Adjusting spatial-entropy measures for scale and resolution effects

In this paper we revisit the concept of entropy as it manifests itself in spatial terms. We focus specifically on the question of how entropy measures applied to different urban contexts can be adjusted to allow for meaningful comparisons between cities with differing geographic dimensions. It is well known that entropy is affected by the number of geographic units over which it is computed. As a result, the size and number of census tracts in an urban area constitute an intervening factor in making direct comparisons. Some authors advocate addressing this problem by normalizing entropy to its maximum value to derive a ‘relative entropy’ measure. We prove that this conventional normalization procedure does not suffice, and we show further that Theil’s decomposition method does provide the proper solution. We then demonstrate how to apply this technique through the use of census data for US cities in 2000, with the empirical results clearly underlying the importance of making these adjustments.