tBayes-MICE: A Bayesian Approach to Multiple Imputation for Time Series Data
Authors: Amuche Ibenegbu, Pierre Lafaye de Micheaux, Rohitash Chandra
Organizations: Transitional Artificial Intelligence Research Group School of Mathematics and Statistics UNSW Sydney Australia · School of Mathematics and Statistics UNSW Sydney Australia
Time-series analysis is often affected by missing data, a common problem across several fields, including healthcare and environmental monitoring. Multiple Imputation by Chained Equations (MICE) has been prominent for imputing missing values through "fully conditional specification". We extend MICE using the Bayesian framework (tBayes-MICE), utilising Bayesian inference to impute missing values via Markov Chain Monte Carlo (MCMC) sampling to account for uncertainty in MICE model parameters and imputed values. We also include temporally informed initialisation and time-lagged features in the model to respect the sequential nature of time-series data. We evaluate the tBayes-MICE method using two real-world datasets (AirQuality and PhysioNet), and using both the Random Walk Metropolis (RWM) and the Metropolis-Adjusted Langevin Algorithm (MALA) samplers. Our results demonstrate that tBayes-MICE reduces imputation errors relative to the baseline methods over all variables and accounts for uncertainty in the imputation process, thereby providing a more accurate measure of imputation error. We also found that MALA mixed better than RWM across most variables, achieving comparable accuracy while providing more consistent posterior exploration. Overall, these findings suggest that the tBayes-MICE framework represents a practical and efficient approach to time-series imputation, balancing increased accuracy with meaningful quantification of uncertainty in various environmental and clinical settings.
Figures & tables
Figure 1: Time-lagged imputation mechanism in univariate MICE, where a missing value at time index t is imputed by conditioning on observed neighbouring values from past and future lags.
Xt,j=fj(Zt,−j⊤θj)+εt,j,εt,j∼N(0,τj2)
Algorithm 2 tBayes-MICE for Multivariate Time Series with MCMC sampling.
Figure 2: The proposed tBayes-MICE imputation framework. The framework consists of temporal pattern detection, placeholder initialisation, MICE loop with lagged predictors and Bayesian modelling, while MCMC (RWM or MALA) provides posterior sampling and parameter updates. The final imputation is generated via posterior predictive draws.
Method
RMSE
MAE
NMAE
NRMSE
MICE (Time-Lagged)
7.8500
5.0708
0.6005
0.9296
Linear Interpolation
8.0579
5.2240
0.6186
0.9542
KNN ( k=5 )
8.4580
5.3885
0.6381
1.0016
Mean Imputation
8.4580
5.3885
0.6381
1.0016
Median Imputation
8.5415
5.1430
0.6090
1.0115
Seasonal Decomposition
8.7597
5.2980
0.6274
1.0373
Table 1: Imputation performance comparison of univariate time-lagged MICE (lag = 1) with established time series baseline methods on a representative variable (WBC) from the PhysioNet dataset . We report RMSE, MAE, NMAE, and NRMSE, Boldface highlights the best value among the methods.
Figure 3: Convergence diagnostics for τ2 (trace, marginal density, and ACF) across two chains and two samplers for one of the variables ( HC0 3 ) from the physionet dataset.
Table 2: Convergence summaries for the bicarbonate ( HCO 3 ) variable from the PhysioNet dataset using tBayes-MICE. Results are shown for two MCMC samplers: MALA (left) and RWM (right). The table reports posterior means, standard deviations (SD), 94% highest density intervals (HDIs), effective sample sizes (ESS), and R^ statistics.
RWM
MALA
Variable
Method
NMAE
NRMSE
NMAE
NRMSE
CO(GT)
MICE
0.2622 (0.0071)
0.3376 (0.0083)
0.2622 (0.0071)
0.3376 (0.0083)
BRITS
0.2646 (0.0473)
0.3870 (0.0653)
0.2646 (0.0473)
0.3870 (0.0653)
tBayes-MICE
0.1349 (0.0013)
0.1788 ( 0.0017)
0.1346 (0.0011)
0.1788 (0.0015)
PT08.S1(CO)
MICE
0.3811 (0.0097)
0.4881 (0.0125)
0.3811 (0.0097)
0.4881 (0.0125)
BRITS
0.3418 (0.0240)
0.6153 (0.0252)
0.3418 (0.0240)
0.6153 (0.0252)
Table 3: Imputation performance of MICE, BRITS and tBayes-MICE under two MCMC samplers (RWM/MALA). Metrics are NMAE and NRMSE (mean, standard deviation) across 30 experimental runs for six AirQuality variables. Boldface highlights the best value per row.
RWM
MALA
Variable
Method
NMAE
NRMSE
NMAE
NRMSE
HCO3
MICE
1.1904 (0.0110)
1.4997 (0.0128)
1.1904 (0.0110)
1.4997 (0.0128)
BRITS
0.7535 (0.0039)
0.9941 (0.0049)
0.7535 (0.0039)
0.9941 (0.0049)
tBayes-MICE
0.7189 (0.0018)
0.9375 (0.0021)
0.7198 (0.0019)
0.9380 (0.0022)
Na
MICE
1.1467 (0.0098)
1.4549 (0.0122)
1.1467 (0.0098)
1.4549 (0.0122)
BRITS
0.7732 (0.0023)
1.5002 (0.0018)
0.7732 (0.0023)
1.5002 (0.0018)
Table 4: Imputation performance of MICE, BRITS and tBayes-MICE under two MCMC samplers (RWM/MALA). Metrics are NMAE and NRMSE (mean, standard deviation) across 30 experimental runs for four PhysioNet variables. Boldface highlights the best value per row.
Figure 4: Prediction accuracy comparison for the CO(GT) variable from the AirQuality dataset (top) and the HCO 3 variable from the physioNet dataset (bottom). Each subplot panel shows:(top-left) predicted versus true values, (top-right) residuals against the true values, (bottom-left) sorted true values with corresponding model predictions, and (bottom-right) box plots of absolute errors.
Figure 5: Imputation error patterns over time index of each method across different datasets, with the AirQuality CO(GT) variable shown in the top panel and the PhysioNet HCO 3 variable shown in the bottom panel
Figure 6: Posterior predictive performance of the proposed tBayes-MICE model over time for the environmental CO(GT) variable (top row) and the clinical Na variable (bottom row), shown across training(left) and test(right) sets. The blue lines represent observed values, the red lines indicate posterior predictive means, and the shaded regions denote the 95% credible intervals.
Figure 7: Comparative imputation accuracy and distribution across runs (box plot) for deterministic and Bayesian methods on the PhsyioNet dataset using NRMSE
Table 5: Convergence summaries for the CO(GT) variable from the AirQuality dataset using tBayes-MICE. Results are shown for two MCMC samplers: MALA (left) and RWM (right). The table reports posterior means, standard deviations (SD), 94% highest density intervals (HDIs), effective sample sizes (ESS), and R^ statistics.
AirQuality
PhysioNet
Method
Variable
NMAE
NRMSE
Variable
NMAE
NRMSE
tBayes-MICE-V1
CO(GT)
0.1346 (0.0011 )
0.1783 (0.0016)
HCO 3
0.7196 (0.0020 )
0.9381 (0.0024)
NMHC(GT)
0.2209 (0.0025)
0.3464 (0.0029)
Na
0.7734 (0.0019)
1.0263 (0.0028)
C 6 H 6 (GT)
0.1058 (0.0005)
0.1556 (0.0015)
Platelet
0.6585 (0.0021)
0.9443 (0.0021)
T
0.1345 (0.0016)
0.1859 (0.0018)
WBC
0.6200 (0.0024)
0.9014 (0.0022)
tBayes-MICE-V2
CO(GT)
0.1349 (0.0013)
0.1788 (0.0017)
HCO 3
0.7189 0.0018)
0.9375 (0.0021)
Table 6: Ablation study comparing tBayes-MICE-V1 and tBayes-MICE-V2 on the AirQuality and PhysioNet datasets under the RWM sampler. Metrics are NMAE and NRMSE (mean, standard deviation) across 30 runs. Boldface indicates the better-performing variant for each variable within each dataset.
Missing data imputation remains a fundamental challenge in modern data science, especially when uncertainty quantification is essential. In this work, we propose MissBGM, an AI-powered missing data imputation method via Bayesian generative modeling that bridges the expressive flexibility of neural networks with the statistical rigor of Bayesian inference. Unlike existing methods that often focus on point estimates or treat the missingness mechanism implicitly, MissBGM explicitly and jointly models the data-generating and missingness mechanisms, providing principled posterior uncertainty over imputations rather than a single point estimate. We develop a stochastic optimization framework with alternating updates among missing values, model parameters, and latent variables until convergence. Our theoretical analysis shows that estimates of missing values from MissBGM converge consistently under mild assumptions. Empirically, we demonstrate that MissBGM achieves superior performance over traditional imputers and recent neural network-based methods across extensive experimental settings. These results establish MissBGM as a principled and scalable solution for modern missing data imputation. The code for MissBGM is open sourced at https://github.com/liuq-lab/MissBGM.
Multivariate time series imputation is fundamental to downstream analysis, yet modeling inter-variable dependencies with incomplete observations remains challenging. Existing methods learn global dependencies across samples or dynamic local dependencies per sample. Global dependencies are stable but adapt poorly to sample variations and temporal non-stationarity, whereas local dependencies are adaptive yet unreliable when observations are insufficient, causing erroneous information propagation. To address these limitations, we propose GLAIM, a Global-Local Adaptive Inter-variable Dependency Modeling framework for multivariate time series imputation. GLAIM comprises two complementary components. The Stable Global Dependency Constructor derives robust global inter-variable dependencies from complementary temporal representations, providing a stable backbone less affected by sample-specific missingness and noise. The Sample-Conditioned Dependency Refiner adapts this backbone to each sample and time step using its temporal state and available observations, enabling reliable local refinement under incomplete observations. Extensive experiments on nine real-world datasets demonstrate that GLAIM achieves state-of-the-art performance under random and block missingness, remains robust to missing-rate shifts, and benefits from its complementary global and local components. Code is available at https://github.com/LuRenjias/GLAIM.
Mingyang Wang, Rongwen Li, Xiao Wang +1
College of Computer Science and Electronic Engineering, Hunan University, Changsha, Hunan, China
Imputing missing values in multivariate time series remains challenging, especially under diverse missing patterns and heavy missingness. Existing methods suffer from suboptimal performance as corrupted temporal features hinder effective cross-variable information transfer, amplifying reconstruction errors. Robust imputation requires both extracting temporal patterns from sparse observations within each variable and selectively transferring information across variables--yet current approaches excel at one while compromising the other. We introduce T1 (Time series imputation with 1-to-1 channel-head binding), a CNN-Transformer hybrid architecture that achieves robust imputation through Channel-Head Binding--a mechanism creating one-to-one correspondence between CNN channels and attention heads. This design enables selective information transfer: when missingness corrupts certain temporal patterns, their corresponding attention pathways adaptively down-weight based on remaining observable patterns while preserving reliable cross-variable connections through unaffected channels. Experiments on 11 benchmark datasets demonstrate that T1 achieves state-of-the-art performance, reducing MSE by 46% on average compared to the second-best baseline, with particularly strong gains under extreme sparsity (70% missing ratio). The model generalizes to unseen missing patterns without retraining and uses a consistent hyperparameter configuration across all datasets. The code is available at https://github.com/Oppenheimerdinger/T1.
Dongik Park, Hyunwoo Ryu, Suahn Bae +2
Graduate School of Data Science, Seoul National University