A system’s heterogeneity (diversity) is 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. 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-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that γ≥α. 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. We therefore describe multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the γ-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that γ≥α for q>0. Conversely, under parametric pooling, γ-heterogeneity can be computed efficiently in closed-form, but the γ≥α condition holds reliably only at q=1. Our findings will further contribute to heterogeneity measurement in continuous systems.

中文翻译：

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连续分布混合中异质性的乘法分解

系统的异质性（多样性）是其事件空间的有效大小，可以使用 Rényi 系列指数（在生态学中也称为希尔数或经济学中的 Hannah-Kay 指数）进行量化，这些指数由弹性参数 q ≥0。在这些指标下，复合系统的异质性（γ-异质性）可分解为由组件子系统内部和之间的变化引起的异质性（分别为α-和β-异质性）。由于组件子系统的平均异质性不应大于池化系统的平均异质性，我们要求 γ≥α。对于具有离散事件空间的复合系统的 Rényi 异质性，存在乘法分解，但对连续设置中的分解关注较少。因此，我们描述了参数和非参数池化假设下连续混合分布的 Rényi 异质性的乘法分解。在非参数池化下，γ-异质性通常必须以数值方式估计，但乘法分解成立，使得 q>0 时 γ≥α。相反，在参数池化下，γ-异质性可以在封闭形式中有效计算，但 γ≥α 条件仅在 q=1 时可靠地成立。我们的发现将进一步有助于连续系统中的异质性测量。在参数池化下，γ-异质性可以以封闭形式有效计算，但 γ≥α 条件仅在 q=1 时可靠地成立。我们的发现将进一步有助于连续系统中的异质性测量。在参数池化下，γ-异质性可以以封闭形式有效计算，但 γ≥α 条件仅在 q=1 时可靠地成立。我们的发现将进一步有助于连续系统中的异质性测量。