Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

• Main Authors: Abraham Nunes, Martin Alda, Thomas Trappenberg
• Format: Article
• Language: English
• Published: MDPI AG 2020-08-01
• Series: Entropy
• Subjects: heterogeneity; diversity; decomposition; Gaussian mixture
• Online Access: https://www.mdpi.com/1099-4300/22/8/858

A system’s heterogeneity (<i>diversity</i>) 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 <inline-formula><math display="inline"><semantics><mrow><mi>q</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Under these indices, the heterogeneity of a composite system (the <inline-formula><math display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-heterogeneity) is decomposable into heterogeneity arising from variation <i>within</i> and <i>between</i> component subsystems (the <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>- and <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that <inline-formula><math display="inline"><semantics><mrow><mi>γ</mi><mo>≥</mo><mi>α</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that <inline-formula><math display="inline"><semantics><mrow><mi>γ</mi><mo>≥</mo><mi>α</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Conversely, under parametric pooling, <inline-formula><math display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-heterogeneity can be computed efficiently in closed-form, but the <inline-formula><math display="inline"><semantics><mrow><mi>γ</mi><mo>≥</mo><mi>α</mi></mrow></semantics></math></inline-formula> condition holds reliably only at <inline-formula><math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Our findings will further contribute to heterogeneity measurement in continuous systems.