用于时变成分数据的扩散模型
A diffusion model for time-dependent compositional data
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中文总结 AI 辅助
该研究提出了名为Dirichlet扩散（DD）的随机过程，用于建模时变成分数据的演化，证明了其解的存在性与聚合性，可用于微生物组演化的联合建模。
中文摘要 AI 辅助
我们引入一种用于建模成分测量随时间演化的随机过程，成分测量即总和为1的非负数值向量。该模型是一种扩散过程，定义为伊藤随机微分方程的解，其稳态分布为Dirichlet分布，我们将此过程命名为Dirichlet扩散（Dirichlet Diffusion, DD）。由于该过程被限制在流形上，且方程系数并非全局Lipschitz，通常的定理无法直接应用，因此建立解的存在性与性质需要较为精细的分析。我们在Dirichlet分布所有参数均大于2的假设下，证明了强解的存在性；对于一般情况，仅能证明弱解的存在性，且这些解无需反射边界即可保持在闭合单纯形内。DD从Dirichlet分布继承的一个有用特性是聚合性：若将成分组合成更粗粒度的成分，所得过程仍为DD。这使得DD可用于联合建模微生物组的演化，例如在不同分类水平上对微生物物种进行分组建模。
英文摘要
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.
