Theory of Probability and Mathematical Statistics
Spatiotemporal covariance functions for Laplacian ARMA fields in higher dimensions
György H. Terdik
Link
Abstract: This paper presents clear formulae of the covariance functions of Laplacian ARMA fields in terms of coefficients and Bessel functions in higher spatial dimensions. Spectral methods are used for the study of spatiotemporal Laplacian ARMA fields in Euclidean spaces and spheres therein with dimension d≥2.
Keywords: Isotropy, homogeneity, stationarity, space-time interaction, spectral density, Whittle–Matérn model, Laplacian ARMA fields in higher spatial dimensions, random fields on sphere
Bibliography: R. J Adler, The geometry of random fields, Classics in Applied Mathematics, vol. 62, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. MR 3396215
V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001), no. 5-6, 1349–1387. MR 1859007
V. V. Anh, N. N. Leonenko, and M. D. Ruiz-Medina, Space-time fractional stochastic equations on regular bounded open domains, Fract. Calc. Appl. Anal. 19 (2016), no. 5, 1161–1199. MR 3571006
V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence, J. Appl. Probab. 41A (2004), 35–53, Stochastic methods and their applications. MR 2057564
G. Arfken and H. J. Weber, Mathematical methods for physicists, Academic Press, HAP, New York, San Diego, London, 2001. MR 1423357
C. Berg and E. Porcu, From Schoenberg coefficients to Schoenberg functions, Constr. Approx. 45 (2017), no. 2, 217–241. MR 3619442
D. R. Brillinger, Fourier analysis of stationary processes, Proc. IEEE 62 (1974), 1628–1643. MR 0362772
—, Time series, Classics in Applied Mathematics, vol. 36, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001, Data analysis and theory, Reprint of the 1981 edition. MR 1853554
F. Dai and Y. Xu, Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3060033
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.17 of 2017-12-22, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Based on notes left by Harry Bateman, Reprint of the 1953 original. MR 698780
D. Fryer, M. Li, and A. Olenko, Rcosmo: R Package for Analysis of Spherical, HEALPix and Cosmological data., R Journal 12 (2020), no. 1, 206–225.
T. Gneiting, Nonseparable, stationary covariance functions for space–time data, J. Amer. Statist. Assoc. 97 (2002), no. 458, 590–600. MR 1941475
—, Strictly and non-strictly positive definite functions on spheres, Bernoulli 19 (2013), no. 4, 1327–1349. MR 3102554
T. Gneiting, W. Kleiber, and M. Schlather, Matérn cross-covariance functions for multivariate random fields, J. Amer. Statist. Assoc. 105 (2010), no. 491, 1167–1177. MR 2752612
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, sixth ed., Academic Press, Inc., San Diego, CA, 2000, Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
J. Jeong, M. Jun, and M. G. Genton, Spherical process models for global spatial statistics, Statist. Sci. 32 (2017), no. 4, 501–513. MR 3730519
R. H. Jones and Y. Zhang, Models for continuous stationary space-time processes, Modelling longitudinal and spatially correlated data, Springer, 1997, pp. 289–298.
M. Ya. Kelbert, N. N. Leonenko, and M. D. Ruiz-Medina, Fractional random fields associated with stochastic fractional heat equations, Adv. in Appl. Probab. 37 (2005), no. 1, 108–133. MR 2135156
S. C. Lim and L. P. Teo, Generalized Whittle-Matérn random field as a model of correlated fluctuations, J. Phys. A 42 (2009), no. 10, 105202, 21. MR 2485858
F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 73 (2011), no. 4, 423–498, With discussion and a reply by the authors. MR 2853727
P. Major, Multiple Wiener-Itô integrals, second ed., Lecture Notes in Mathematics, vol. 849, Springer, Cham, 2014, With applications to limit theorems. MR 3155040
B. Matern, Spatial variation, second ed., Lecture Notes in Statistics, vol. 36, Springer-Verlag, Berlin, 1986, With a Swedish summary. MR 867886
I. Mirouze, E. W. Blockley, D. J. Lea, M. J. Martin, and M. J. Bell, A multiple length scale correlation operator for ocean data assimilation, Tellus A: Dynamic Meteorology and Oceanography 68 (2016), no. 1, 29744.
G. R. North, J. Wang, and M. Genton, Correlation models for temperature fields, Journal of Climate 24 (2011), no. 22, 5850–5862.
E. Porcu, A. Alegria, and R. Furrer, Modelling temporally evolving and spatially globally dependent data, Int. Stat. Rev. 86 (2018), no. 2, 344–377. MR 3852415
E. Porcu, M. Bevilacqua, and M. G. Genton, Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere, J. Amer. Statist. Assoc. 111 (2016), no. 514, 888–898. MR 3538713
E. Porcu, R. Furrer, and D. Nychka, 30 years of space-time covariance functions, Wiley Interdiscip. Rev. Comput. Stat. 13 (2021), no. 2, Paper No. e1512, 24. MR 4218945
E. Porcu, J. Mateu, and F. Saura, New classes of covariance and spectral density functions for spatio-temporal modelling, Stoch. Environ. Res. Risk Assess. 22 (2008), no. suppl. 1, 65–79. MR 2418413
T. Subba Rao and Gy. Terdik, Statistical analysis of spatio-temporal models and their applications, Time Series Analysis: Methods and Applications (Suhasini Subba Rao Tata Subba Rao and C.R. Rao, eds.), Handbook of Statistics, vol. 30, Elsevier, 2012, pp. 521 – 540. MR 3295420
T. Subba Rao and Gy. Terdik, A new covariance function and spatio-temporal prediction (kriging) for a stationary spatio-temporal random process, J. Time Series Anal. 38 (2017), no. 6, 936–959. MR 3714117
M. D. Ruiz-Medina, V. V. Anh, and J. M. Angulo, Fractional generalized random fields of variable order, Stochastic Anal. Appl. 22 (2004), no. 3, 775–799. MR 2047278
F. Sigrist, H. R Künsch, and W. A Stahel, Stochastic partial differential equation based modelling of large space-time data sets, J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015), no. 1, 3–33. MR 3299397
E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
T. Subba Rao and Gy. Terdik, Statistical analysis of spatio-temporal models and their applications, Handbook of statistics: time series analysis: methods and applications (T. Subba Rao, S. Subba Rao, and C. R.. Rao, eds.), vol. 30, Elsevier, 2012, pp. 521–540. MR 3295420
—, On the frequency variogram and on frequency domain methods for the analysis of spatio-temporal data, J. Time Series Anal. 38 (2017), no. 2, 308–325. MR 3611746
Gy. Terdik, Trispectrum and higher order spectra for non-Gaussian homogeneous and isotropic random field on the 2D-plane, Publ. Math. Debrecen 90 (2017), no. 3-4, 471–492. MR 3666643
—, Covariance functions for Gaussian Laplacian fields in higher dimension, Theory and Applications of Time Series Analysis (Cham) (Olga Valenzuela, Fernando Rojas, Luis Javier Herrera, Héctor Pomares, and Ignacio Rojas, eds.), Springer International Publishing, 2020, pp. 19–29. MR 4295457
Gy. Terdik and L. Nadai, Bispectrum and nonlinear model for non-Gaussian homogeneous and isotropic field in 3D, Teor. Ĭmovīr. Mat. Stat. (2016), no. 95, 138–156. MR 3631649
A. V. Vecchia, A general class of models for stationary two-dimensional random processes, Biometrika 72 (1985), no. 2, 281–291. MR 801769
A. T. Weaver and I. Mirouze, On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation, Quarterly Journal of the Royal Meteorological Society 139 (2012), no. 670, 242–260.
P. White and E. Porcu, Towards a complete picture of stationary covariance functions on spheres cross time, Electron. J. Stat. 13 (2019), no. 2, 2566–2594. MR 3988087
P. Whittle, On stationary processes in the plane, Biometrika 41 (1954), 434–449. MR 67450
M. Ĭ. Yadrenko, Spectral Theory of Random Fields (in Russian), Vishcha Shkola, Kiev, 1980. MR 590889
—, Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983, Translated from the Russian. MR 697386
A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. I, Springer Series in Statistics, Springer-Verlag, New York, 1987, Basic results. MR 893393
T.-C. J. Yeh, L. W. Gelhar, and A. L. Gutjahr, Stochastic analysis of unsaturated flow in heterogeneous soils: 1. Statistically isotropic media, Water Resources Research 21 (1985), no. 4, 447–456.
