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A note on smoothness measures for space–time surfaces


The differentiability of a random field has a direct relationship with the differentiability of its covariance function. We review the concept of differentiability of space–time covariance models and random fields, and its implications on predictions. We analyze the change of behavior of the covariance function at the origin and at different space–time lags away from the origin, by using the concept of smoothness which can be considered the geometrical view of the differentiability. We propose a way to measure the smoothness of any covariance function, and apply it to purely spatial and space–time covariance functions.

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  • Adler RJ (2009) The geometry of random fields. Siam, Philadelphia

  • Banerjee S, Gelfand A (2003) On smoothness properties of spatial processes. J Multivar Anal 84(1):85–100

    Google Scholar 

  • Cressie N, Huang H (1999) Classes of nonseparable, spatio-temporal stationary covariance functions. J Am Stat Assoc 94(448):1330–1340

    Google Scholar 

  • Cressie N, Wikle C (2011) Statistics for spatio-temporal data, vol 465. Wiley, New York

  • De Iaco S, Myers D, Posa D (2002) Nonseparable space–time covariance models: some parametric families. Math Geol 34(1):23–42

    Google Scholar 

  • Diggle P, Ribeiro P (2007) Model-based geostatistics, vol 13. Springer, New York

  • Eriksson M, Siska P (2000) Understanding anisotropy computations. Math Geol 32(6):683–700

    Google Scholar 

  • Fernández-Avilés G, Montero JM, Porcu E, Schlather M (2012) Space–time processes and geostatistics. Advances and challenges in space–time modelling of natural events. Springer, New York, pp 1–23

  • Gerharz LE, Pebesma E (2013) Using geostatistical simulation to disaggregate air quality model results for individual exposure estimation on gps tracks. Stoch Environ Res Risk Assess 27:223–234

    Google Scholar 

  • Gneiting T (2002) Nonseparable, stationary covariance functions for space–time data. J Am Stat Assoc 97(458):590–600

    Google Scholar 

  • Hristopulos DT, Elogne SN (2007) Analytic properties and covariance functions for a new class of generalized gibbs random fields. IEEE Trans Inf Theory 53(12):4667–4679

    Article  Google Scholar 

  • Hristopulos DT, Žukovič M (2011) Relationships between correlation lengths and integral scales for covariance models with more than two parameters. Stoch Environ Res Risk Assess 25(1):11–19

    Article  Google Scholar 

  • De Iaco S, Myers D, Posa D (2001) Space–time analysis using a general product–sum model. Stat Probab Lett 52(1)21–28

    Google Scholar 

  • Kent J, Mohammadzadeh M, Mosammam A (2011) The dimple in gneiting’s spatial-temporal covariance model. Biometrika 98(2):489–494

    Google Scholar 

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

    Article  Google Scholar 

  • Mateu J, Porcu E, Gregori P (2008) Recent advances to model anisotropic space–time data. Stat Methods Appl 17(2):209–223

    Google Scholar 

  • Mateu J, Fernández-Avilés G, Montero J (2011) On a class of non-stationary, compactly supported spatial covariance functions. Stoch Environ Res Risk Assess 27(2):297–309

    Google Scholar 

  • Mehlum E, Tarrou C (1998) Invariant smoothness measures for surfaces. Adv Comput Math 8(1):49–63

    Google Scholar 

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

    Google Scholar 

  • Porcu E, Mateu J, Zini A, Pini R (2007) Modelling spatio-temporal data: a new variogram and covariance structure proposal. Stat Prob Lett 77(1):83–89

    Google Scholar 

  • Porcu E, Mateu J, Saura F (2008) New classes of covariance and spectral density functions for spatio-temporal modelling. Stoch Environ Res Risk Assess 22:65–79

    Article  Google Scholar 

  • Porcu E, Mateu J, Christakos G (2009) Quasi-arithmetic means of covariance functions with potential applications to space–time data. J Multivar Anal 100(8):1830–1844

    Article  Google Scholar 

  • Stein M (1999) Interpolation of spatial data: some theory for kriging. Springer, New York

  • Stein M (2005) Space–time covariance functions. J Am Stat Assoc 100(469):310–321

    Google Scholar 

  • Stoker J (1969) Differential geometry. Wiley Interscience, New York

  • Xue Y, Xiao Y (2011) Fractal and smoothness properties of space–time Gaussian models. Front Math China 6(6):1217–1248

    Google Scholar 

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Work partially funded and supported by Grants MTM 2010-14961 and P1-1B2012-52.

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Correspondence to M. Bohorquez.

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Bohorquez, M., Mateu, J. & Diaz, L. A note on smoothness measures for space–time surfaces. Stoch Environ Res Risk Assess 28, 1011–1022 (2014).

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  • Geostatistics
  • Random fields
  • General second-order processes
  • Continuity and differentiation questions
  • Surfaces in Euclidean space
  • Graphical methods