What is Fixed-Lag Smoother?
A Fixed-Lag Smoother is a state estimator that delays each output by a chosen lag so measurements from a bounded future window can revise recent state estimates.
Quick Facts
| Specification | Official Specification |
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How It Works
Use a bounded future window
At time k, a lag-L estimator can update states in the window x_(k-L:k) using measurements through k, then emit or freeze the oldest state. In a linear Gaussian model, one construction augments the Kalman state with delayed copies and their cross-covariances; another runs local backward smoothing over stored filter quantities.
Caltech's fixed-lag notes derive the augmented-state formulation and show why cross-covariances are required. Keeping only independent per-time variances loses the information needed to propagate a new measurement backward.
Marginalize old states without discarding evidence
Factor-graph implementations keep variables and factors inside a moving time window. When a state becomes older than the lag, marginalization summarizes its influence as a prior over retained variables. Deleting the variable and its factors without this summary changes the posterior and can make the estimator overconfident or inconsistent.
Nonlinear marginalization also fixes a linearization point for eliminated variables. Repeated relinearization inside the active window cannot recover information discarded or approximated at the boundary, so lag length, ordering, sparsity, and marginalization policy need explicit tests.
Budget delay, computation, and evaluation leakage
The GTSAM tutorial distinguishes filtering, full smoothing, and fixed-lag estimation by retaining only a subset of recent poses and marginalizing older ones. Bounded state count does not imply constant latency under every graph topology or solver; measure update time and memory on the real factor pattern.
Report lag in both steps and wall-clock time, output timestamp, late-measurement policy, marginalization cost, trajectory error, consistency, and tail latency. Evaluate each estimate only against information permitted by its declared delay. Comparing a lagged output with a zero-delay filter without accounting for future evidence is data leakage.
Key Characteristics
- Uses observations from a bounded future window to revise recent states
- Emits estimates with an explicit delay measured in steps or time
- Bounds active history through windowing and marginalization
- Can be implemented with augmented Kalman states or factor graphs
- Occupies the latency-accuracy space between filtering and full smoothing
- Requires timestamp, late-data, marginalization, and leakage policies
Common Use Cases
- GNSS and IMU fusion with a small permitted output delay
- Robot localization with delayed or out-of-sequence measurements
- Online trajectory refinement in a sliding factor graph
- Signal processing where recent estimates may be revised before release
- Evaluation of accuracy versus latency across smoothing-window lengths
Example
Loading code...Frequently Asked Questions
How is a Fixed-Lag Smoother different from a Kalman Filter?
A filter estimates the current state using observations available through the current time. A Fixed-Lag Smoother waits for a bounded number of future observations, revises recent states, and emits an older timestamp with an explicit delay.
How is fixed-lag smoothing different from RTS smoothing?
An RTS fixed-interval smoother normally processes a completed sequence with a full backward pass. A Fixed-Lag Smoother only revises a moving recent window, bounding history and delay while giving up information beyond the selected lag.
How should the lag be chosen?
Measure error, consistency, compute time, memory, and deadline misses across candidate lags using realistic sensor delays and dynamics. Choose the smallest lag that captures useful delayed evidence while satisfying the product's timestamp and latency contract.
What happens to states outside the active window?
A correct implementation freezes emitted estimates and summarizes eliminated variables through marginalization or an equivalent sufficient statistic. Simply deleting old variables and factors discards evidence and changes the inferred distribution.
Can fixed-lag results be reported as real-time filter accuracy?
Only if the declared output delay is included. The estimate for time `k-L` uses observations through `k`, so it has access to future data relative to its timestamp. Comparing it with zero-delay outputs without accounting for that advantage is leakage.