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BayOTIDE: Bayesian Online Multivariate Time Series Imputation with Functional Decomposition
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Proposes BayOTIDE, a Bayesian online probabilistic method for multivariate time series imputation that models series as weighted low-rank GP-based functional factors converted to state-space form and updated via moment matching.
Problem
Multivariate time series imputation with irregular timestamps, global patterns beyond local fixed-length windows, uncertainty quantification, and streaming/online updates, where existing methods are offline and discrete-time.
Approach
Decompose the series into trend and seasonal low-rank temporal factors with Matérn and periodic GP priors, convert the GPs to LTI-SDE state-space priors, and perform closed-form online inference via conditional expectation propagation with Kalman filtering and RTS smoothing, enabling continuous-time imputation.
Key results
- On Traffic-GuangZhou at 50% observed, BayOTIDE reaches RMSE 3.820 and CRPS 0.055 versus 4.708 RMSE for TIDER and 4.790 RMSE for CSBI.
- On Solar-Power at 50% observed, TIDER attains the best RMSE (1.679) while BayOTIDE gets 1.699 RMSE and the best CRPS of 0.122 among probabilistic methods.
- BayOTIDE-fix weight performs very poorly (e.g., RMSE 11.032 on Traffic-GuangZhou at 50%), indicating the learned weights are crucial.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add learned channel-wise weights instead of fixing all weights to one (ablation) | BayOTIDE · Traffic-GuangZhou, 50% observed | RMSE: 11.032 -> 3.820 | Helped |
| add learned channel-wise weights instead of fixing all weights to one (ablation) | BayOTIDE · Solar-Power, 50% observed | RMSE: 5.245 -> 1.699 | Helped |
| add learned channel-wise weights instead of fixing all weights to one (ablation) | BayOTIDE · Uber-Move, 50% observed | RMSE: 5.950 -> 1.901 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · Traffic-GuangZhou, 50% observed | RMSE: 4.188 -> 3.820 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · Solar-Power, 50% observed | RMSE: 1.789 -> 1.699 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · Uber-Move, 50% observed | RMSE: 2.052 -> 1.901 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · Traffic-GuangZhou, 70% observed | RMSE: 4.002 -> 3.724 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · Solar-Power, 70% observed | RMSE: 1.651 -> 1.621 | Helped |
| add periodic factors to trend factors (ablation) | BayOTIDE · UberLondon, 70% observed | RMSE: 2.015 -> 1.832 | Helped |
| increase trend-factor dimension from 5 to 40 with seasonality dimension 5 (ablation) | BayOTIDE · Traffic-GuangZhou, 70% observed | Test CRPC: 0.067 -> 0.055 | Helped |
| increase seasonality-factor dimension from 5 to 20 with trend-factor dimension 5 (ablation) | BayOTIDE · Traffic-GuangZhou, 70% observed | Test CRPC: 0.067 -> 0.081 | Hurt |
| use BayOTIDE instead of CSBI | BayOTIDE · Traffic-GuangZhou, 50% observed | RMSE: 4.790 -> 3.820 | Helped |
Surprising findings
- On Solar-Power, the deterministic TIDER baseline gives lower RMSE than BayOTIDE, so the online method does not win on every metric.
- Fixing the factor weights to one collapses performance, showing the trivial GP-SSM usage is insufficient.
Limitations
- Hyperparameters (factor counts, kernel lengthscales, periodicity) were fine-tuned per dataset, and the sensitivity to kernel hyperparameters is notable for Matérn nu=3/2.
- Deterministic accuracy lags TIDER on Solar-Power RMSE.
- Evaluated only on imputation with stationary kernels; long-term non-stationary forecasting left for future work.
Experimental setup
Datasets: Traffic-GuangZhou Solar-Power Uber-Move UberLondon synthetic four-channel series
Benchmarks: Traffic-GuangZhou Solar-Power Uber-Move
Metrics: RMSE MAE CRPS NLLK
Baselines: SimpleMean BRITS NAOMI SAITS TIDER Multi-Task GP GP-VAE CSDI CSBI BayOTIDE-fix weight BayOTIDE-trend only
Builds on: TIDER GP-SSM (state-space GPs) conditional expectation propagation CSDI
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