Abstract
Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angulo JM, Anh VV, McVinish R, Ruiz-Medina MD (2005) Kinetic equation driven by Gaussian or infinitely divisible noise. Adv Appl Probab 37:366–392
Bosq D (2000) Linear processes in function spaces. Springer, New York
Bosq D, Blanke D (2007) Inference and predictions in large dimensions. Wiley, Paris
Brelot M (1960) Lectures on potential theory. Tata Institute of Fundamental Research, Bombay
Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent, part II. Geophys J R Astron Soc 13:529–539
Christakos G (1992) Random field models in earth science. Dover Publications, New York
Christakos G (2000) Modern spatiotemporal geostatistics. Oxford University Press, New York
Christakos G, Raghu VR (1996) Dynamic stochastic estimation of physical variables. Math Geol 28:341–365
Dautray R, Lions JL (1985) Mathematical analysis and numerical methods for science and technology. Functional and variational methods, vol 2. Springer, New York
Edwards RE (1965) Functional analysis. Holt, Rinehart and Winston, New York
Erdély A, Magnus W, Obergettinger F, Tricomi FG (1955) Higher transcendental functions, vol 3. McGraw-Hill, New York
Fuglede B (2005) Dirichlet problems for harmonic maps from regular domains. Proc Lond Math Soc 91:249–272
Gelfand A, Banerjee S, Gamerman D (2005) Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16:1–15
Gneiting T (2002) Compactly supported correlation functions. J Multivar Anal 83:493–508
Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177
Grebenkov DS, Nguyen BT (2013) Geometrical structure of Laplacian eigenfunctions. arXiv:1206.1278v2
Kelbert M, Leonenko NN, Ruiz-Medina MD (2005) Fractional random fields associated with stochastic fractional heat equations. Adv Appl Probab 108:108–133
Leonenko NN, Ruiz-Medina MD (2006) Scaling laws for the multidimensional Burgers equation with quadratic external potentials. J Stat Phys 124:191–205
Mardia KV, Goodall C (1993) Spatial-temporal analysis of multivariate environmental monitoring data. In: Patil GP, Rao CR (eds) Multivariate environmental statistics. North Holland, Amsterdam, pp 347–386
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Porcu E, Gregori P, Mateu J (2006) Nonseparable stationary anisotropic space–time covariance functions. Stoch Environ Res Risk Assess 21:113–122
Porcu E, Zastavnyi V (2011) Characterization theorems for some classes of covariance functions associated to vector-valued random fields. J Multivar Anal 102:1293–1301
Ruiz-Medina MD, Angulo JM (2007) Functional estimation of spatio-temporal heterogeneities. Environmetrics 18:775–792
Ruiz-Medina MD, Fernández-Pascual R (2010) Spatiotemporal filtering from fractal spatial functional data sequences. Stoch Environ Res Risk Assess 24:527–538
Ruiz-Medina MD, Salmerón R (2010) Functional maximum-likelihood estimation of ARH(p) models. Stoch Environ Res Risk Assess 24:131–146
Stein ML (2005) Space–time covariance functions. J Am Stat Assoc 100:310–321
Acknowledgments
This work has been supported in part by Projects MTM2012-32674 and MTM2012-32666 (both Projects co-funded by FEDER) of the DGI, MINECO, Spain. We would like to thank the referee for his/her helpful comments.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Ruiz-Medina, M.D., Angulo, J.M., Christakos, G. et al. New compactly supported spatiotemporal covariance functions from SPDEs. Stat Methods Appl 25, 125–141 (2016). https://doi.org/10.1007/s10260-015-0333-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-015-0333-8