Heterogeneity versus homogeneity: A conceptual and mathematical theory in terms of scale-invariant and scale-covariant distributions
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Ecologists often use the words heterogeneity and homogeneity when they are describing the distribution of a variable, yet there has been no formal exploration into the relation of these two states to one another. This work formalizes their relationship within statistical theory with a conceptual framework that includes five parameters: L, K, S, D, and R. These parameters provide an increasing degree of specificity about a distribution as they are enumerated and accord with measurements of system extent, richness, evenness, variance, and scale-covariance. Moreover, a mathematical and physical basis for understanding heterogeneity and homogeneity is outlined in terms of Brownian motion, where the Hurst exponent H and the notion of variance are utilized to delimit these two states. Spatial and temporal patterns are quantified and classified according to the mathematical and physical basis at multiple scales, as well as within the conceptual framework that can be applied to other metrics. Concepts of scale-invariance and scale-covariance are discussed in terms of hierarchy theory. It is argued that in hierarchical distributions, we should expect more than simple scale-covariance; we should expect division by a transitional state that divides heterogeneity from homogeneity at some scale. By revisiting the underlying statistical theory behind these concepts, a more efficient approach to quantifying ecological distributions can result. 2005 Elsevier B.V. All rights reserved.
