BAMIFun: Bayesian Multiple Imputation for Functional Data

Technical Summary of BAMIFun: Bayesian Multiple Imputation for Functional Data

Problem Statement
Modern functional datasets are frequently characterized by sparse and irregular observations, where trajectories are not fully observed across a continuum (e.g., time or space). While Functional Principal Component Analysis (FPCA) is a standard tool for reconstructing incomplete curves, existing approaches—specifically the "PACE" (Principal Components Analysis through Conditional Expectation) framework—typically rely on single imputation. This treats reconstructed trajectories as known truths, failing to quantify the uncertainty inherent in the imputation process. Consequently, downstream statistical inferences (e.g., regression coefficients) often suffer from overly optimistic confidence intervals and poor coverage rates, particularly under severe missingness. Furthermore, existing multiple imputation methods for functional data often require auxiliary scalar outcomes or covariates, and none currently address the specific challenges of multiway functional data (tensor data where one mode is a continuum) without imposing smoothness constraints.

Methodology
The authors propose BAMIFun, a Bayesian multiple imputation framework designed for both single-level and multiway functional data. The methodology is built on three core components:

1. Modeling Framework:

   • Single-Level Data: The method employs a Bayesian low-rank model where the functional data matrix $X$ is approximated as $X \approx VU^T + E$. Crucially, the eigenfunctions $U$ are represented via penalized spline bases ($U^T = B\Theta$). A penalty term on the second derivative of the eigenfunctions is incorporated into the log-likelihood, which is equivalent to placing a multivariate normal prior on the spline coefficients $B$. This enforces trajectory smoothness while jointly quantifying uncertainty in the eigenfunctions and scores.
   • Multiway Data: The framework extends to tensor data ($N \times J \times K$) using a Functional Tensor Singular Value Decomposition (FTSVD) model. This model decomposes the tensor into a sum of rank-one components involving subject vectors, feature/visit vectors, and smooth functional curves. Unlike the multilevel FPCA (MFPCA) which uses additive mixed-effects, FTSVD utilizes a multiplicative CP-type factorization suitable for coherent tensor structures.

2. Inference Algorithm:

   • The authors derive an efficient Gibbs sampler for posterior computation. The algorithm iteratively samples the principal component scores ($V$), spline coefficients ($B$), error variance ($\sigma^2$), and smoothing parameters ($\sigma^2_B$) from their full conditional distributions.
   • Missing values are imputed by drawing from the posterior predictive distribution (Gaussian with mean equal to the reconstructed trajectory and variance equal to the error variance) at each iteration.
   • A cross-validation procedure is proposed to select the number of principal components ($R$) and the smoothing parameter, though the authors note that estimating the smoothing parameter within the Bayesian framework yields comparable performance with lower computational cost.

3. Downstream Inference:

   • The framework generates $S$ completed datasets. For downstream analyses (e.g., Scalar-on-Function Regression), point estimates are pooled, and variance is calculated using Rubin's rules, which combine within-imputation and between-imputation variances to produce valid confidence intervals that account for imputation uncertainty.

Key Contributions

• Novel Framework: BAMIFun is presented as the first multiple imputation procedure specifically designed for multiway functional data that incorporates smoothness constraints via penalized splines.
• Uncertainty Quantification: Unlike single-imputation FPCA methods, BAMIFun explicitly models imputation uncertainty, leading to valid statistical inference in downstream analyses.
• Algorithmic Efficiency: The development of a Gibbs sampler that integrates penalized splines within a low-rank tensor decomposition allows for the handling of sparse and irregular grids without requiring auxiliary scalar data.
• Software Availability: The authors provide an open-source implementation at https://github.com/ZirenJiang/BAMIFun.

Results
The paper evaluates BAMIFun through extensive simulation studies and two real-world case studies:

• Single-Level Simulations: Compared to the PACE approach and the tensor-based BAMITA algorithm (which lacks smoothness constraints), BAMIFun achieved imputation accuracy comparable to PACE. However, BAMIFun demonstrated substantially improved coverage rates (close to the nominal 95% level) for confidence intervals, whereas PACE exhibited significant under-coverage. BAMIFun also outperformed BAMITA in accuracy due to the enforcement of smoothness.
• Multiway Simulations: In settings with low-rank structures, BAMIFun outperformed the frequentist MFPCA (PACE) approach in both accuracy and coverage. In non-low-rank settings, BAMIFun showed slightly higher Mean Squared Error (MSE) than PACE but maintained robust, near-nominal coverage rates, highlighting its reliability across diverse data structures.
• Case Studies:
  • NHANES (Physical Activity): Under severe missingness (up to 97.5%), BAMIFun maintained coverage near 92–94%, while the FPCA-based PACE method dropped to as low as 53–71%.
  • Infant Gut Microbiome: In a dataset with 91.1% missingness, BAMIFun consistently outperformed BAMITA in both MSE and coverage, successfully handling the complex multiway structure where standard MFPCA methods were inapplicable.

Significance and Claims
The authors claim that BAMIFun addresses critical methodological gaps in the analysis of sparse functional data. By moving beyond single imputation, the method prevents the "overly optimistic" inferences that plague current FPCA-based approaches. The paper emphasizes that while the imputation accuracy (MSE) is often comparable to existing frequentist methods, the primary advantage lies in the validity of downstream inference. The ability to handle multiway functional data without auxiliary variables and to provide proper uncertainty quantification makes BAMIFun a robust tool for modern applications involving wearable devices, genomics, and neuroimaging where data are inherently sparse and high-dimensional. The authors modestly note that performance depends on the choice of spline basis and suggest future work could explore alternative parameterizations, such as those on the Stiefel manifold.