Academic journal article Cartography and Geographic Information Science

Toward a Differential Calculus for Temporal Map Analysis

Academic journal article Cartography and Geographic Information Science

Toward a Differential Calculus for Temporal Map Analysis

Article excerpt


Current literature contains a great many examples of researchers evaluating how spatial phenomena change over time (Guan et al. 2003; Palandro et al. 2003; Ruiz-Luna and Berlanga-Robles 2003). Given the frequency with which such analyses are being conducted, it is important to investigate spatio-temporal analysis procedures, and to develop new procedures when current approaches are found to be lacking. The ability to perform temporal analysis is often cited as one of the strengths of GIS, but conversely, the abilities of current GIS digital data models and software to analyze the temporal component of spatial data are often seen as limited and inadequate (Peuquet 2001; Hornsby and Egenhofer 2000; Wachowicz 1999). This situation is analogous to one encountered in the field of mathematics prior to the latter half of the 17th century. In both cases, it was widely believed that existing intellectual approaches (i.e., mathematics prior to the second half of the 17th century and contemporary spatial data analysis techniques) had the potential of being able to handle temporal problems, but existing techniques and methodologies were not adequate to actually perform the desired temporal analyses (Hahn 1998).

The most obvious way of handling the temporal aspect of spatial data is to consider time to be an additional dimension of the data (Peuquet 2002; Langran 1993). This approach has its roots in physics, where the idea of space-time is central to relativity theory (Slowik 2002). Considering time to be an additional dimension of spatial data has a natural attraction for geospatial scientists accustomed to thinking in multidimensional terms, and this approach is made even more appealing by its linkage to relativity theory, which is generally held in very high regard. Unfortunately, while the idea of recording time as an additional dimension of spatial data provides a very elegant method for recording the temporal component of spatial data, it does not lead to any obvious ways of analyzing temporal spatial data (Peuqnet 2002).

In the field of mathematics, the ability to conduct temporal analysis was obtained with the development of differential calculus by Newton and Leibniz in the late 17th century. (1) Differential calculus gave mathematicians the ability to analyze the instantaneous rate of change of any function. Virtually all contemporary efforts to analyze temporal change in spatial data make use of the concepts of differential calculus, although the differential roots of these analyses are seldom explicitly recognized.

This paper will evaluate how the techniques of differential calculus are commonly applied to spatial data. It will be argued that differential techniques are usually applied to the wrong components of spatial data, resulting in overly simplistic characterizations of temporal change. More appropriate methods for analyzing temporal change are presented, and the results of applying these methods and conventional techniques to a simple spatial data set are compared.

Contemporary Analyses of Temporal Change in Spatial Data

Contemporary spatio-temporal analyses frequently apply the techniques of differential calculus to relatively simple spatial metrics. Arguably the nlost common of these metrics is area. Current literature is replete with examples of empirical studies which evaluate how the area occupied by various land covers have changed over time in response to natural and/or anthropogenic causes (Guan et al. 2003; Palandro et al. 2003; Ruiz-Lnna and Berlanga-Robles 2003). Many (but not all) of these studies are based upon raster data (frequently derived from digital processing of remotely sensed imagery), where calculating area involves little more than counting the number of raster cells falling into the land cover category of interest. This is a perfectly viable method of computing area that is compatible with contemporary spatio-temporal analysis techniques. …

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