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eduKateSG · Why Mathematics? · Cubature Kalman Filtering for Nonlinear Reliability Sensors: Spherical-Radial Points, Moment Integrals, Prediction and Update
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 5–8
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 9–12
Check assumptions
See where design and calibration matter.Route 4 · Chapters 13–16
Connect to condition monitoring
Connect the method to a condition-monitoring investigation.Route 5 · Chapters 17–20
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes visible when a nonlinear filter must approximate Gaussian-weighted moment integrals in several dimensions. The cubature Kalman filter, or CKF, uses a third-degree spherical-radial cubature rule with 2n equally weighted points in n state dimensions. It maps those points through nonlinear state and sensor functions, then reconstructs approximate means, covariances and cross covariance for a Kalman-style update.
In one dimension, the standardised cubature points are +1 and −1 with weights one half. Let a state have mean 2 and variance 1, so the points become 3 and 1. Passing them through h(x)=x² gives 9 and 1. The approximated transformed mean is (9+1)/2=5, and the approximated variance is [(9−5)²+(1−5)²]/2=16. The exact Gaussian moments for X~N(2,1) give E[X²]=5 and Var(X²)=18, so this low-order rule gets the mean exactly here but not the fourth-moment-dependent variance.
Full chapter index
Chapters 1–4 · Understand the question
Chapters 5–8 · Reproduce the example
Chapters 9–12 · Check assumptions
Chapters 13–16 · Connect to condition monitoring
Chapters 17–20 · Study and verify
Start with the Gaussian integral the filter must approximate. Nonlinear Bayesian prediction requires expectations of functions under a probability distribution. A squared sensor needs E of X squared and related cross moments. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Evaluating only at the mean misses curvature. Identify the required moments.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use why moment integrals appear as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Did You Know? The one-dimensional squared example shows a useful subtlety: a third-degree rule can reproduce the transformed mean while missing variance because variance depends on a fourth moment. The method becomes easier to trust when this hidden structure is made visible.
Each cubature point is one symmetric probe of the nonlinear function. The CKF derives symmetric points from a cubature rule for Gaussian-weighted integrals. Points lie on coordinate axes at a dimension-dependent radius. The calculation matters only when its probability statement, units and observation boundary remain attached to it. The rule has a stated polynomial degree. Do not call it exact for arbitrary functions.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use spherical-radial rule as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
The update depends on moments reconstructed after nonlinear transformation. An n-dimensional third-degree CKF uses 2n points with equal weights. A five-state model uses ten points. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Linear growth in points does not remove matrix costs. Report state dimension and factorisation.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use number of cubature points as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
A deterministic point rule still makes an approximation. For the teaching rule, points plus one and minus one have weights one half. Symmetry makes their mean zero and variance one. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Different scaling conventions belong to different quadrature rules. Keep the derivation consistent.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use standardised one-dimensional points as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Primary authoritative source: https://doi.org/10.1109/TAC.2009.2019800. Check its state definition, approximation and numerical conditions before transferring the method to reliability work.
Start with the Gaussian integral the filter must approximate. A covariance square root scales and rotates standard points around the mean. Mean two and standard deviation one produce three and one. The calculation matters only when its probability statement, units and observation boundary remain attached to it. A failed factorisation signals numerical or modelling trouble. Use a stable square root.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use mapping through covariance as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Did You Know? The one-dimensional squared example shows a useful subtlety: a third-degree rule can reproduce the transformed mean while missing variance because variance depends on a fourth moment. The method becomes easier to trust when this hidden structure is made visible.
Each cubature point is one symmetric probe of the nonlinear function. Every point passes through the full nonlinear function. Squaring three and one yields nine and one. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Point propagation samples the function, not the true posterior density. Preserve approximation language.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use nonlinear propagation as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
The update depends on moments reconstructed after nonlinear transformation. Equal weights average transformed values. Nine plus one divided by two equals five. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Using the function at mean two would give four. Compare both to see curvature.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use worked transformed mean as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
| Quantity | Value |
|---|---|
| Standard points | +1, -1 |
| State points | 3, 1 |
| Squared values | 9, 1 |
| Approximate mean | 5 |
| Cubature variance | 16 |
| Exact Gaussian variance | 18 |
A deterministic point rule still makes an approximation. Average squared deviations from the cubature mean. Two deviations of magnitude four give variance sixteen. The calculation matters only when its probability statement, units and observation boundary remain attached to it. This needs fourth-order information the third-degree rule does not exactly preserve. Compare with an analytic benchmark.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use worked transformed variance as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Start with the Gaussian integral the filter must approximate. For X normal with mean two and variance one, E[X squared] is five and Var(X squared) is eighteen. The mean matches while variance differs by two. The calculation matters only when its probability statement, units and observation boundary remain attached to it. One matching moment does not validate all moments. Report approximation error.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use exact gaussian benchmark as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Did You Know? The one-dimensional squared example shows a useful subtlety: a third-degree rule can reproduce the transformed mean while missing variance because variance depends on a fourth moment. The method becomes easier to trust when this hidden structure is made visible.
Each cubature point is one symmetric probe of the nonlinear function. Transition-transformed points reconstruct predicted state mean and covariance, then Q is added. Nonlinear degradation bends the point cloud. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Process-noise structure still requires justification. Separate cubature and noise contributions.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use state prediction as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
The update depends on moments reconstructed after nonlinear transformation. Predicted-state cubature points map through the sensor function and R is added. A nonlinear voltage response yields predicted sensor moments. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Calibration drift changes the function itself. Version sensor models.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use measurement prediction as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
A deterministic point rule still makes an approximation. Paired state and observation deviations estimate how sensor evidence should adjust the state. High-condition points co-vary with high readings. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Marginal variances alone are insufficient. Preserve point pairing.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use cross covariance as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Second authoritative source: https://ieeexplore.ieee.org/document/4982682/. Compare its notation and scope with the teaching example, especially where computation and information boundaries matter.
Start with the Gaussian integral the filter must approximate. Cross covariance and innovation covariance produce the gain and update. Observed minus predicted reading shifts condition. The calculation matters only when its probability statement, units and observation boundary remain attached to it. The update remains a Gaussian approximation. Check posterior simulations.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use kalman-style update as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Did You Know? The one-dimensional squared example shows a useful subtlety: a third-degree rule can reproduce the transformed mean while missing variance because variance depends on a fourth moment. The method becomes easier to trust when this hidden structure is made visible.
Each cubature point is one symmetric probe of the nonlinear function. Both use deterministic points, but CKF points and weights follow a particular spherical-radial rule while UKF has scaling parameters. One-dimensional examples look similar. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Names do not guarantee identical finite-dimensional behaviour. Benchmark on the actual model.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use comparison with ukf as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
The update depends on moments reconstructed after nonlinear transformation. CKF avoids Jacobians and approximates moments by point propagation; EKF uses local derivatives. A nonsmooth or difficult derivative motivates points. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Derivative-free does not mean more accurate in every case. Test accuracy and cost.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use comparison with ekf as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
A deterministic point rule still makes an approximation. A factor-based implementation uses QR and triangular operations for improved numerical stability. Covariance roots remain positive-definite more reliably. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Numerical stability and cubature accuracy are different issues. Report both algorithm layers.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use square-root ckf as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Start with the Gaussian integral the filter must approximate. 2n points scale linearly, yet Gaussian summaries and covariance matrices can still be large. A hundred-state model uses two hundred points. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Strong non-Gaussianity may defeat moment closure. Consider ensembles or particles.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use high-dimensional limits as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Did You Know? The one-dimensional squared example shows a useful subtlety: a third-degree rule can reproduce the transformed mean while missing variance because variance depends on a fourth moment. The method becomes easier to trust when this hidden structure is made visible.
Each cubature point is one symmetric probe of the nonlinear function. Innovations, covariance consistency, analytic tests and simulated coverage reveal approximation failure. The x-squared benchmark exposes variance error. The calculation matters only when its probability statement, units and observation boundary remain attached to it. A single benchmark covers only one function. Use a suite of nonlinear cases.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use diagnostics as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
The update depends on moments reconstructed after nonlinear transformation. Filtered distributions support threshold probabilities and expected-loss choices. A nonlinear sensor updates wear probability. The calculation matters only when its probability statement, units and observation boundary remain attached to it. CKF output does not identify root cause. Pair estimates with engineering investigation.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use maintenance interpretation as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Useful next reading: extended Kalman filtering, unscented Kalman filtering, ensemble Kalman filtering, Mathematics Learning Hub, eduKate Mathematics Learning System, approach to learning mathematics, Bukit Timah Mathematics curriculum overview, cross-subject learning approach. Choose the route that answers the next question rather than treating the links as a checklist.
A deterministic point rule still makes an approximation. CKF study connects symmetry, weighted averages, variance, matrices and nonlinear functions. Students can reproduce points three and one and compare variance sixteen with eighteen. The calculation matters only when its probability statement, units and observation boundary remain attached to it. Cubature points are deterministic integration points, not random particles. Explain the rule and its degree.
A dependable audit follows four moves. Write the conditioning information in words, preserve the intermediate quantities, check dimensions and probability totals, then compare the direction of the result with the physical story. Save state dimension, cubature points and weights, nonlinear functions, Q, R, transformed points, moments, cross covariance and innovations. Reproduction is useful because it reveals whether a surprising conclusion came from data, approximation, code or interpretation.
For practice, use practice, misconceptions and next reading as the focus and map cubature points plus and minus one to state points three and one, square them to nine and one, then calculate mean five and cubature variance sixteen. Before calculating, predict which term should rise or fall. Change one defensible input and explain the response without jargon. End with one supported statement and one limit. This turns mathematics education into transferable reasoning rather than formula recall.
Quick answers
Why is mathematics important here? It exposes how uncertainty moves through a model, which evidence changes the estimate, and where numerical or sampling approximation enters.
Does a successful update prove the physical model? No. It shows what follows from the declared model and data; external validation remains necessary.
Which mathematics transfers from school? Functions, weighted averages, probability, variance, graphs and matrices all reappear, joined by careful explanation.
How should a student practise? Reproduce the small example, keep intermediate values, predict directions and explain one limitation.
What should a practitioner report? State equations, data boundaries, initialization, uncertainty settings, numerical method, diagnostics, sensitivity checks and decision rules.
How can a parent help? Ask the learner what each symbol represents and what assumption would change the conclusion. Clear explanation is part of mathematical mastery.
Extended practice laboratory
Practice case 1: Why moment integrals appear
Setup: Use this chapter’s illustrative case: A squared sensor needs E of X squared and related cross moments.
Task: Identify the required moments.
Checking target: Evaluating only at the mean misses curvature.
Practice case 2: Spherical-radial rule
Setup: Use this chapter’s illustrative case: Points lie on coordinate axes at a dimension-dependent radius.
Task: Do not call it exact for arbitrary functions.
Checking target: The rule has a stated polynomial degree.
Practice case 3: Number of cubature points
Setup: Use this chapter’s illustrative case: A five-state model uses ten points.
Task: Report state dimension and factorisation.
Checking target: Linear growth in points does not remove matrix costs.
Practice case 4: Standardised one-dimensional points
Setup: Use this chapter’s illustrative case: Symmetry makes their mean zero and variance one.
Task: Keep the derivation consistent.
Checking target: Different scaling conventions belong to different quadrature rules.
Practice case 5: Mapping through covariance
Setup: Use this chapter’s illustrative case: Mean two and standard deviation one produce three and one.
Task: Use a stable square root.
Checking target: A failed factorisation signals numerical or modelling trouble.
Practice case 6: Nonlinear propagation
Setup: Use this chapter’s illustrative case: Squaring three and one yields nine and one.
Task: Preserve approximation language.
Checking target: Point propagation samples the function, not the true posterior density.
Practice case 7: Worked transformed mean
Setup: Use this chapter’s illustrative case: Nine plus one divided by two equals five.
Task: Compare both to see curvature.
Checking target: Using the function at mean two would give four.
Practice case 8: Worked transformed variance
Setup: Use this chapter’s illustrative case: Two deviations of magnitude four give variance sixteen.
Task: Compare with an analytic benchmark.
Checking target: This needs fourth-order information the third-degree rule does not exactly preserve.
Practice case 9: Exact Gaussian benchmark
Setup: Use this chapter’s illustrative case: The mean matches while variance differs by two.
Task: Report approximation error.
Checking target: One matching moment does not validate all moments.
Practice case 10: State prediction
Setup: Use this chapter’s illustrative case: Nonlinear degradation bends the point cloud.
Task: Separate cubature and noise contributions.
Checking target: Process-noise structure still requires justification.