monitoring in freshwater. J Limnol 60(Suppl 1):33–38

    Google Scholar 

  • Jones R, Zhang Y (1997) Models for continuous stationary space–time processes. In: Gregoire TG, Brillinger DR, Diggle PJ, Russek-Cohen E, Warren WG, Wolfinger RD (eds). Modelling longitudinal and spatially correlated data. Lecture Notes in Statistics, vol 122. Springer, New York, pp 289–298

    Google Scholar 

  • Kolovos A, Christakos G, Hristopulos DT, Serre ML (2004) Methods for generating non-separable spatiotemporal covariance models with potential environmental applications. Adv Water Resources 27:815–830

    Article  Google Scholar 

  • Kyriakidis PC, Journel AG (1999) Geostatistical space–time models: a rewiew. Math Geol 31:651–684

    Article  MathSciNet  MATH  Google Scholar 

  • Levinson SJ, Beall JM, Powers EJ, Bengtson RD (1984) Space–time statistics of the turbulence in a Tokamak edge plasma. Nucl Fusion 24:527–540

    Google Scholar 

  • Lu N, Zimmerman DL (2005) Testing for directional symmetry in spatial dependence using the periodogram. J Stat Plan Inference 129(1–2):369–385

    Article  MathSciNet  MATH  Google Scholar 

  • Ma C (2002) Spatio-temporal covariance functions generated by mixtures. Math Geol 34:965–974

    Article  MathSciNet  MATH  Google Scholar 

  • Ma C (2003) Families of spatio-temporal stationary covariance models. J Stat Plan Inference 116:489–501

    Article  MATH  Google Scholar 

  • Ma C (2005a) Spatio-temporal variograms and covariance models. Adv Appl Prob 37(3):706–725

    Article  MATH  Google Scholar 

  • Ma C (2005b) Linear combinations of spatio-temporal covariance functions and variograms. IEEE Trans Signal Process 53(3):857–864

    Article  MathSciNet  Google Scholar 

  • Matheron G (1962) Traité de Géostatistique Apliquée, Tome I, Mémoires du Bureau de Recherches Géologiques et Miniès, no. 14. Editions Technip, Paris

  • Matheron G (1963a) Traité de Géostatistique Apliquée, Tome II. Le Krigéage, Mémoires du Bureau de Recherches Géologiques et Miniès, no. 14. Editions Bureau de Recherches Géologiques et Miniès, Paris

  • Matheron G (1963b) Principles of geostatistics. Econ Geogr 58:1246–1266

    Google Scholar 

  • Matheron G (1965) Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature. Masson, Paris

    Google Scholar 

  • Mitchell MW, Genton MG, Gumpertz ML (2005) Testing for separability of space–time covariances. Environmetrics 16(8):819–831

    Article  MathSciNet  Google Scholar 

  • Nelsen R (1999) An introduction to Copulas lecture notes in statistics. Springer, Heidelberg

    Google Scholar 

  • Porcu E, Mateu J (2007) Covariance functions which are stationary or nonstationary in space and stationary in time. Statistica Neerlandica (to appear)

  • Porcu E, Gregori P, Mateu J (2006) Nonseparable stationary anisotropic space–time covariance functions. Stochast Environ Res Risk Assess 21(2):113–122

    Article  MathSciNet  Google Scholar 

  • Porcu E, Gregori P, Mateu J (2007) La descente et la montée étendues: the spatially d-anisotropic and the spatio-temporal case. Stochast Environ Res Risk Assess. doi:10.1007/s00477-006-0079-9

  • Rouhani S, Hall TJ (1989) Space–time kriging of groundwater data, geostatistics, vol 2. Kluwer, Dordrecht, pp 639–651

    Google Scholar 

  • Sampson PD, Guttorp P (1992) Nonparametric estimation of nonstationary spatial covariance structures. J Am Stat Assoc 87:108–119

    Article  Google Scholar 

  • Scaccia L, Martin RJ (2005) Testing axial symmetry and separability of lattice processes. J Stat Plan Inference 131(1):19–39

    Article  MathSciNet  MATH  Google Scholar 

  • Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841

    Article  MathSciNet  Google Scholar 

  • Shapiro A, Botha JD (1991) Variogram fitting with a conditional class of conditionally nonnegative definite functions. Comput Stat Data Anal 11:87–96

    Article  Google Scholar 

  • Shkarofsky IP (1968) Generalized turbulence space–correlation and wave–number spectrum–function pairs. Can J Phys 46:2133–2140

    Google Scholar 

  • Stein ML (1999) Interpolation of spatial data. Some theory of kriging. Springer, New York

    MATH  Google Scholar 

  • Stein ML (2005) Space–time covariance functions. J Am Stat Assoc 100:310–321

    Article  MATH  Google Scholar 

  • Xu Z-W, Wu J, Huo W-P, Wu Z-S (2003a) Temporal skewness of electromagnetic pulsed waves propagating through random media with embedded irregularity slab. Chin Phys Lett 20:370–373

    Article  Google Scholar 

  • Xu Z-W, Wu J, Huo W-P, Wu Z-S (2003b) Statistical temporal behaviour of pulse wave propagation through continuous random media. Waves Random Media 13:59–73

    Article  MATH  Google Scholar 

  • Yakhot V, Orszag SA, She Z-S (1989) Space–time correlations in turbulence—Kinematical versus dynamical effects. Phys Fluids 1:184–186

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Universitat Jaume I, 12071, Castellon, Spain

    J. Mateu, E. Porcu & P. Gregori

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  1. J. Mateu
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  2. E. Porcu
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  3. P. Gregori
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Correspondence to P. Gregori.

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Mateu, J., Porcu, E. & Gregori, P. Recent advances to model anisotropic space–time data. Stat. Meth. & Appl. 17, 209–223 (2008). https://doi.org/10.1007/s10260-007-0056-6

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