What is Rauch-Tung-Striebel Smoother?
A Rauch-Tung-Striebel Smoother is a fixed-interval Bayesian algorithm that runs a Kalman Filter forward and then recursively uses future observations to refine every past state estimate.
Quick Facts
| Specification | Official Specification |
|---|
How It Works
Store the complete forward Kalman pass
Run a Kalman Filter from the first through final observation and retain m_k|k, P_k|k, plus the next-step predictions m_(k+1)|k and P_(k+1)|k. Keeping only the final filtered state is insufficient because every backward correction needs a local filtered covariance and its paired prediction.
Rauch, Tung, and Striebel's 1965 paper derives filtering and smoothing equations for linear dynamic systems with additive Gaussian noise. The fixed interval and model parameters must be declared before interpreting the recursion as a particular posterior.
Propagate future information backward
Initialize the final smoothed state with the final filtered state. For k = T-1 ... 0, compute smoother gain G_k = P_k|k F_k^T (P_(k+1)|k)^-1, then correct the mean with G_k(m_(k+1)|T - m_(k+1)|k) and the covariance with G_k(P_(k+1)|T - P_(k+1)|k)G_k^T.
This is not simply a Kalman Filter run on reversed observations: the backward gain depends on the forward conditional covariance and transition model. Use linear solves instead of explicit inverses and preserve the exact indexing relationship among filtered, predicted, and smoothed arrays.
Prevent future leakage in evaluation
Berkeley's state-space notes distinguish filtering, forecasting, and smoothing by the observation horizon. RTS can improve reconstructed historical trajectories, missing-value estimates, and sufficient statistics for parameter learning, but it cannot produce the same result in a real-time system before future measurements arrive.
Report interval boundaries, boundary initialization, stored precision, filtered versus smoothed RMSE on simulated or delayed truth, covariance coverage, missing-data policy, and latency. For deployment, compare fixed-interval smoothing with a declared fixed-lag window rather than leaking the entire test future into online metrics.
Key Characteristics
- Performs a forward Kalman pass followed by a backward recursion
- Conditions each state on all observations in a fixed interval
- Uses stored filtered and one-step predicted moments
- Is exact for a specified linear Gaussian state-space model
- Typically reduces retrospective covariance relative to filtering
- Requires future data and additional trajectory storage
Common Use Cases
- Offline trajectory reconstruction from noisy measurements
- Retrospective sensor-data denoising and gap filling
- State estimation for completed scientific experiments
- Expectation-Maximization sufficient-statistic computation
- Reference smoothing for Gaussian Process state-space models
Example
Loading code...Frequently Asked Questions
What is the difference between Kalman filtering and RTS smoothing?
Kalman filtering estimates state `k` from observations through `k`. RTS smoothing estimates the same state after observations through the final interval time are available, using a backward correction built from the forward filter outputs.
Is an RTS Smoother just a Kalman Filter run backward?
No. Its smoother gain uses the forward filtered covariance, transition matrix, and next predicted covariance. Reversing observations without deriving the corresponding reverse-time model does not reproduce the RTS conditional distribution.
What must be stored for RTS smoothing?
At minimum, retain each filtered mean and covariance and each one-step predicted mean and covariance required by the next backward gain. Time-varying transition matrices and any factorization needed for stable solves must also remain aligned by index.
Can RTS smoothing be used in a real-time benchmark?
Only if the benchmark explicitly permits future observations or a fixed delay. Full fixed-interval RTS uses the entire sequence, so comparing it with an online filter without disclosing that information advantage is future leakage.
Does RTS smoothing always reduce actual state error?
Its posterior covariance does not increase under the correctly specified linear Gaussian model, but realized error on one trajectory can still vary. Model misspecification, numerical errors, and wrong interval boundaries can also produce misleading confidence.