location at different times. This violates a key assumption central to much statistical theory—that observations are mutually independent.
Attempts to deal with these challenges have stimulated the development of a new subfield of statistics. Although this work began in ecology and biostatistics and has lately attracted the interest of statisticians, many of these developments were pioneered by geographers. For example, geographers have developed methods for estimating the degree and nature of spatial autocovariance in point, line, and area data. They have also addressed such complicating features as periodicity or waves in spatial patterns. Geographers have also played an important role in developing multivariate statistical analysis methods to deal with the spatial and temporal autocovariance of much spatially referenced data.
Spatial data pose special problems that are subjects for research by geographers. Because geographic data often fail to meet distributional assumptions necessary for classical statistical procedures, geographers have been at the center of attempts to develop distribution-free methods for estimating statistical relationships among variables. They also have been involved in the development of methods for estimating prior probability distributions, either through Monte Carlo simulations that generate reference distributions unique to each locality or by developing Bayesian methods that allow investigators to incorporate knowledge of known relationships in statistical investigations.
Methods for evaluating data for spatial dependencies have received recent attention from geographers (e.g., Getis and Ord, 1992). Such measures are used both to identify spatial patterns in data and to allow analysts to understand spatial relationships in their data so that appropriate analytical techniques can be chosen. When measures of spatial dependency are applied to real world datasets of the magnitude needed to address the societal problems identified in Chapter 2, spatial dependency measures can become intractable. One promising solution to the dilemma is the application of massively parallel processing methods (Armstrong and Marciano, 1995).
Recent research in GISs aims to develop "data models" that facilitate the routine analysis of spatial dependence, spatial heterogeneity and spatially referenced diagnosis of regression models. These spatial data models are used to prepare data in the special forms needed to efficiently accomplish these spatial analysis methods. The intent is to bring the same level of enabling technology to spatial analysis that spreadsheets and statistical packages have brought to statistical analysis (Anselin et al., 1993).
Research in spatial statistics is believed by many to be in its infancy because many research questions are likely to yield to computational approaches. There are enormous complexities in the analysis of spatially referenced data that require additional research. For instance, research is needed to understand the scale dependence of statistical methods and sampling designs. This work is necessary to understand why different aggregations of spatially referenced observations give statistical results that vary dramatically and unpredictably from one spatial