Kalman Filter

Concept

Theoretically, Kalman Filter is a recursive state estimation algorithm that estimates the unobserved state of a dynamic system by combining information from a mathematical process model and sequential noisy observations. It is based on Bayesian estimation under linear system dynamics and Gaussian error assumptions, producing minimum mean squared error estimates of the underlying state variables. In health economics, the Kalman filter is used in dynamic state-space models, longitudinal data analysis and disease progression modelling when health states cannot be observed directly.

Mathematically, Kalman Filter is formulated as a recursive prediction?correction algorithm consisting of a state equation describing system evolution and an observation equation relating the hidden state to observed measurements. At each time step, the filter predicts the next system state and updates that prediction using new observations while accounting for uncertainty in both the model and the measurements. The algorithm recursively updates the state estimate and its associated covariance matrix.

In practice, Kalman Filter is applied to estimate latent disease severity, monitor patient trajectories, model healthcare utilisation and analyse longitudinal health data. It is implemented within state-space modelling frameworks using specialised statistical software and is particularly valuable when observations are incomplete, noisy or collected sequentially over time. Although less common than Markov models in health economic evaluation, the Kalman filter is used in dynamic modelling and parameter estimation where recursive state estimation is required.

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Purpose

Used to estimate unobserved dynamic system states from sequential noisy observations, enabling longitudinal disease progression and dynamic health system processes to be modelled accurately.

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Mathematical Formulae

Primary Formula

Prediction step:

x??|??? = Ax????|??? + Bu?

Update step:

x??|? = x??|??? + K?(z? ? Hx??|???)

where:

• x??|??? is the predicted state estimate
• x??|? is the updated state estimate
• A is the state transition matrix
• B is the control matrix
• u? is the control input
• z? is the observation vector
• H is the observation matrix
• K? is the Kalman gain.

Supporting Formulae

Kalman gain:

K? = P?|???H?(HP?|???H? + R)??

Covariance prediction:

P?|??? = AP???|???A? + Q

where:

• P is the state covariance matrix
• Q is the process noise covariance matrix
• R is the observation noise covariance matrix.

Related Mathematical Methods

• State-space modelling
• Bayesian estimation
• Recursive estimation
• Linear dynamic systems
• Time-series analysis
• Hidden Markov models
• Maximum likelihood estimation

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Example

A disease progression model estimates an unobserved disease severity score from quarterly biomarker measurements collected over three years.

The state equation models progression between visits, while observed biomarker values are incorporated through the observation equation. At each quarter, the Kalman filter combines the predicted disease state with the new biomarker measurement to produce an updated estimate with reduced uncertainty. These estimated disease states are subsequently used to predict healthcare costs and quality-adjusted life-years within the economic model.

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Excel Implementation

Function	Example Formula	Health Economics Application
MMULT	=MMULT(A1:C3,D1:D3)	Perform matrix multiplication during prediction and update steps.
MINVERSE	=MINVERSE(A1:C3)	Calculate matrix inverses required for the Kalman gain.
TRANSPOSE	=TRANSPOSE(A1:C3)	Calculate matrix transposes in covariance updates.
SUM	=SUM(B2:D2)	Aggregate state estimates or covariance components.
LET	=LET(...)	Organise recursive matrix calculations in advanced Excel implementations.

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VBA (Optional)

Automate recursive prediction and updating of latent health states from sequential observations using the Kalman filtering algorithm.

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Sources

• Kalman RE. A new approach to linear filtering and prediction problems. Transactions of the ASME: Journal of Basic Engineering. 1960;82(1):35?45.
• Durbin J, Koopman SJ. Time Series Analysis by State Space Methods. 2nd ed. Oxford University Press; 2012.
• Harvey AC. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press; 1989.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.