Scale effect on principal component analysis for vector random function

Principal component analysis (PCA) is commonly applied without looking at t he "spatial support" (size and shape, of the samples and the field), and th e cross-covariance structure of the explored attributes. This paper shows t hat PCA can depend on such spatial features. If the spatial random function s for attributes correspond to largely dissimilar variograms and cross-vari ograms, the scale effect will increase as well. On the other hand, under co nditions of proportional shape of the variograms and cross-variograms (i.e. , intrinsic coregionalization), no scale effect may occur. The theoretical analysis leads to eigenvalue and eigenvector functions of the size of the d omain and sample supports. We termed this analysis "growing scale PCA," whe re spatial (or time) scale refers to the size and shape of the domain and s amples. An example of silt, sand, and clay attributes for a second-order st ationary vector random function shows the correlation matrix asymptotically approaches constants at two or three times the largest range of the spheri cal variogram used in the nested model. This is contrary to the common beli ef that the correlation structure between attributes become constant at the range value. Results of growing scale PCA illustrate the rotation of the o rthogonal space of the eigenvectors as the size of the domain grows. PCA re sults are strongly controlled by the multivariate matrix variogram model. T his approach is useful for exploratory data analysis of spatially autocorre lated vector random functions.