Abstract
Problems related to weather forecast, forest attributes estimation and prediction, disease propagation, among others, are commonly approximated in the framework of multivariate Gaussian random field modeling. This paper deals with the equivalence condition of two zero-mean Gaussian infinite-dimensional vector measures defined on the finite product of separable Hilbert spaces. In particular, sufficient conditions are provided. The results derived are applied to obtain the equivalence of Gaussian measures associated with two stationary zero-mean Gaussian vector random fields. Classical problems related to, for example, asymptotic properties of maximum likelihood vector Gaussian random field parameter estimators from tapered multivariate covariance functions, often arising in Multivariate Geostatistics, can be solved as direct application of the results derived.
Similar content being viewed by others
References
Apanasovich TV, Genton MG, Sun Y (2012) A valid Matern class of cross-covariance functions for multivariate random fields with any number of components. J Am Stat Assoc 107:180–193
Bosq D (2000) Linear processes in function spaces. Springer, Berlin
Christakos G (1992) Random field models in earth science. Dover Publications, Mineola
Christakos G (2000) Modern spatiotemporal geostatistics. Oxford University Press, Oxford
Daley DJ, Porcu E, and Bevilacqua M (2014) Classes of compactly supported covariance functions for multivariate random field. Technical Report, Universidad Federico Santa Maria (submitted for publication)
Da Prato G, Zabczyk J (2002) Second order partial differential equations in Hilbert spaces. Cambridge University Press, Cambridge
Dautray R, Lions J-L (1990) Mathematical analysis and numerical methods for science and technology volume 3: spectral theory and applications. Springer, Berlin
Du J, Zhang H, Mandrekar V (2009) Infill asymptotic properties of tapered maximum likelihood estimators. Ann Stat 37:3330–3361
Feldman J (1958) Equivalence and perpendicularity of Gaussian processes. Pac J Math 8:699–708
Furrer R, Genton M, Nychka D (2006) Covariance tapering for interpolation of large spatial datasets. J Comput Graph Statist 15:502–523
Gaek Y (1958) On a property of normal distributions of an arbitrary stochastic process. Czech Math J 8:610–618 (in Russian)
Gikhman II, Skorokhod AV (1966) On densities of probability measures in functional spaces. Uspekhi Mat Nauk 21:83–152
Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105:1167–1177
Golosov JI, Tempelman AA (1969) On equivalence of measures corresponding to Gaussian vector-valued functions. Sov Math Dokl 10:228–232
Kaufman C, Schervish M, Nychka D (2008) Covariance tapering for likelihood-based estimation in large spatial datasets. J Am Stat Assoc 103:1545–1555
Kullback S (1959) Information theory and statistics. Wiley, New York
Matheron G (1970) Random functions and their applications in geology. In: Merriam DF (Ed) Geostatistics, pp 79–88
Porcu E, Zastavnyi V (2011) Characterization theorems for some classes of covariance functions associated to vector valued random fields. J Multivar Anal. doi:10.1016/j.jmva.2011.04.013
Ramm AG (1990) Random fields estimation. World Scientific, Singapore
Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, Massachusetts Institute of Technology. http://www.GaussianProcess.org/gpml
Rozanov YA (1962) On the density of one Gaussian measure with respect to another. Theory of Probab Appl 7:82–87
Rozanov YuA (1964) On probability measures in functional spaces corresponding to stationary Gaussian processes. Theory of Probab Appl 9:404–420
Ruiz-Medina MD, Fernndez-Pascual RM (2010) Spatiotemporal Filtering from fractal spatial functional data sequence. 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
Salmerón R, Ruiz-Medina MD (2009) Multispectral decomposition of functional autoregressive models. Stoch Environ Res Risk Assess 23:289–297
Skorokhod AV, Yadrenko MI (1973) On absolute continuity of measures corresponding to homogeneous Gaussian fields. Theory of Probab Appl 18:27–40
Stein ML (1988) Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann Stat 16:55–63
Stein ML (1988) An application of the theory of equivalence of Gaussian measures to a prediction problem. IEEE Trans Inf Theory 34:580–582
Stein ML (1990) Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure. Ann Stat 18:850–872
Stein ML (2004) Equivalence of Gaussian measures for some nonstationary random fields. J Stat Plan Inference 123:1–11
Acknowledgments
This work has been supported in part by projects MTM2012-32674 of the DGI, MEC, and P09-FQM-5052 of the Andalousian CICE, Spain. Emilio Porcu is supported by Proyecto Fondecyt Regular from the Chilean Government.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ruiz-Medina, M.D., Porcu, E. Equivalence of Gaussian measures of multivariate random fields. Stoch Environ Res Risk Assess 29, 325–334 (2015). https://doi.org/10.1007/s00477-014-0926-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-014-0926-z