Multivariate Spatial Modeling for Geostatistical Data Using Convolved Covariance Functions
- 384 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.
Keywordsconvolution coregionalization Fourier transforms Gaussian spatial process hierarchical model Markov chain Monte Carlo spectral density
Unable to display preview. Download preview PDF.
- 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
- Daniels, M., Zhou, Z., and Zou, H., 2004, Conditionally specified space-time models for multivariate processes: submitted.Google Scholar
- 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
- 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
- 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
- Sain, S., and Cressie, N., 2002, Multivariate lattice models for spatial environmental data: American Statistical Association Proceedings, p. 2820–2825.Google Scholar
- Stein, M. L., 1999, Interpolation of spatial data: Some theory for kriging: Springer Verlag, New York, p. 24–25.Google Scholar
- Wackernagel, H., 2003, Multivariate geostatistics: An introduction with applications, 2nd ed.: Springer Verlag, Berlin.Google Scholar