Representation and Spatial Analysis in Geographic Information Systems

Abstract

A common—perhaps modal—representation of geography in spatial analysis and geographic information systems is native (unexamined) objects interacting based on simple distance and connectivity relationships within an empty Euclidean space. This is only one possibility among a large set of geographic representations that can support quantitative analysis. Through the vehicle of GIS, many researchers are adopting this representation without realizing its assumptions or its alternatives. Rather than locking researchers into a single representation, GIS could serve as a toolkit for estimating and exploring alternative geographic representations and their analytical possibilities. The article reviews geographic representations, their associated analytical possibilities and relevant computational tools in the combined spatial analysis and GIScience literatures. The discussion identifies several research and development frontiers, including analytical gaps in current GIS software.

Key Words:

• geographic information systems
• geographic representation
• spatial analysis

Notes

1. “Analysis” is a multifaceted term. We refer here to its precise mathematical definition as the class of sciences that examines exact relations between quantities or magnitudes ( Webster's Unabridged Dictionary 1998). Analysis can only examine real-world properties that are measurable, countable, or formally comparable. This definition is consistent with the spatial analytic tradition in modern geography (Taaffe 1974). While there are other valid forms of inquiry, including narratives and graphics, these are not the direct concern of the discussion in this article.

2. Following standard usage, we use the term “geographic information systems” (GIS) to refer to the technology and “geographic information science” (GIScience) to refer to the theories and methods that underlie the technological implementation.

3. We purposely avoid naming any commercial or public-license GIS software directly. When we use the phrase “most GIS software” (or a variation), we imply precisely that: there is one or a very small number of software packages to which this observation does not apply. We decline to name the specific GIS software, either in a positive or a negative light.

4. “Countable” means that we can derive a one-to-one correspondence between the set in question and the set of natural numbers. In 1873, Georg Cantor proved that different “sizes” of infinite sets exist, with the natural numbers being the smallest. The set of irrational numbers is a larger and uncountable infinite set (Borowski and Borwein 1991). The term “σ-bounded” is often used to designate “countably bounded.”

5. See Flake (2000) for an excellent discussion of the relationships between computability and natural (including human-made) systems.

6. More precisely, this set is the Borelσ-algebra associated with (X, d _L ). This is the smallest σ-algebra that contains the open subsets defined by (X,d_L ). The σ-algebra of a set is a collection of subsets that contains: (1) the set itself; (2) the empty set; (3) the complements of all members of the set; and (4) all countable unions of members of the set (Borowski and Borwein 1991; see Beguin and Thisse 1979 for an alternative definition). Any Borel set is measurable, meaning that we can define a “measure” as indicated in the main text (see Haaser and Sullivan 1971).

7. Note that the concept of “area” does not require a metric space topology; see Casati, Smith, and Varzi (1998).

8. O ( ) or “big oh” notation indicates the order or general complexity class of the algorithm in the worst case. For example, O(n ²) states that the algorithm will require no more than n ² operations, subject to a proportionality constant. For more information on complexity analysis, see Garey and Johnston (1979) and Sipser (1997).

9. The law of transportation refraction assumes a cost-density field with no points of concentration or other impenetrable objects that can “block” shortest paths. Incorporating these objects requires an analogous concept of diffusion. Thanks to Mike Goodchild for pointing this out.

10. Or, more precisely, cells. A cell is a connected two-dimensional region with no “holes.” See Worboys (1995) for a more precise definition.

11. A simple line is a one-dimensional spatial object that is topologically equivalent to a straight line (Worboys 1995): in other words, the line does not “cross” itself.