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Technology / Technical guide
Kalman Estimation: Prediction, Innovation & Covariance
Kalman estimation combines a dynamic prediction with an observation under a stated mathematical model. It keeps an estimate of the internal state and a model of estimation error. A smoother-looking trace is not its defining feature, and the method’s name cannot establish accuracy when the underlying assumptions are wrong.
Prediction and observation carry different information.
Follow their difference into the correction, then distinguish modelled covariance from measured certainty.
01 / Two models
Motion and observation have separate relationships.
A basic discrete linear model describes x[k] from the previous state through a transition matrix, with a process-noise term.
A basic discrete linear model describes x[k] from the previous state through a transition matrix, with a process-noise term. An observation model maps the current state through H into z[k], with a separate measurement-noise term.
MathWorks presents these prediction and correction relationships. The process uncertainty concerns the state evolution; measurement uncertainty concerns the reported observation. They are not two labels for the same noise source, nor are they automatically independent in every physical system.
02 / Prediction
The estimate advances through the stated dynamics.
The predicted state follows from the preceding estimate and known model inputs.
The predicted state follows from the preceding estimate and known model inputs. Its covariance also evolves, including the process-noise contribution Q. This represents uncertainty within the assumed model rather than newly observed information about physical truth.
A prediction can remain smooth while drifting from an unrepresented change. Conversely, a modelled increase in uncertainty can be appropriate even where the predicted trace is visually steady. Smoothness and justified confidence answer different questions.
03 / Innovation
Correction starts from a measured-versus-predicted difference.
The innovation is z−H x_predicted: the observation minus its predicted measurement.
The innovation is z−H x_predicted: the observation minus its predicted measurement. A gain maps this difference into the state correction. When state and observation have different units, that gain must perform the corresponding dimensional conversion.
A large innovation is not uniquely a failed sensor. It can arise from model mismatch, a genuine change, bias or an observation problem. The observability guide explains why some internal distinctions cannot be recovered from the chosen outputs alone.
04 / Scalar example
The weighting follows invented uncertainty inputs.
Hypothetical scalar model: predicted state is 4 units with variance 4 units²; an independent observation is 6 units with measurement variance 1 unit², and H = 1.
Hypothetical scalar model: predicted state is 4 units with variance 4 units²; an independent observation is 6 units with measurement variance 1 unit², and H = 1. The gain is 4/(4+1) = 0.8.
The corrected estimate is 4+0.8(6−4) = 5.6 units, with modelled posterior variance (1−0.8)4 = 0.8 units². These values illustrate one ideal update. They are not filter settings, product measurements or proof that the physical error is now exactly known.
05 / Noise assumptions
Covariance describes spread and shared variation.
The standard linear interpretation retains the declared means, covariances and relationships of process and measurement errors.
The standard linear interpretation retains the declared means, covariances and relationships of process and measurement errors. Correlated errors require their proper model; ignoring a common disturbance can treat repeated information as independent evidence.
Covariance diagonal entries use squared units of their variables; off-diagonal entries use products of the corresponding units. A variance is not a tolerance envelope. A small computed covariance can coexist with a biased physical estimate if that bias is absent from the model.
06 / Scope
A model-based estimate is not a universal noise cure.
Kalman estimation differs from a fixed low-pass transformation because prediction, observation mapping and uncertainty govern its correction.
Kalman estimation differs from a fixed low-pass transformation because prediction, observation mapping and uncertainty govern its correction. Nonlinear dynamics, outliers and missing observations raise additional modelling questions; the simple equations do not automatically resolve them.
No Kalman filter, covariance value or accuracy improvement is confirmed for MARSTON. The explanation provides no implementation, covariance tuning or live-data adjustment, and it does not label an unverified controller as advanced merely by naming an algorithm.
Related technical guides: Dynamic States & Observability: What Outputs Can Reveal · Cross-Correlation: Lag Sign, Similarity & Time Delay · Ratiometric Sensing: Common Reference & Residual Errors.
Extend the estimation review with estimator bias, variance and mean-squared error, least-squares residuals and input errors, and model residuals and ambiguous causes.
General background: MathWorks: Linear prediction, innovation and covariance equations. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Predicted state | Model-propagated internal estimate | Not a direct new observation. |
| Innovation | z−H x_predicted | Its sign and measurement units are explicit. |
| Q | Process-noise covariance | Applies to the stated state-evolution model. |
| R | Measurement-noise covariance | Separate from process uncertainty. |
| Posterior covariance | Modelled estimation-error spread | Not a guaranteed physical error bound. |
Engineering questions.
Is Kalman estimation merely smoothing?
No. Dynamics, observations and uncertainty determine the correction.
Does a large innovation identify a faulty sensor?
Not uniquely; several model or physical causes can contribute.
Is covariance a permitted tolerance?
No. It describes modelled statistical spread.
Can a small covariance coexist with bias?
Yes, if the model fails to represent that bias.
Are 0.8 and 5.6 MARSTON results?
No. They follow from an invented independent scalar example.
Does this page prescribe filter tuning?
No implementation or uncertainty setting is supplied.
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Kalman estimation combines a dynamic prediction with an observation under a stated mathematical model. Share the model and applicable operating or design information for an engineering discussion.