Modeling Time-course Gene Expression Data through Bayesian Partition Functional Principal Component Analysis

Technical Summary: Modeling Time-Course Gene Expression Data Through Bayesian Partition Functional Principal Component Analysis

Problem Statement
High-dimensional biomarker data, such as longitudinal gene expression levels, present a dual challenge: reducing dimensionality while simultaneously uncovering shared temporal structures and quantifying inter-individual variability. Existing methods typically address these objectives separately. Standard Multivariate Functional Principal Component Analysis (MFPCA) assumes all variables share a single underlying latent process, an assumption that is often unrealistic in high-dimensional settings where only subsets of variables exhibit coordinated dynamics. Conversely, clustering approaches often treat variables as independent or fail to explicitly model between-individual variability, leading to clusters driven by idiosyncratic features of single subjects rather than robust, shared biological patterns. There is a lack of a principled framework that jointly partitions variables into groups based on shared latent temporal processes and performs dimensionality reduction within those groups.

Methodology
The authors introduce Partition Functional Principal Component Analysis (PFPCA), a Bayesian hierarchical model that simultaneously partitions variables into groups driven by shared latent processes and performs MFPCA within each group.

• Model Structure: The model assumes that $p$ random functions (variables) are partitioned into $Q$ latent groups. Within a group $q$, variables share a common set of $L$ principal component scores ($\zeta^{(q)}_i$) but possess distinct mean functions and eigenfunctions. The model utilizes a semi-parametric regression approach, representing mean functions and eigenfunctions via O'Sullivan penalized splines.
• Inference Strategy: Given the high dimensionality of the data, the authors develop a mean-field variational Bayes (MFVB) algorithm. To address the multimodality of the posterior distribution and avoid convergence to poor local modes, this is coupled with a simulated annealing procedure (Annealed Mean-Field Variational Bayes or AMFVB).
• Parameter Learning: The framework includes a mechanism to learn the optimal number of mixture components ($Q$) and FPCA components ($L$) directly from the data. This is achieved by setting a conservatively large initial number of components and selecting the effective number based on the proportion of variance explained (targeting 95% cumulative variance) and the count of non-empty groups in an overfitted mixture model.
• Post-Processing: To ensure the uniqueness of the FPCA decomposition, a post-inference orthogonalization procedure is applied within each group using the variational posterior means of scores and eigenfunctions.

Key Contributions

1. Joint Inference Framework: PFPCA is the first approach to jointly learn variable partitions and latent functional structures within a single Bayesian framework, enabling information sharing across variables within groups.
2. Scalable Algorithm: The development of a scalable AMFVB algorithm allows for the analysis of high-dimensional longitudinal data where Markov Chain Monte Carlo (MCMC) methods (e.g., Hamiltonian Monte Carlo) are computationally prohibitive.
3. Automatic Model Selection: The method provides a principled way to determine the number of groups and principal components without relying on external model selection criteria or pre-specification.

Results

• Simulation Studies:
  • Accuracy: In simulations involving 100 variables across 4 groups, PFPCA significantly outperformed a two-step baseline (Univariate FPCA followed by K-medoids clustering). PFPCA recovered the true partition in 27% of replicates compared to 1% for the two-step baseline.
  • Estimation: Joint inference led to more accurate recovery of individual scores, eigenfunctions, and mean functions compared to univariate FPCA, particularly in large groups where information borrowing was effective.
  • Efficiency: The variational algorithm was orders of magnitude faster than MCMC (less than 2 minutes vs. over 14 hours for a small dataset), making it feasible for large-scale applications.
• Application to Influenza Data:
  • Applied to longitudinal gene expression data from 17 individuals infected with H3N2 influenza, PFPCA identified distinct groups of genes.
  • Two groups showed significant pathway enrichment: "Thermogenesis" and "Influenza A."
  • Analysis of the "Influenza A" group revealed that the first principal component scores effectively discriminated between symptomatic and asymptomatic individuals.
  • The model identified that differences in gene expression evolution between symptomatic and asymptomatic individuals emerged only at later stages of infection, not immediately post-inoculation.

Significance and Claims
The paper claims that PFPCA offers a robust solution for extracting biological insights from high-dimensional longitudinal data by addressing the limitations of existing methods that either ignore group structure or fail to model inter-individual variability. The authors emphasize that joint inference yields more reliable parameter estimates and partitioning than sequential approaches. The application to the influenza dataset demonstrates the method's ability to identify biologically coherent gene groups and link their temporal dynamics to clinical outcomes (symptom status). The authors note that while the method supports irregular time grids, it currently requires at least one observation per individual and variable, and future work could extend the framework to handle informative dropout and survival processes.