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Fractals and Chaos

One well-known isomorphism is the “self-similarity” between scales exhibited by fractal structures (Mandelbrot 1977) which may provide another approach to the problem of scaling. This self-similarity implies a regular and predictable relationship between the scale of measurement (here meaning the resolution of measurement) and the measured phenomenon. For example, the regular relationship between the measured length of a coastline and the resolution at which it is measured is a fundamental, empirically observable one. It can be summarized in the following equation:

where:

L = the length of the coastline or other “fractal” boundary

s= the size of the fundamental unit of measure or the resolution of the measurement k = a scaling constant

D = the fractal dimension

Primary questions concern the range of applicability of fractals and chaotic systems dynamics to the practical problems of modeling ecological economic systems. The influence of scale, resolution, and hierarchy on the mix of behaviors one observes in systems has not been fully investigated, and this remains a key question for developing coherent models of complex ecological economic systems.

Resolution and Predictability

The significant effects of nonlinearities raise some interesting questions about the influence of resolution (including spatial, temporal, and component) on the performance of models, and in particular their predictability. Costanza and Maxwell (1994) analyzed the relationship between resolution and predictability and found that while increasing resolution provides more descriptive information about the patterns in data, it also increases the difficulty of accurately modeling those patterns. There may be limits to the predictability of natural phenomenon at particular resolutions, and “fractal like” rules that determine how both “data” and “model” predictability change with resolution.

Some limited testing of these ideas was done by resampling land use map data sets at several different spatial resolutions and measuring predictability at each. Colwell (1974) used categorical data to defi predictability as the reduction in uncertainty (scaled on a 0–1 range) about one variable given knowledge of others. One can defi spatial auto-predictability (Pa) as the reduction in uncertainty about the state of a pixel in a scene, given knowledge of the state of adjacent pixels in that scene, and spatial cross-predictability (Pc) as the reduction in uncertainty about the state of a pixel in a scene, given knowledge of the state of corresponding pixels in other scenes. Pa is a measure of the internal pattern in the data, while Pc is a measure of the ability of some other model to represent that pattern.

A strong linear relationship was found between the log of Pa and the log of resolution (measured as the number of pixels per square kilometer). This fractal-like characteristic of “self-similarity” with decreasing resolution implies that predictability, like the length of a coastline, may be best described using a unitless dimension that summarizes how it changes with resolution. One can define a “fractal predictability dimension” (DP) in a manner analogous to the normal fractal dimension (Mandelbrot 1977, 1983). The resulting DP allows convenient scaling of predictability measurements taken at one resolution to others.

Cross-predictability (Pc) can be used for pattern matching and testing the fit between scenes. In this sense it relates to the predictability of models versus the internal predictability in the data revealed by Pa. While Pa generally increases with increasing resolution (because more information is being included), Pc generally falls or remains stable (because it is easier to model aggregate results than fine grain ones). Thus we can define an optimal resolution for a particular modeling problem that balances the benefit in terms of increasing data predictability (Pa) as one

Fig. 1.1 Relationship between resolution and predictability for data and models (From Costanza and Maxwell 1994)

increases resolution, with the cost of decreasing model predictability (Pc). Figure 1.1 shows this relationship in generalized form.

 
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