RESOLUTION AND PREDICTABILITY - AN APPROACH TO THE SCALING PROBLEM

We analyzed the relationship between resolution and predictability and found that while increasing resolution provides more descriptive info rmation about the patterns in data, it also increases the difficulty o f accurately modeling those patterns. There are limits to the predicta bility of natural phenomenon at particular resolutions, and ''fractal- like'' rules determine how both ''data'' and ''model'' predictability change with resolution. We analyzed land use data by resampling map da ta sets at several different spatial resolutions and measuring predict ability at each. Spatial auto-predictability (P(a)) is the reduction i n uncertainty about the state of a pixel in a scene given knowledge of the state of adjacent pixels in that scene, and spatial cross-predict ability (P(c)) is the reduction in uncertainty about the state of a pi xel in a scene given knowledge of the state of corresponding pixels in other scenes. P(a) is a measure of the internal pattern in the data w hile P(c) is a measure of the ability of some other ''model'' to repre sent that pattern.We found a strong linear relationship between the lo g of P(a) and the log of resolution (measured as the number of pixels per square kilometer). This fractal-like characteristic of ''self-simi larity'' with decreasing resolution implies that predictability may be best described using a unitless dimension that summarizes how it chan ges with resolution. While P(a) generally increases with increasing re solution (because more information is being included), P(c) generally falls or remains stable (because it is easier to model aggregate resul ts than fine grain ones). Thus one can define an ''optimal'' resolutio n for a particular modeling problem that balances the benefit in terms of increasing data predictability (P(a)) as one increases resolution, with the cost of decreasing model predictability (P(c)).