×
Ma, Chunsheng

Stationary and isotropic vector random fields on spheres. (English) Zbl 1321.62117

Math. Geosci. 44, No. 6, 765-778 (2012).
Summary: This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer’s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a \(\chi^2\) or \(\log\)-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres.

Cited in 19 Documents

MSC:

62M40 Random fields; image analysis

Keywords:

absolutely monotone function; cross covariance; covariance matrix function; direct covariance; elliptically contoured random field; Gaussian random field; Gegenbauer’s polynomials; positive definite matrix
Cite Review PDF
Full Text: DOI

References:

[1] Bapat RB, Raghavan TES (1997) Nonnegative matrices and applications. Cambridge University Press, Cambridge · Zbl 0879.15015
[2] Bingham NH (1973) Positive definite functions on spheres. Proc Camb Philos Soc 73:145–156 · Zbl 0244.43002 · doi:10.1017/S0305004100047551
[3] Calder CA (2007) Dynamic factor process convolution models for multivariate space–time data with application to air quality assessment. Environ Ecol Stat 14:229–247 · doi:10.1007/s10651-007-0019-y
[4] Chen D, Menegatto A, Sun X (2003) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc Am Math Soc 131:2733–2740 · Zbl 1125.43300 · doi:10.1090/S0002-9939-03-06730-3
[5] Chilés, JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York · Zbl 0922.62098
[6] Choi J, Reich B, Fuentes M, Davis J (2009) Multivariate spatial-temporal modeling and predication of speciated fine particles. J Statist Theory Pract 3:407–418 · Zbl 1211.62091 · doi:10.1080/15598608.2009.10411933
[7] Christensen JPR, Ressel P (1978) Functions operating on positive definite matrices and a theorem of Schoenberg. Trans Am Math Soc 243:89–95 · Zbl 0394.15011 · doi:10.1090/S0002-9947-1978-0502895-2
[8] Cohen A, Johns RH (1969) Regression on a random field. J Am Stat Assoc 64:1172–1182 · doi:10.1080/01621459.1969.10501048
[9] Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York
[10] Dimitrakopoulos RD, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42:65–99 · Zbl 1184.86012 · doi:10.1007/s11004-009-9258-9
[11] Du J, Ma C (2011) Spherically invariant vector random fields in space and time. IEEE Trans Signal Process 59:5921–5929 · Zbl 1391.60115 · doi:10.1109/TSP.2011.2166391
[12] Du J, Ma C (2012) Variogram matrix functions for vector random fields with second-order increments. Math Geosci 44:411–425 · Zbl 1321.86020 · doi:10.1007/s11004-011-9377-y
[13] Feller W (1971) An introduction to probability theory and its applications, vol. II, 2nd edn. Wiley, New York · Zbl 0219.60003
[14] Gaspari G, Cohn SE (1999) Construction of correlations in two and three dimensions. Q J R Meteorol Soc 125:723–757 · doi:10.1002/qj.49712555417
[15] Gaspari G, Cohn E, Guo J, Pawson S (2006) Construction and application of covariance functions with variable length-fields. Q J R Meteorol Soc 132:1815–1838 · doi:10.1256/qj.05.08
[16] Haas TC (1998) Multivariate spatial prediction in the presence of non-linear trend and covariance non-stationarity. Environmetrics 7:145–165 · doi:10.1002/(SICI)1099-095X(199603)7:2<145::AID-ENV200>3.0.CO;2-T
[17] Hannan EJ (1966) Spectral analysis for geophysical data. Geophys J R Astron Soc 11:225–236 · doi:10.1111/j.1365-246X.1966.tb03502.x
[18] Hannan EJ (1969) Fourier methods and random processes. Bull Internat Statist Inst 42:475–496 · Zbl 0192.54103
[19] Hannan EJ (1970) Multiple time series. Wiley, New York · Zbl 0211.49804
[20] Huang C, Yao Y, Cressie N, Hsing T (2009) Multivariate intrinsic random functions for cokriging. Math Geosci 41:887–904 · Zbl 1178.86021 · doi:10.1007/s11004-009-9218-4
[21] Huang C, Zhang H, Robeson SM (2011) On the validity of commonly used covariance and variogram functions on the sphere. Math Geosci 43:721–733 · Zbl 1219.86015 · doi:10.1007/s11004-011-9344-7
[22] Johns RH (1963a) Stochastic processes on a sphere. Ann Math Stat 34:213–218 · Zbl 0209.49002 · doi:10.1214/aoms/1177704149
[23] Johns RH (1963b) Stochastic processes on a sphere as applied to meteorological 500-millibar forecasts. Proceedings of the symposium of time series analysis, vol 119. Wiley, New York
[24] Jun M (2011) Non-stationary cross-covariance models for multivariate processes on a globe. Scand J Stat 38:726–747 · Zbl 1246.91113 · doi:10.1111/j.1467-9469.2011.00751.x
[25] Le ND, Zidek JV (2006) Statistical analysis of environmental space–time processes. Springer, New York
[26] Ma C (2011a) Vector random fields with second-order moments or second-order increments. Stoch Anal Appl 29:197–215 · Zbl 1213.60071 · doi:10.1080/07362994.2011.532039
[27] Ma C (2011b) Covariance matrices for second-order vector random fields in space and time. IEEE Trans Signal Process 59:2160–2168 · Zbl 1221.70044 · doi:10.1109/TSP.2011.2112651
[28] Ma C (2011c) Covariance matrix functions of vector {\(\chi\)} 2 random fields in space and time. IEEE Trans Commun 59:2254–2561 · doi:10.1109/TCOMM.2011.060911.100686
[29] Mangulis V (1965) Handbook of series for scientists and engineers. Academic Press, New York · Zbl 0141.06201
[30] Mardia KV (1988) Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. J Multivar Anal 24:265–284 · Zbl 0637.60065 · doi:10.1016/0047-259X(88)90040-1
[31] Matérn B (1986) Spatial variation, 2nd edn. Springer, New York · Zbl 0608.62122
[32] Matheron G (1989) The internal consistency of models in geostatistics. In: Armstrong M (ed) Geostatistics, vol 1. Kluwer Academic, Dordrecht, pp 21–38
[33] McLeod MG (1986) Stochastic processes on a sphere. Phys Earth Planet Inter 43:283–299 · doi:10.1016/0031-9201(86)90018-X
[34] Roy R (1972) Spectral analysis for random field on the circle. J Appl Probab 9:745–757 · Zbl 0248.62041 · doi:10.2307/3212612
[35] Roy R (1973) Estimation of the covariance function of a homogeneous process on the sphere. Ann Stat 1:780–785 · Zbl 0263.62052 · doi:10.1214/aos/1176342475
[36] Roy R (1976) Spectral analysis for a random process on the sphere. Ann Inst Stat Math 28:91–97 · Zbl 0368.62084 · doi:10.1007/BF02504732
[37] Sain R, Cressie N (2007) A spatial model for multivariate lattice data. J Econom 140:226–259 · Zbl 1418.62368 · doi:10.1016/j.jeconom.2006.09.010
[38] Sain R, Furrer R, Cressie N (2011) A spatial analysis of multivariate output from regional climate models. Ann Appl Stat 5:150–175 · Zbl 1220.62152 · doi:10.1214/10-AOAS369
[39] Schoenberg I (1942) Positive definite functions on spheres. Duke Math J 9:96–108 · Zbl 0063.06808 · doi:10.1215/S0012-7094-42-00908-6
[40] Szegö G (1959) Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ., vol 23. Amer. Math. Soc., Providence
[41] Tebaldi C, Lobell DB (2008) Towards probabilistic projections of climate change impacts on global crop yields. Geophys Res Lett 35:L08705. doi: 10.1029/2008GL033423 · doi:10.1029/2008GL033423
[42] Trenberth KE, Shea DJ (2005) Relationships between precipitation and surface temperature. Geophys Res Lett 32:L14703. doi: 10.1029/2005GL022760 · doi:10.1029/2005GL022760
[43] Watson GN (1944) A treatise on the theory of Bessel functions, 2nd edn. Cambridge University Press, London · Zbl 0063.08184
[44] Weaver A, Courtier P (2001) Correlation modelling on the sphere using a generalized diffusion equation. Q J R Meteorol Soc 127:1815–1846 · doi:10.1002/qj.49712757518
[45] Widder DV (1946) The Laplace transform. Princeton Univ. Press, Princeton
[46] Yadrenko AM (1983) Spectral theory of random fields, Optimization Software, New York.
[47] Yaglom AM (1987) Correlation theory of stationary and related random functions, vol. I. Springer, New York · Zbl 0685.62078
[48] Zidek JV, Sun W, Le ND (2000) Designing and integrating composite networks for monitoring multivariate Gaussian pollution fields. J R Stat Soc, Ser C, Appl Stat 49:63–79 · Zbl 0974.62109 · doi:10.1111/1467-9876.00179
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.