Abstract
Many applications involving spatial data require several layers of information to be simultaneously analyzed in relation to underlying geography and topographic detail. This in turn generates a need for forms of multivariate analysis particularly oriented to spatial problems and designed to handle spatial structure and dependency both within and between spatially indexed multivariate responses. In this paper we focus on one group of such methods sometimes referred to as “spatial factor analysis.” Use of these techniques has so far been mostly restricted to applications in the geosciences and in some forms of image processing, but the methods have potential for wider use outside these fields. They are concerned with identifying components of a multivariate data set with a spatial covariance structure that predominantly acts over a particular spatial range or zone of influence. We review the various forms of spatial factor analysis that have been proposed and emphasize links between them and with the linear model of coregionalization employed in geostatistics. We then introduce extensions to such methods that may prove useful in exploratory spatial analysis, both generally and more specifically in the context of multivariate spatial prediction. Application of our proposed exploratory techniques is demonstrated on a small but illustrative geochemical data set involving multielement measurements from stream sediments.
Similar content being viewed by others
REFERENCES
Armstrong, M., ed., 1988, Geostatistics: Kluwer Academic Publisher, Dordrecht, 255 p.
Bourgault, G., and Marcotte, D., 1991, Multivariable variogram and its application to the linear model of coregionalization: Math. Geology, v. 23, p. 899–928.
Brown, P. J., Le, N. D., and Zidek, J. V., 1994, Multivariate spatial interpolation and exposure to air pollutants: Can. Jour. Stats., v. 22, p. 489–509.
Campbell, N. A., and Atchley, W. R., 1981, The geometry of canonical variate analysis: Systematic Zoology, v. 30, p. 268–280.
Cleveland, W. S., Grosse, E., and Shyu, W. M., 1992, Local regression models, in Chambers, J. M., and Hastie, T. J., eds., Statistical models in S:Wadsworth and Brooks, Pacific Grove, CA, p. 309–376.
Cressie, N., 1991, Statistics for spatial data: John Wiley, New York, 900 p.
Deutsch, C., and Journel, A., 1992, GSLIB Geostatistical Software Library and User's Guide: Oxford University Press, New York, 225 p.
Flury, B. N., 1988, Common principal components and related multivariate models: John Wiley, New York, 258 p.
Goovaerts, P., 1992, Factorial kriging analysis, a useful tool for exploring the structure of multivariate spatial information: Jour. Soil Science, v. 43, p. 597–619.
Goovaerts, P., 1993, Spatial orthogonality of the principal components computed from coregionalized variables: Math. Geology, v. 25, p. 281–302.
Goovaerts, P., 1994, Study of spatial relationships between two sets of variables using multivariate geostatistics: Geoderma, v. 62, p. 93–107.
Goulard, M., and Voltz, M., 1992, Linear coregionalization model, tools for estimation and choice of multivariate variograms: Math. Geology, v. 24, p. 269–286.
Grunsky, E. C., and Agterberg, F. P., 1988, Spatial and multivariate analysis of geochemical data from metavolcanic rocks in the Ben Nevis area, Ontario: Math. Geology, v. 20, p. 825–861.
Grunsky, E. C., and Agterberg, F. P., 1991, SPFAC, a Fortran 77 program for spatial multivariate analysis: Computers & Geosciences, v. 17, p. 133–160.
Grunsky, E. C., and Agterberg, F. P., 1992, Spatial relationships of multivariate data: Math. Geology, v. 24, p. 731–758.
Haas, T. C., 1996, Multivariate spatial prediction in the presence of nonlinear trend and covariance nonstationarity: Environmetrics, v. 7, p. 145–165.
Journel, H., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, New York, 600 p.
Krzanowski, W. J., 1990, Principles of multivariate analysis: Oxford University Press, Oxford, UK, 553 p.
Myers, D. E., 1982, Matrix formulation of co-kriging: Math. Geology, v. 14, p. 249–257.
Myers, D. E., 1984, Cokriging-new developments, in Verly, G., ed., Geostatistics for natural resources characterization: Reidel, Dordrecht, Holland, p. 295–305.
Myers, D. E., 1991, Pseudo-cross variograms, positive-definiteness and cokriging: Math. Geology, v. 23, p. 805–816.
Papritz, A., Künsch H. R., and Webster, R., 1993, On the pseudo cross-variogram: Math. Geology, v. 15, p. 1015–1026.
Rouhani, S., and Wackernagel, H., 1990, Multivariate geostatistical approach to space-time data analysis: Water Resources Res., v. 26, p. 585–591.
Searle, S. R., Casella, G., and McCulloch, C. E., 1992, Variance components: John Wiley, New York, 623 p.
Switzer, P., 1985, Min/Max autocorrelation factors for multivariate spatial imagery: Computer Science and Statistics, Proceedings of the 16th Symposium on the Interface, p. 13–16.
Switzer, P., and Green, A., 1984, Min/Max autocorrelation factors for multivariate spatial imagery: Technical Report No. 6, Department of Statistics, Stanford University, Stanford, CA, 23 p.
Ver Hoef, J. M., and Cressie, N., 1993, Multivariable spatial prediction: Math. Geology, v. 25, p. 219–240.
Wackernagel, H., 1988, Geostatistical techniques for interpreting multivariate spatial information, in Chung, C. F., Fabri, A. G., and Sinding-Larsen, R., eds., Quantitative analysis of mineral and energy resources: Reidel, Dordrecht, p. 393–409.
Wackernagel, H., 1995, Multivariate geostatistics an introduction with applications: Springer-Verlag, Berlin, 255 p.
Wackernagel, H., Petitgas, P., and Touffait, Y., 1989, Overview of methods for coregionalisation analysis, in Armstrong, M., ed., Geostatistics: Kluwer Academic Publisher, Amsterdam, Holland, p. 409–420.
Webster, R., 1978, Optimally partitioning soil transects: Jour. Soil Science, v. 45, p. 205–218.
Yao, T., and Journel, A. G., 1998, Automatic modeling of (cross)covariance tables using Fast fourier transform: Math. Geology, v. 30, p. 589–616.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bailey, T.C., Krzanowski, W.J. Extensions to Spatial Factor Methods with an Illustration in Geochemistry. Mathematical Geology 32, 657–682 (2000). https://doi.org/10.1023/A:1007589505425
Issue Date:
DOI: https://doi.org/10.1023/A:1007589505425