Spatial relationships of multivariate data

Abstract

Spatial factor analysis (SFA) is a multivariate method that determines linear combinations of variables with maximum autocorrelation at a given lag. This is achieved by deriving estimates of auto-/cross-correlations of the variables and calculating the corresponding eigenvectors of the covariance quotient matrix. A two-point spatial factor analysis model derives factors by the formation of transition matrixU comparing auto-/cross-correlations at lag “0,”R ₀, with those at a specified lag “d,”R _d, expressed asU _d=R ⁻¹₀ R_d. The matrixU _d can be decomposed into its spectral components which represent the spatial factors. The technique has been extended to include three points of reference. Spatial factors can be derived from the relationship:

$$\left[ {\begin{array}{*{20}c} {U_{d1} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {U_{d2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_0 R_{d3} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {R_{d3} R_0 } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {R_{d1} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {R_{d2} } \\ \end{array} } \right]$$

where the factors ofU _d1 andU _d2 predict the relationships of the variables over three lag distances and orientations from matricesR _d1,R _d2, andR _d3. Estimates of auto-/cross-correlation can be achieved using irregularly spaced or gridded data from experimental correlograms or function estimates determined from the experimental correlograms. The technique has applications in studies where several variables have different spatial ranges or zones of influence.