MODELS OF COVARIANCE FUNCTIONS OF GAUSSIAN RANDOM FIELDS ESCAPING FROM ISOTROPY, STATIONARITY AND NON NEGATIVITY
DOI:
https://doi.org/10.5566/ias.v33.p75-81Keywords:
anisotropy, covariance model, Gaussian random field, non negativity, non stationarityAbstract
This paper represents a survey of recent advances in modeling of space or space-time Gaussian Random Fields (GRF), tools of Geostatistics at hand for the understanding of special cases of noise in image analysis. They can be used when stationarity or isotropy are unrealistic assumptions, or even when negative covariance between some couples of locations are evident. We show some strategies in order to escape from these restrictions, on the basis of rich classes of well known stationary or isotropic non negative covariance models, and through suitable operations, like linear combinations, generalized means, or with particular Fourier transforms.References
Bochner S (1933). Monotone funktionen, Stieltjes integrale und harmonische analyse. Math Ann, 108, 378--410.
Cressie N, Wikle C (2011). Statistics for Spatio-Temporal Data. Wiley & Sons.
Daley DJ, Porcu E (2013). Dimension walks through Schoenberg spectral measures. P Am Math Soc. In press.
Gneiting T (1999). On the derivatives of radial positive definite functions. J Math Anal Appl 236: 86--93.
Gneiting T (2002). Compactly supported correlation functions.
J Multivariate Anal 83:493--508.
Gregori P, Porcu E, Mateu J, Sasvari Z (2008). On potentially negative space time covariances obtained as sum of products of marginal ones. Ann I Stat Math 60:865--882.
P. Guttorp P and Schmidt AM (2013). Covariance structure of spatial and spatio-temporal processes. WIREs. Computational Stat 5:279--287.
Higdon DM, Swall J, Kern J (1999). Non-stationary spatial modeling. In Bayesian Statistics 6. Proceedings of the Sixth Valencia International Meeting 761--768. Oxford University Press.
Ma C (2005). Linear combinations of space-time covariance functions and variograms. IEEE Trans Signal Proc 53:857--864.
Matérn B (1960). Spatial Variation. Meddelanden fran Statens Skogsforskningsinstitut, 49, No.5. Almänna Förlaget, Stockholm. Second edition (1986), Berlin: Springer-Verlag.
Matheron G (1965). Les variables régionalisées et leur estimation. Paris: Masson.
Matheron G (1973). The intrinsic random functions and their applications. Adv Appl Prob 5:439--468.
Paciorek C, Schervish MJ (2004). Nonstationary covariance functions for Gaussian process regression. In Thrun, S., Saul, L., and Schölkopf, B., editors, Advances in Neural Information Processing Systems 16. MIT Press.
Pintore A, Holmes C (2004). Non-stationary covariance functions via spatially adaptive spectra. A revised version ``tentatively accepted'' in J Am Stat Assoc. http://www.stats.ox.ac.uk/~cholmes/Reports/spectral_tempering.pdf
Porcu E, Gregori P, Mateu J (2007). La descente et la montée étendues: the spatially d-anisotropic and the spatiotemporal case. Stoch Env Res Risk A 21(6):683--693.
Porcu E, Gregori P, Mateu J (2008). Recent advances to model anisotropic space-time data. Stat Method Appl 17(2):209--223
Porcu E, Gregori P, Mateu J (2009). Archimedean spectral densities for nonstationary space-time geostatistics. Stat Sinica 19(1):273--286.
Sampson PD, Guttorp P (1992). Nonparametric estimation of nonstationary covariance structure. J Am Stat Assoc 87:108--119.
Schoenberg IJ (1938). Metric Spaces and Positive Definite Functions. T Am Math Soc 44(3):522--536.
Stein ML (1999). Interpolation of Spatial Data. New York: Springer-Verlag.
Wendland H (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4:389--396.
Wendland H (2005). Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press.
Winkler G (2006). Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction.
Stochastic Modelling and Applied Probability. New York: Springer-Verlag.
Yaglom AM (1987). Correlation Theory of Stationary and Related Random Functions Volume I: Basic Results. Springer Verlag.
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