On strict positive definiteness of product and product-sum covariance models

Journal of Statistical Planning and Inference

Volumn 141, Issue 3, 2011, Pages 1132-1140

  De Iaco, S ^a   Myers, D E ^b   Posa, D ^a  

^a UNIVERSITY OF SALENTO   (Italy)

^b The University of Arizona   (United States)

Author keywords

Kronecker product; Marginal variograms; Product model; Product sum model; Separability; Space time random field

Indexed keywords

EID: 78549252017     PISSN: 03783758     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jspi.2010.09.014     Document Type: Article

References (33)

1. Bochner S. Lectures on Fourier Integrals 1959, Princeton University Press, Princeton, NJ.
2. Chang K.-F. Strictly positive definite functions. Journal of Approximation Theory 1996, 87(2):148-158.
3. 2 in Milan District. Geostatistics Wollongong '96 1997, vol. 2:1031-1042. Kluwer Academic Publishers.
4. De Cesare L., Myers D.E., Posa D. Product-sum covariance for space-time modeling. Environmetrics 2001, 12:11-23.
5. De Cesare L., Myers D.E., Posa D. Estimating and modeling space-time correlation structures. Statistics and Probability Letters 2001, 51(1):9-14.
6. De Iaco S. Space-time correlation analysis: a comparative study. Journal of Applied Statistics 2010, 37(6):1027-1041.
7. De Iaco S., Myers D.E., Posa D. Total air pollution and space-time modeling. geoENV III: Geostatistics for Environmental Applications 2000, 45-52. Kluwer Academic Publisher, Dordrecht. P. Monestiez, D. Allard, R. Froidevaux (Eds.).
8. De Iaco S., Myers D.E., Posa D. Space-time analysis using a general product-sum model. Statistics and Probability Letters 2001, 52(1):21-28.
9. De Iaco S., Myers D.E., Posa D. Nonseparable space-time covariance models: some parametric families. Mathematical Geology 2002, 34(1):23-41.
10. De Iaco S., Myers D.E., Posa D. Space-time variograms and a functional form for total air pollution measurements. Computational Statistics and Data Analysis 2002, 41(2):311-328.
11. Dimitrakopoulos R., Luo X. Spatiotemporal modeling: covariances and ordinary kriging systems. Geostatistics for the Next Century 1994, 88-93. Kluwer Academic Publishers. R. Dimitrakopoulos (Ed.).
12. Fernandez-Casal R., Gonzalez-Manteiga W., Febrero-Bande M. Space-time dependency modeling using general classes of flexible stationary variogram models. Journal of Geophysical Research D: Atmospheres 2003, 108(24). STS 6-1-STS 6-12.
13. Gething P.W., Noor A.M., Gikandi P.W., Ogara E.A.A., Hay S.I., et al. Improving imperfect data from health management information systems in Africa using space-time geostatistics. PLoS Medicine 2006, 3(6):825-831. e271. 10.1371/journal.pmed.0 030271.
14. Genton M.G. Separable approximations of space-time covariance matrices. Environmentrics 2007, 18:681-695.
15. Graham A. Kronecker Products and Matrix Calculus with Applications 1981, Ellis Horwood Limited Publishers, New York, 128pp.
16. Gregori P., Porcu E., Mateu J., Sasvári Z. On potentially negative space-time covariances obtained as sum of products of marginal ones. Annals of the Institute of Statistical Mathematics 2008, 60:865-882.
17. Matern, B., 1980. Spatial Variation. Lecture Notes in Statistics, vol. 36, second ed., Springer Verlag, New York, 151pp. First edition published in Meddelanden fran Statens Skogsforskningsinstitute Swed., Band 49, no. 5, 1960.
18. Mathias M. Über positive Fourier-Integrale. Mathematische Zeitschrift 1923, 16:103-125.
19. Myers D.E. Space-time correlation models and contaminant plumes. Environmetrics 2002, 13(5-6):535-553.
20. Myers D.E. Anisotropic radial basis functions. International Journal of Pure and Applied Mathematics 2008, 42:197-203.
21. Myers D.E., Journel A. Variograms with zonal anisotropies and non-invertible kriging systems. Mathematical Geology 1990, 22(7):779-785.
22. Posa D. A simple description of spatial-temporal processes. Computational Statistics & Data Analysis 1993, 15(4):425-438.
23. Rodriguez-Iturbe I., Meija J.M. The design of rainfall networks in time and space. Water Resources Research 1974, 10(4):713-728.
24. Rouhani S., Hall T.J. Space-time kriging of groundwater data. Geostatistics 1989, vol. 2:639-651. Kluwer Academic Publishers.
25. Schoenberg I.J. Metric spaces and completely monotone functions. Annals of Mathematics 1938, 39(4):811-841.
26. Schoenberg I.J. Metric spaces and positive definite functions. Transactions of the American Mathematical Society 1938, 44(3):522-536.
27. Skien J.O., Blöschl G. Catchments as space-time filters-a joint spatio-temporal geostatistical analysis of runoff and precipitation. Hydrology and Earth System Sciences 2006, 10(5):645-662.
28. Spadavecchia L., Williams M. Can spatio-temporal geostatistical methods improve high resolution regionalisation of meteorological variables?. Agricultural and Forest Meteorology 2009, 149(6-7):1105-1117.
29. Stewart J. Positive definite functions and generalizations, a historical survey. Rocky Mountain Journal of Mathematics 1976, 6(3):409-434.
30. Strauss H. On interpolation with products of positive definite functions. Numerical Algorithms 1997, 15:153-165.
31. Wendland H. Scattered Data Approximation 2005, Cambridge University Press, New York, 336pp.
32. Zimmerman D.L. Computationally exploitable structure of covariance matrices and generalized covariance matrices in spatial models. Journal of Statistical Computation and Simulation 1989, 32:1-15.
33. zu Castell W., Filbir F., Szwarc R. Strictly positive definite functions in Rd. Journal of Approximation Theory 2005, 137:277-280.