Resolution and predictability: An approach to the scaling problem

Abstract

We 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 are limits to the predictability 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 data sets at several different spatial resolutions and measuring predictability at each. Spatial auto-predictability (P_a) is 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 (P_c) is the reduction in uncertainty about the state of a pixel 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 while P_c is a measure of the ability of some other “model” to represent that pattern. We found a strong linear relationship between the log of P_a 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 may be best described using a unitless dimension that summarizes how it changes with resolution. While P_a generally increases with increasing resolution (because more information is being included), P_c generally falls or remains stable (because it is easier to model aggregate results than fine grain ones). Thus one can define an “optimal” resolution 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).