Principal component analysis for zero-inflated compositional data

Recent advances in DNA sequencing technology have led to a growing interest in microbiome data. Since the data are often high-dimensional, there is a clear need for dimensionality reduction. However, the compositional nature and zero-inflation of microbiome data present many challenges in developing new methodologies. New PCA methods for zero-inflated compositional data are presented, based on a novel framework called principal compositional subspace. These methods aim to identify both the principal compositional subspace and the corresponding principal scores that best approximate the given data, ensuring that their reconstruction remains within the compositional simplex. To this end, the constrained optimization problems are established and alternating minimization algorithms are provided to solve the problems. The theoretical properties of the principal compositional subspace, particularly focusing on its existence and consistency, are further investigated. Simulation studies have demonstrated that the methods achieve lower reconstruction errors than the existing log-ratio PCA in the presence of a linear pattern and have shown comparable performance in a curved pattern. The methods have been applied to four microbiome compositional datasets with excessive zeros, successfully recovering the underlying low-rank structure.