Sparse compositional microbiome data
Hellinger-Riemann intrinsic coordinates represent microbiome compositions directly on the simplex, retaining boundary compositions while supporting diversity summaries and ordination.
Research structure
The site is organized as a compact research narrative: the compositional geometry problem, the HRIC alpha-gamma-beta construction, and the classical diversity theory that motivates the partition.
01 Introduction
Microbiome tables are constrained compositions. The introduction explains why zero handling and sequencing depth can distort CLR-based ordination, and why Simplex Hellinger geometry offers a stable coordinate system.
02 HRIC framework
HRIC maps each sample into intrinsic coordinates. Alpha diversity is an evenness score, gamma summarizes the cohort center, and beta is the remaining among-sample turnover.
03 Theory review
Whittaker's alpha, beta, and gamma terminology separates local diversity, regional diversity, and compositional differentiation. The review clarifies how that ecological idea connects to HRIC.
Central idea
For a sample composition $\pi_i$ with $p$ taxa, HRIC gives intrinsic coordinates $Z_i = \mathrm{HRIC}(\pi_i)$. The same coordinate system supports within-sample evenness, cohort-level gamma diversity, and between-condition turnover without requiring artificial replacement of exact zeros.
HRIC coordinates support cohort-level beta structure, condition-relevant turnover, and formal group comparison in the same geometric system.
References
The pages cite the ecological diversity literature, compositional data analysis, Hellinger transformations in community ecology, and the HRIC software implementation.