Regionalized Variables

Geographically distributed phenomena over a one-, twoor three-dimensional metric space can be concentrated in some subset of discrete points, be collected into mutually exclusive and collectively exhaustive areal unit aggregates, or be continuous. A surface formed by one of these phenomena, such as a population density map, usually is too irregular to be described analytically by simple, smooth mathematical functions. Rather, a single variable function, called a regionalized variable, can be specified in terms of an underlying geographic coordinate system in order to describe the phenomenon in a mixed deterministic and stochastic way, by including (a) a structural component that captures geographic trends (e.g., population density declining with increasing distance from a city center), (b) a structured random component that captures spatial autocorrelation effects (e.g., nearby population densities tend to be similar), and (c) an uncorrelated random noise component (e.g., a cemetery here, an uninhabited house there).

Because regionalized variables are intermediate between completely deterministic and truly random variables, they allow an observed map to be interpreted as one of many possible realizations of the phenomenon under study; the two stochastic components are random quantities with potentially infinite possible values. Thus, a regionalized variable can be referred to synonymously as a random function, stochastic process, or random field. These variables constitute the core data of geographic information science (GISci).

The first task in describing a regionalized variable is to specify suitable mathematical functions for each of its three components. Frequently, the structural component is specified as either a constant mean or some polynomial function of the underlying coordinate system across a geographic landscape. This functional [Page 364]form captures deterministic trends in geographic variation from place to place. The spatially structured random component is modeled as a function of distance between places. This specification rests on two distance-covariation relationship assumptions (i.e., the intrinsic hypothesis), stating that this function is (1) of distance separation, not of absolute location, and (2) constant across a geographic landscape (i.e., stationary). These assumptions allow observations for a single map realization to be treated as though they were exchangeable (i.e., a given value could have materialized at any of the observed locations), compensating for the absence of more than one geographic realization (e.g., population density maps for 1990 and 2000). Coupled with a finite variance, they also establish second-order stationarity (i.e., only a constant mean and variance are needed across a geographic landscape).

The resulting spatial covariation must yield positive definite sample covariance matrices (e.g., variances cannot be zero or negative) for the function to be valid, limiting the possible mathematical function specifications that can be posited. Virtually all valid functions are nonlinear, with some of the more popular ones being the spherical, power, rational quadratic, stable, and Bessel. For population density calculated with census tract data across the Cusco region of Peru,

where RESS denotes “relative error sum of squares,” C0 is an intercept term that theoretically should be 0, C1 is the net variance (i.e., minus C0) in the absence of spatial autocorrelation, r is a range parameter indicating the distance at which spatial autocorrelation becomes negligible or zero, a is an exponent, distm denotes average distance separation for some distance interval (intervals need to be established to ensure that they all contain a minimum number of distances, say 30), exp denotes the base of the natural logarithm, and B1 denotes a Bessel function of the first order and second kind. These estimation results suggest that most of the semivariogram models furnish similar mathematical descriptions of a regionalized variable.

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