ABSTRACT
This chapter introduces two distinct specifications of covariance structure to be incorporated into the spatial linear model. For geostatistical data, the covariance structure is specified by a spatial covariance function, semivariogram, or generalized covariance function, which directly model the covariance or some other second-order moment of two observations as a function of their corresponding locations. For areal data, the covariance structure is more commonly specified indirectly through a spatial weights matrix that assigns positive weights to a site's neighbors. The chapter begins with a review of moments, up to second order, of random variables, vectors, and processes. It then proceeds to describe, for geostatistical data, many types of spatial structure such as second-order stationarity, isotropy, and separability; and for areal data, different types of neighborhoods and weights, and ways to generate a covariance matrix from the spatial weights matrix via spatial autoregressive or moving average models. The chapter concludes by revisiting the spatial linear model, in particular a general parametric form for its covariance matrix resulting from the covariance structures described earlier.
