Data increasingly take the form of a multi-way array, or tensor, in several biomedical domains. Such tensors are often incompletely observed. For example, we are motivated by longitudinal microbiome studies in which several timepoints are missing for several subjects. There is a growing literature on missing data imputation for tensors. However, existing methods give a point estimate for missing values without capturing uncertainty. We propose a multiple imputation approach for tensors in a flexible Bayesian framework, that yields realistic simulated values for missing entries and can propagate uncertainty through subsequent analyses. Our model uses efficient and widely applicable conjugate priors for a CANDECOMP/PARAFAC (CP) factorization, with a separable residual covariance structure. This approach is shown to perform well with respect to both imputation accuracy and uncertainty calibration, for scenarios in which either single entries or entire fibers of the tensor are missing. For two microbiome applications, it is shown to accurately capture uncertainty in the full microbiome profile at missing timepoints and used to infer trends in species diversity for the population. Documented R code to perform our multiple imputation approach is available at https://github.com/lockEF/MultiwayImputation .

翻译：在多个生物医学领域中，数据日益呈现为多路数组（即张量）的形式。此类张量常存在观测不完整的情况。例如，我们受纵向微生物组研究的启发，该研究中多个受试者的若干时间点数据存在缺失。关于张量缺失数据插补的研究文献正不断增长。然而，现有方法仅提供缺失值的点估计，未能捕捉其不确定性。我们在灵活的贝叶斯框架中提出一种张量多重插补方法，该方法能为缺失条目生成逼真的模拟值，并能将不确定性传递至后续分析中。我们的模型采用高效且广泛适用的共轭先验，对CANDECOMP/PARAFAC（CP）分解进行建模，并具有可分离的残差协方差结构。实验表明，在张量单一条目或整条纤维缺失的场景下，该方法在插补精度和不确定性校准方面均表现优异。在两个微生物组应用中，该方法被证明能准确捕捉缺失时间点完整微生物组谱的不确定性，并用于推断种群物种多样性的变化趋势。用于执行本多重插补方法的R代码文档已发布于https://github.com/lockEF/MultiwayImputation。