Estimate a changing state by balancing a model with noisy measurements
A state-space model separates system dynamics from observation error; the Kalman filter predicts, measures the innovation, updates the state and carries quantified uncertainty forward.
A sensor rarely measures the quantity we care about without noise, delay or partial coverage. At the same time, a physical or behavioural model is never exact. State estimation combines both: the transition model predicts how a hidden state changes, and the observation model explains how measurements arise from it.
For a linear system with Gaussian errors and known parameters, the Kalman filter gives recursive filtered means and covariances. Its innovation residuals also provide a disciplined way to diagnose drift, bias, outliers and model misspecification.
Chapter contents
- 1. Define the latent state and measurement
- 2. Write the transition equation and process noise
- 3. Write the observation equation and measurement noise
- 4. Predict the state mean and covariance
- 5. Compute the innovation and its covariance
- 6. Use Kalman gain to update state and uncertainty
- 7. Worked example: estimate position and velocity
- 8. Handle missing, delayed and irregular observations
- 9. Smooth past states with future observations
- 10. Estimate parameters without hiding identifiability problems
- 11. Diagnose the model with innovation residuals
- 12. Know when linear Gaussian filtering is not enough
- 13. Transfer the estimator with observability and consequence checks
Choose a reading route
- Route 1: Specify the system — state, transition and observation equations
- Route 2: Run the filter — predict, innovate and update
- Route 3: Work the example — position and velocity estimation
- Route 4: Look backward — smoothing and parameter learning
- Route 5: Diagnose transfer — residuals, nonlinearity and change
How to use this guide
Start with the chapters that match your immediate decision, then return to the worked example and audit chapters. Each chapter separates a model from the evidence needed to use it responsibly. A claim is not strengthened merely by being technical; it is strengthened when assumptions, observations, alternatives and limits can be inspected.
| Layer | Question | Minimum record |
|---|---|---|
| Concept | What exactly is being represented? | Definitions and boundary |
| Method | How will the reasoning be carried out? | Steps, inputs and assumptions |
| Transfer | Would it work in a changed setting? | A deliberately altered case |
| Audit | What could prove it wrong? | Evidence, limits and revision trigger |
Chapter 1. Define the latent state and measurement
The idea
The state contains the smallest variables needed to propagate the system; the measurement is what an instrument observes. They can have different dimensions and units.
A disciplined method
Name each state component, time step and unit, then write how observations map to state components and which quantities remain unobserved.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whether the chosen state is sufficient for future prediction. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Position alone is not sufficient under constant velocity; position and velocity form a more useful state while GPS measures position.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Change the sampling interval and decide whether acceleration or time-varying inputs must enter the state.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: A useful state is model-dependent and may not correspond to one directly measurable physical object.
Evidence to retain: variable definitions, units, measurement protocol and sufficiency tests
Common misconception: The state should contain every available variable.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 2. Write the transition equation and process noise
The idea
The transition matrix propagates the previous state; control inputs add known influence; process noise represents unmodelled dynamics and uncertainty in evolution.
A disciplined method
Derive the matrix for the chosen time step, define control inputs, choose process-noise covariance and test implied trajectories.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which departures from the transition model are treated as process noise. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A constant-velocity model advances position by velocity times dt; acceleration uncertainty enters through process covariance.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Double dt and rebuild both transition and process covariance rather than reusing values.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Process noise is a modelling choice, not a bucket that automatically fixes wrong dynamics.
Evidence to retain: equations, units, covariance rationale, simulations and sensitivity results
Common misconception: Process-noise covariance is simply the sensor error.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 3. Write the observation equation and measurement noise
The idea
The observation matrix maps state variables into sensor space. Measurement covariance describes conditional sensor error, including correlations among channels.
A disciplined method
Calibrate sensors, define the matrix and covariance on the measurement scale, inspect bias and missingness, and record time alignment.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which state combinations each sensor actually measures. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A position sensor observes the first state component; a velocity sensor observes the second with different variance.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Add a correlated sensor derived from the first and avoid double-counting information.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Gaussian zero-mean error may fail under saturation, censoring, bias or heavy tails.
Evidence to retain: calibration data, bias checks, timestamps, covariance estimates and missing-data rules
Common misconception: More sensors always reduce uncertainty regardless of dependence.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 4. Predict the state mean and covariance
The idea
The prediction step advances the filtered state and expands uncertainty through transition dynamics and process noise before seeing the new measurement.
A disciplined method
Compute x minus = F x plus B u and P minus = F P plus F transpose plus Q; verify symmetry and units.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is how uncertainty transforms through the dynamics. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Uncertain velocity increases future position variance even when current position is well known.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Set process noise to zero and examine unrealistic overconfidence after repeated predictions.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: A precise model can still be wrong; covariance represents assumed uncertainty, not guaranteed error bounds.
Evidence to retain: predicted states, covariance matrices, eigenvalue checks and simulation coverage
Common misconception: Prediction covariance should shrink because the model supplies information.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 5. Compute the innovation and its covariance
The idea
The innovation is measurement minus predicted measurement. Its covariance combines projected state uncertainty and measurement noise. It is the filter’s surprise signal.
A disciplined method
Compute v = y minus Hx and S = HPH transpose plus R; standardise residuals and retain them in time order.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whether the new measurement is plausible under the predictive distribution. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A GPS reading is three metres above prediction; its importance depends on S, not metres alone.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Increase measurement variance during poor satellite geometry and observe the same residual become less surprising.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Innovation tests depend on correct covariance and can miss structured bias when the model adapts.
Evidence to retain: raw measurement, prediction, innovation, S, standardised residual and quality flag
Common misconception: A large raw residual always means the sensor failed.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 6. Use Kalman gain to update state and uncertainty
The idea
The Kalman gain balances model prediction with the new measurement according to their uncertainties. The update moves the state and reduces covariance in observed directions.
A disciplined method
Compute K = P H transpose S inverse, update x plus = x minus + K v, and use a numerically stable covariance update such as Joseph form when appropriate.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is how strongly the measurement should influence each state component. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A precise position reading creates a high gain for position and also adjusts velocity when covariance links them.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Change R from small to large and explain why the filter follows the model more closely.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: A high gain is not inherently good; it reflects relative uncertainty and model structure.
Evidence to retain: gain, updated state, Joseph covariance, symmetry and positive-semidefinite checks
Common misconception: The Kalman gain is a manually chosen smoothing constant.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 7. Worked example: estimate position and velocity
The idea
A simple constant-velocity tracker makes every matrix step visible and shows how repeated position measurements can improve an unobserved velocity estimate.
A disciplined method
Start with state [0,1], covariance, dt=1, one position observation per step, and stated Q and R. Calculate prediction, innovation, gain and update for three readings.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is how cross-covariance transfers position evidence to velocity. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Measurements 1.2, 2.0 and 3.1 pull the position estimate while velocity stabilises near one unit per step.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Change the middle reading to 5.0 and compare an ordinary update with an outlier-aware rule.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: The values are illustrative and not a calibrated navigation system.
Evidence to retain: complete matrices, hand arithmetic, code output, units and random seed for simulations
Common misconception: Because velocity is not measured, the filter cannot estimate it.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
| Stage | Mean | Covariance role |
|---|---|---|
| Predict | F x | Carries prior uncertainty through F and Q |
| Innovate | y-Hx | Measures new surprise |
| Gain | P Hᵀ S⁻¹ | Balances prediction and measurement |
| Update | x+Kv | Corrects state and contracts uncertainty |
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 8. Handle missing, delayed and irregular observations
The idea
State propagation can continue without an update, but uncertainty grows. Delayed or out-of-sequence measurements require explicit time handling rather than insertion at the wrong step.
A disciplined method
Use actual dt, skip unavailable measurement updates, record data latency, and apply an out-of-sequence method or rerun when required.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which time the measurement describes versus when it arrived. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A sensor misses two intervals; the filter predicts twice and reports wider covariance before the next update.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Deliver an old measurement after two newer ones and compare discard, fixed-lag rerun and specialised update choices.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Long gaps magnify model error, and irregular sampling can invalidate fixed matrices.
Evidence to retain: event time, arrival time, missingness cause, dt-specific matrices and rerun log
Common misconception: Carrying the previous estimate forward is equivalent to a proper prediction.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 9. Smooth past states with future observations
The idea
Filtering conditions on observations up to time t; smoothing also uses later data. Rauch–Tung–Striebel smoothing revises past estimates through stored filtered and predicted quantities.
A disciplined method
Run the forward filter, store means and covariances, then recurse backward with the smoother gain; keep filtered and smoothed outputs distinct.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is whether the task is live estimation or retrospective reconstruction. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A noisy early position estimate becomes more precise after later measurements reveal a consistent trajectory.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Transfer to fraud monitoring where using future data in a live performance test would create leakage.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Smoothing improves retrospective estimates under the same model but cannot repair biased dynamics.
Evidence to retain: stored forward pass, smoother gain, backward results, timestamps and evaluation protocol
Common misconception: A smoothed value was available to the real-time decision maker.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 10. Estimate parameters without hiding identifiability problems
The idea
Unknown Q, R or dynamic parameters can be estimated with likelihood or expectation–maximisation, but different combinations may explain the same data.
A disciplined method
Use calibrated information where possible, constrain parameters, try multiple starts, validate out of sample and inspect whether Q and R trade off unrealistically.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which parameters the observations can identify. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A model with larger process noise and smaller measurement noise fits similarly to the reverse until calibration data constrain R.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Remove direct velocity observations and examine which dynamic parameters become weakly identified.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Maximum likelihood can converge locally and produce numerically plausible but physically absurd covariances.
Evidence to retain: initial values, constraints, likelihood trace, calibration data and profile or sensitivity plots
Common misconception: EM discovers the true Q and R automatically.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 11. Diagnose the model with innovation residuals
The idea
Under a well-specified linear Gaussian model, standardised innovations should be roughly zero-mean, appropriately scaled and temporally uncorrelated. Patterns identify specific failures.
A disciplined method
Plot residuals and squared values, test autocorrelation, compare empirical coverage, stratify by operating condition and inspect runs or bias.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which residual pattern points to dynamics, sensor bias or variance error. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Positive residual runs suggest underpredicted motion; variance spikes during rain suggest condition-dependent R.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Move the filter to a faster regime and see autocorrelation emerge before state error becomes obvious.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Formal tests can overreact in large samples or miss rare safety-critical failures; plots and domain review remain essential.
Evidence to retain: innovation sequence, S, whiteness tests, coverage, condition labels and incidents
Common misconception: Small average residual proves the filter is correct.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 12. Know when linear Gaussian filtering is not enough
The idea
Nonlinear dynamics, non-Gaussian noise, constraints, mode switching or multimodal uncertainty can make the ordinary Kalman filter misleading.
A disciplined method
Inspect equations and residuals, compare extended, unscented, ensemble or particle methods, and retain the simplest method that passes validation.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which assumption fails and what replacement addresses it. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
A bearing-only tracker has nonlinear observation geometry; linearisation may fail when uncertainty is large.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Change the system to two operating modes and compare one broad Gaussian with a switching model.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: More sophisticated filters add tuning, computation and new failure modes; none compensates for unobservable states.
Evidence to retain: nonlinearity analysis, observability checks, benchmark results, computation and failure tests
Common misconception: The extended Kalman filter makes any nonlinear problem safe.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Chapter 13. Transfer the estimator with observability and consequence checks
The idea
A filter that works in one site or sensor package may fail when dynamics, calibration, timing or decision costs change. Transfer requires structural and outcome validation.
A disciplined method
Recheck observability, units, Q and R, residuals, coverage and downstream error costs; set drift and recalibration triggers.
A useful working record makes the sequence visible: first state the question and unit of analysis; next record the observations and their provenance; then apply the stated rule; finally compare the result with plausible alternatives. For this chapter, the decisive choice is which states remain observable in the new configuration. Keeping that choice explicit prevents a later reader from mistaking a convenient convention for a fact about the world.
Worked example
Removing the velocity sensor still permits estimation through repeated positions, but long gaps make velocity uncertainty unacceptable for control.
Work the example twice. On the first pass, follow the stated assumptions. On the second, reverse one assumption, remove one piece of evidence and ask whether the conclusion, ranking or boundary changes. This small stress test matters because polished outputs can hide sensitivity. The objective is not to force certainty; it is to locate the evidence on which the conclusion actually turns.
Changed-context transfer check
Change the use from display to automatic control and tighten latency, coverage and fail-safe requirements.
A transfer succeeds only when the structure of the problem survives the change. Match roles, dependencies and measurement conditions rather than surface vocabulary. Record what was preserved, what changed, and why the method still applies—or why it does not. If the new context changes the data-generating process or the people bearing the burden, re-specify the model instead of importing the old answer.
Limits, evidence and revision
Limit: Good average accuracy does not ensure safe tail performance or fair human impact.
Evidence to retain: observability matrix, external validation, residual dashboard, tail errors, decision costs and override logs
Common misconception: Matching variable names means a saved filter can be reused unchanged.
A gold-standard conclusion is therefore conditional. It should name the observation period, excluded cases, uncertainty and the event that would trigger review. Negative findings and unresolved contradictions belong in the record. They protect future readers from repeating a hidden assumption and make correction possible without reconstructing the entire analysis from memory.
Practical evidence check
Can another reader recover the input, reproduce the intermediate decision and identify at least one observation that would change the result? If not, the chapter has produced an explanation but not yet an auditable finding.
Field workbook
Implement the two-state position–velocity example in a spreadsheet or script. Verify one prediction and update by hand, then plot standardised innovations. Change Q and R separately and explain the different effects on gain, lag and residuals.
- Write the decision question in one sentence and name the decision owner.
- List observations separately from interpretations, with dates and provenance.
- Record at least two plausible alternatives before choosing a preferred account.
- Run the changed-context test and note which assumptions fail.
- State the limit, confidence and next review trigger in plain language.
| Check | Evidence | Failure signal |
|---|---|---|
| Structure | State and observation equations | Unit or dimension mismatch |
| Numerics | Stable covariance update | Negative variance or asymmetry |
| Fit | White calibrated innovations | Bias or autocorrelation |
| Transfer | New-context validation | Coverage or cost drift |
Continue through the verified ecosystem
- Hidden Markov models — discrete latent states and sequence inference
- Bayesian networks — conditional independence and diagnostic inference
- Partially observable MDPs — belief states connected to actions
- Factor graphs and message passing — graphical-model computation
- Conditional random fields — discriminative sequence labelling
Frequently asked questions
What is a state-space model?
It specifies how a latent state evolves and how observations are generated from that state.
What does the Kalman gain do?
It weights the innovation according to predicted-state and measurement uncertainty, updating all correlated state components.
What is an innovation?
The new measurement minus its predicted value, with a covariance that states how surprising it should be.
What is the difference between filtering and smoothing?
Filtering uses data available up to the current time; smoothing also uses later observations to revise past states.
Why inspect innovation residuals?
Bias, autocorrelation or wrong variance can reveal misspecified dynamics, sensor problems or unmodelled regimes.
When should I not use an ordinary Kalman filter?
When linearity, Gaussian noise, observability or a single-mode state distribution is seriously violated.
Authoritative sources and further reading
- Kalman (1960), A New Approach to Linear Filtering and Prediction Problems — original paper
- NASA: Discovery of the Kalman Filter as a Practical Tool — state-estimation history and practice
- NASA Technical Reports Server record — bibliographic record
- Kalman and Bucy (1961), New Results in Linear Filtering — continuous-time theory
- UC Berkeley state-space model notes — current academic worked equations
This educational guide explains statistical estimation. Safety-critical deployment requires validated dynamics, calibrated sensors, numerical assurance, domain engineering and fail-safe review.