Kalman Filter

Kalman Filter: Concept Architecture

Estimate an evolving state from noisy observations

The Kalman filter is a recursive method for estimating an unobserved, changing state from measurements that contain noise. At each time point it first predicts the state using a model of how it evolves, then adjusts that prediction when a new observation arrives. It carries forward both an estimate and its uncertainty. Under a correctly specified linear Gaussian state-space model, the resulting filtered state distribution is the exact conditional distribution given observations available through that time.

The state might represent an underlying disease-severity trajectory, a latent biomarker level, a service-demand trend, or a changing economic parameter. A filter does not discover such a state without a model: its interpretation depends on the transition and measurement equations, their noise assumptions, and the observation times. Filtering uses current and past data; smoothing uses later observations to revise an earlier state and is therefore a different task.

Specify the state and measurement equations

For a state vector $x_t$ and observation vector $y_t$ at time $t$, a linear model can be written as:

$$x_t=A_tx_{t-1}+B_tu_t+w_t,\qquad y_t=H_tx_t+v_t.$$

Here $A_t$ describes how the latent state changes, $u_t$ is an observed input such as an intervention and $B_t$ its influence, and $H_t$ maps the state to measured outcomes. The process disturbance $w_t$ has mean zero and covariance $Q_t$; the measurement disturbance $v_t$ has mean zero and covariance $R_t$. The standard formulation assumes appropriately independent disturbances across time and between the two equations, together with a specified initial state distribution. Dimensions must align: if $x_t$ has $p$ components and $y_t$ has $m$, $A_t$ is $p\times p$, $H_t$ is $m\times p$, $Q_t$ is $p\times p$, and $R_t$ is $m\times m$. Covariance matrices must be valid, and the observations must mean what the measurement equation says they mean.

For an exact Gaussian interpretation, the initial state and noise terms are jointly Gaussian under the stated independence structure. Without Gaussianity, the conventional linear filter can still be understood as a best linear mean-square estimator under appropriate second-moment conditions, but its reported Gaussian distribution or likelihood may no longer be exact. A linear Kalman filter is not automatically appropriate for counts, bounded utilities, discrete disease states, or strongly nonlinear response curves.

Predict before looking at the new measurement

Let $m_{t-1\mid t-1}$ and $P_{t-1\mid t-1}$ be the state mean and covariance after processing observations through $t-1$. The one-step prediction is:

$$m_{t\mid t-1}=A_tm_{t-1\mid t-1}+B_tu_t,\qquad P_{t\mid t-1}=A_tP_{t-1\mid t-1}A_t^{\mathrm{T}}+Q_t.$$

The predicted covariance increases by $Q_t$ because the evolving state has process uncertainty in addition to uncertainty carried from the last estimate. A high value of $Q_t$ lets the state move more freely; a low value implies a relatively predictable transition. Process variance is not interchangeable with observation variance: one describes real evolution not explained by the transition equation, the other describes noise in measuring the current state.

When a new observation is unavailable, the filter can propagate this prediction without a measurement update. Its uncertainty often grows while data are absent. Missingness that depends on unobserved clinical deterioration can invalidate a simple analysis that treats missing observations as ignorable, even though the recursion itself can skip a measurement.

Correct the prediction with the innovation

When $y_t$ arrives, compare it with the measurement predicted from the state. The innovation $e_t$, innovation covariance $S_t$, and Kalman gain $K_t$ are:

$$e_t=y_t-H_tm_{t\mid t-1},\qquad S_t=H_tP_{t\mid t-1}H_t^{\mathrm{T}}+R_t,\qquad K_t=P_{t\mid t-1}H_t^{\mathrm{T}}S_t^{-1}.$$

The update is:

$$m_{t\mid t}=m_{t\mid t-1}+K_te_t,\qquad P_{t\mid t}=(I-K_tH_t)P_{t\mid t-1}.$$

The gain balances model prediction against the measurement according to their uncertainties. In the scalar case with direct measurement, a more precise observation (smaller $R_t$) receives greater weight; a noisier observation receives less. For multidimensional states, $K_t$ is a matrix, and correlations between components matter. The simple covariance formula above is exact algebraically under the model; implementations may use a numerically more robust equivalent, often called the Joseph form, to better preserve symmetry and nonnegative covariance under finite precision.

Do not compute a general matrix inverse explicitly when solving for $K_t$ in a numerical implementation. Use a suitable linear system solve or factorization and check whether $S_t$ is well-conditioned. Model parameters such as $A_t,Q_t,H_t,R_t$ are usually specified or estimated separately; the filter equations alone do not determine them from one observation.

Work a one-dimensional update

Suppose the previous estimate of an unobserved level is 10 with variance 4. For a one-step random-walk transition, set $A=H=1$, no input, process variance $Q=1$, and measurement variance $R=4$. The predicted mean remains 10, and predicted variance is $4+1=5$. A new measurement is 12, giving innovation 2, innovation variance $5+4=9$, and gain $K=5/9$.

The corrected mean and variance are:

$$m_{t\mid t}=10+\frac{5}{9}(12-10)=\frac{100}{9}\approx11.11,\qquad P_{t\mid t}=\left(1-\frac{5}{9}\right)5=\frac{20}{9}\approx2.22.$$

The estimate moves from 10 toward 12 but does not simply become the new reading, since that reading is noisy. Under this model the updated variance is smaller than the predicted variance of 5 because the measurement adds information. The standard deviation is approximately $\sqrt{20/9}\approx1.49$ in the state's units; variance 2.22 is not a standard deviation. The figures are illustrative, and their meaning depends on a well-calibrated choice of $Q$ and $R$.

If the observation were missing, the time-$t$ estimate would stay at the predicted 10 with variance 5. If measurements became much more precise relative to process uncertainty, the gain would rise. If the state changed unexpectedly but $Q$ were fixed too low, the filter might respond too slowly and provide overconfident estimates.

Fit, check, and extend the model

Choose an initial mean and covariance, specify observation timing and any known inputs, and estimate or justify transition and noise parameters. Innovations should be examined for systematic bias, remaining serial correlation, and miscalibrated variability under the model. Check whether estimates or forecasts change substantially under plausible $Q$, $R$, initial conditions, structural alternatives, and missing-data assumptions. In a fitted state-space model, the sequence of innovation distributions supports likelihood-based parameter estimation; fitting and validating on the same data can still overstate prospective predictive performance.

Filtering answers what could be inferred at time $t$ from data then available. Fixed-interval smoothing conditions the earlier state on the entire observed sequence; it can improve retrospective reconstruction but would leak future measurements if presented as a real-time prediction. Forecasting extends the state transition forward beyond the last observation, usually with growing uncertainty. Keep these outputs labeled separately, especially in a clinical decision tool.

For nonlinear transitions or measurement functions, an extended Kalman filter linearizes locally and an unscented filter approximates propagated uncertainty by selected points; neither is an exact solution to every nonlinear model. Particle filtering can represent broader, possibly non-Gaussian state distributions at greater computational cost. Model choice should follow the data-generating mechanism and decision question, not merely the availability of an algorithm.

Use with care in health economics

Longitudinal health data often include irregular visits, measurement error, missing measurements, treatment changes, and latent disease progression. A suitable state-space model may estimate a continuous trajectory that feeds risk, resource-use, or outcome predictions in an economic analysis. A filtered estimate at a visit is conditional on observations through that visit and on the model; it should not be mistaken for a directly observed clinical state or for a future treatment effect.

A cohort Markov model commonly uses discrete health states and transition probabilities, whereas the basic Kalman filter above uses a continuous Gaussian latent state with linear dynamics. Both can have a Markov property, but the filter does not supply a valid disease transition matrix or a cost-effectiveness result by itself. If filtered states inform future event probabilities, costs, or QALYs, validate that mapping, propagate parameter and state uncertainty, and avoid using smoothed future information in decisions that would have been made earlier.

Report state and observation definitions, time interval, $A_t,B_t,H_t,Q_t,R_t$, initial distribution, parameter estimation, handling of missing and irregular data, diagnostics, and whether results are predicted, filtered, or smoothed. Interpret the gain and uncertainty in the units of the actual health or economic measurement. A credible application makes its assumptions and prospective use explicit.

Sources and further reading

• Kalman RE. A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering (1960). Original state-space filtering formulation.
• statsmodels: Custom state-space models. Primary software documentation for transition and observation equations.
• statsmodels: Kalman filter results. Primary documentation for forecast innovations, states, and covariances.
• Cauchi M, et al. Individualized survival predictions using state space model with longitudinal and survival data. Journal of the Royal Society Interface (2024). Applied research connecting longitudinal state estimation with health outcomes.