On the Multiplicative Decomposition of Heterogeneity in Continuous Assemblages
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A system's heterogeneity (equivalently, diversity) amounts to the effective size of its event space, and can be quantified using the Rényi family of indices (also known as Hill numbers in ecology or Hannah-Kay indices in economics), which are indexed by an elasticity parameter q ≥ 0. Importantly, under these indices, the heterogeneity of a composite system (the γ-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the α- and β-heterogeneities, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled assemblage, we require that α≤γ. There exists a multiplicative decomposition for Rényi heterogeneity of composite systems with discrete event spaces, but less attention has been paid to decomposition in the continuous setting. This paper therefore describes multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. We show that under non-parametric pooling (where γ-heterogeneity must typically be estimated numerically), the multiplicative decomposition holds such that γ≥α for all values of the elasticity parameter q > 0, and β-heterogeneity amounts to the discrete number of distinct mixture components in the system. Conversely, under parametric pooling (as in a Gaussian mixed-effects model), which facilitates efficient analytical computation of γ-heterogeneity, we show that the γ≥α condition holds only at q=1. By providing conditions under which the decomposability axiom of heterogeneity measurement holds, our findings further advance the understanding of heterogeneity measurement in non-categorical systems.
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