Multi-way overlapping clustering by Bayesian tensor decomposition

Abstract

The development of modern sequencing technologies provides great opportunities to measure gene expression of multiple tissues from different individuals. The three-way variation across genes, tissues, and individuals makes statistical inference a challenging task. In this paper, we propose a Bayesian multi-way clustering approach to cluster genes, tissues, and individuals simultaneously. The proposed model adaptively trichotomizes the observed data into three latent categories and uses a Bayesian hierarchical construction to further decompose the latent variables into lower-dimensional features, which can be interpreted as overlapping clusters. With a Bayesian nonparametric prior, i.e., the Indian buffet process, our method determines the cluster number automatically. The utility of our approach is demonstrated through simulation studies and an application to the Genotype-Tissue Expression (GTEx) RNA-seq data. The clustering result reveals some interesting findings about depression-related genes in human brain, which are also consistent with biological domain knowledge. The detailed algorithm and some numerical results are available in the online Supplementary Material, http://intlpress.com/site/pub/files/-supp/sii/2024/0017/0002/sii-2024-0017-0002-s001.pdf.

Keywords: Bayesian nonparametric prior; Gene expression data; Indian buffet process; Low-rank tensor; Mixture model; Primary 62H30; secondary 62F15.

Figure 1.

The average simulation results based on 50 replicates in the overlapped case for BayMC. The green, black, and red cells represent 1, 0, and −1, respectively. The first row: plots of the true values of ${\mathit{C}}_1,{\mathit{C}}_2$, and ${\mathit{C}}_3$. The second row: plots of the average estimations of ${\mathit{C}}_1,{\mathit{C}}_2$, and ${\mathit{C}}_3$ with the complete observations. The third row: plots of the average estimations of ${\mathit{C}}_1,{\mathit{C}}_2$, and ${\mathit{C}}_3$ with 50% proportion of missing observations.

Figure 2.

The trace plot of the number of clusters for the proposed BayMC method.

Figure 3.

From left to right are the estimated membership matrices of tissues and genes using BayMC. The green, black, and red cells represent 1,0, and −1, respectively. The membership matrix of donors is presented in Section S.3 of the Supplementary Material.

Figure 4.

From left to right are the estimated membership matrices of tissues and genes using HLloyd. The green and black cells represent 1 and 0, respectively.

Figure 5.

From left to right are the estimated membership matrices of tissues and genes using MultiCluster. The green and black cells represent 1 and 0, respectively.