A diffusion model for time-dependent compositional data

Abstract

We introduce a stochastic process for modeling the evolution in time of compositional measurements (i.e., a vector of non-negative values that add up to a total of 1). This model is a diffusion, as it is defined as the solution for a stochastic differential equation in the Ito sense, and it has a Dirichlet distribution as its steady distribution. We have named this process Dirichlet Diffusion (DD). As the process is confined to a manifold and the coefficients of the equation are not globally Lipschitz, the usual theorems do not apply directly and establishing the existence and properties of solutions requires a somewhat delicate analysis. We establish the existence of strong solutions under the assumption that all the parameters of the Dirichlet distribution are greater than 2; for the general case we were only able to establish the existence of weak solutions, but also that these solutions remain confined to the closed simplex without need for reflecting boundaries. A useful feature of DD that it inherits from the Dirichlet distribution is the property of aggregation: If components are combined to create a coarser composition, the resulting process is also a DD. This makes it useful, for example, for jointly modeling the evolution of a microbiome grouping the microbe species at different taxonomic levels.