Mathematical Geology

, Volume 39, Issue 2, pp 225–245

Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions

Article
  • 304 Downloads

Soil pollution data collection typically studies multivariate measurements at sampling locations, e.g., lead, zinc, copper or cadmium levels. With increased collection of such multivariate geostatistical spatial data, there arises the need for flexible explanatory stochastic models. Here, we propose a general constructive approach for building suitable models based upon convolution of covariance functions. We begin with a general theorem which asserts that, under weak conditions, cross convolution of covariance functions provides a valid cross covariance function. We also obtain a result on dependence induced by such convolution. Since, in general, convolution does not provide closed-form integration, we discuss efficient computation.

We then suggest introducing such specification through a Gaussian process to model multivariate spatial random effects within a hierarchical model. We note that modeling spatial random effects in this way is parsimonious relative to say, the linear model of coregionalization. Through a limited simulation, we informally demonstrate that performance for these two specifications appears to be indistinguishable, encouraging the parsimonious choice. Finally, we use the convolved covariance model to analyze a trivariate pollution dataset from California.

Keywords

convolution coregionalization Fourier transforms Gaussian spatial process hierarchical model Markov chain Monte Carlo spectral density 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M., and Stegun, I. A., 1965, Handbook of mathematical functions with formulas, graphs and mathematical tables: Dover, New York, p. 374–379.Google Scholar
  2. Daniels, M., Zhou, Z., and Zou, H., 2004, Conditionally specified space-time models for multivariate processes: submitted.Google Scholar
  3. Diggle, P. J., Tawn, J. A., and Moyeed, R. A., 1998, Model based Geostatistics(with discussion): Appl. Stat., v. 47, no. 3, p. 299–350.Google Scholar
  4. Gaspari, G., and Cohn, S. E., 1999, Construction of correlation functions in two and three dimensions: Q. J. R. Meteorol. Soc., v. 125, p. 723–757.CrossRefGoogle Scholar
  5. Gelfand A. E., Kim H. J., Sirmans, C. F., and Banerjee, S. K., 2003, Spatial modeling with spatially varying coefficient processes: J. Am. Stat. Assoc., v. 98, no. 462, p. 387–396.CrossRefGoogle Scholar
  6. Gelfand, A. E., Schmidt A. M., Banerjee, S., and Sirmans, C. F., 2004, Nonstationary multivariate process modeling through spatially varying coregionalization (with discussion): Test, v. 13, no. 2, p. 1–50.CrossRefGoogle Scholar
  7. Gelfand, A.E., and Vounatsou, P., 2002, Proper multivariate conditional autoregressive models for spatial data analysis: Biostatistics, v. 4, p. 11–25.CrossRefGoogle Scholar
  8. Higdon, D. M., 2001, Space and Space-time modeling using process convolutions: Technical reports, 01-03, Duke University, Institute of Statistical and Decision Sciences.Google Scholar
  9. Mardia, K.V., and Goodall, C., 1993, Spatiotemporal analyses of multivariate environmental monitoring data, in Patil, G. P., and Rao, C. R., eds., Multivariate environmental statistics: Elsevier, Amsterdam, p. 347–386.Google Scholar
  10. Myers, D., 1991, Pseudo-cross variograms, positive-definiteness, and co-kriging: Math. Geol., v. 23, no. 6, p. 805–816.CrossRefGoogle Scholar
  11. Sain, S., and Cressie, N., 2002, Multivariate lattice models for spatial environmental data: American Statistical Association Proceedings, p. 2820–2825.Google Scholar
  12. Schmidt, A. M., and Gelfand, A. E., 2003, A Bayesian coregionalization approach for multivariate pollutant data: J. Geophys. Res. Atmos., v. 108, no. D24, p. 8783.CrossRefGoogle Scholar
  13. Stein, M. L., 1999, Interpolation of spatial data: Some theory for kriging: Springer Verlag, New York, p. 24–25.Google Scholar
  14. Stein, A., and Corsten, L. C. A., 1991, Universal kriging and cokriging as a regression procedure: Biometrics, v. 47, no. 2, p. 575–587.CrossRefGoogle Scholar
  15. Ver Hoef, J. M., and Barry, R. P., 1998, Constructing and fitting models for cokriging and multivariate spatial prediction: J. Stat. Plan. Inference, v. 69, no. 2, p. 275–294.CrossRefGoogle Scholar
  16. Wackernagel, H., 2003, Multivariate geostatistics: An introduction with applications, 2nd ed.: Springer Verlag, Berlin.Google Scholar
  17. Xie, T., Myers, D. E., and Long A. E., 1995, Fitting matrix-valued variogram models by simultaneous diagonalization (Part I: Theory): Math. Geol., v. 27, no. 7, p. 867–876.CrossRefGoogle Scholar
  18. Xie, T., Myers, D. E., and Long A. E., 1995, Fitting matrix-valued variogram models by simultaneous diagonalization (Part II: Application): Math. Geol., v. 27, no. 7, p. 877–888.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeArizona
  2. 2.Institute of Statistics and Decision SciencesDuke UniversityDurhamDuke

Personalised recommendations