Kalman Filtering

Kalman filtering is the closed-form solution to the Bayesian inference problem for the linear Gaussian state-space model: a hidden Markov state $x_t$ that evolves linearly in time with Gaussian shocks, observed only through linear measurements corrupted by Gaussian noise. The setup is
$x_t=\Phi x_{t-1}+w_t,y_t=A_t{x}_t+v_t,w_t\sim \mathcal{N}(0,Q),v_t\sim \mathcal{N}(0,R),$
with initial state $x_0\sim \mathcal{N}({\mu }_0,{\Sigma }_0)$. The inference question (what does the data $y_{1:s}$ tell me about the state $x_t$?) has a Gaussian answer at every $t$ because the model is linear and Gaussian throughout, and the answer can be updated sequentially in $O(p^3)$ work per step rather than recomputed from scratch.

The two algorithms in this section solve different versions of the question depending on what data is available:

• The Kalman Filter computes the posterior $p(x_t\mid y_{1:t})$, the filter distribution, using only data up through the current time. This is the right tool for online estimation, real-time control, and forecasting from the current moment.
• The Kalman Smoother computes the posterior $p(x_t\mid y_{1:T})$, the smoother distribution, using the entire record (past and future). This is the right tool for retrospective analysis and missing-data interpolation.

Both algorithms are the textbook example of conjugate sequential Bayesian inference. The prior on $x_t$ given the past is Gaussian; the likelihood of the new observation given $x_t$ is Gaussian; the posterior is Gaussian by the standard Gaussian conjugacy formula. Everything else (the prediction-update split, the Kalman gain, the smoother’s backward pass) is bookkeeping that makes the recursion efficient.

Prerequisites

• 6.2 Multivariate Gaussian Distribution: the conditioning identity that produces every step of the recursion
• 3.1 Bayesian Framework: prior, likelihood, posterior; the language of sequential updating
• 2.6.2 Dynamic Linear Model: the time-series state-space model that the filter operates on
• 4.7 Stationarity: the filter does not require stationarity; in fact it is most useful when the state evolves non-stationarily

Contents

• 3.12.1 Kalman Filter: the forward recursion: prediction step, update step, Kalman gain.
• 3.12.2 Kalman Smoother: the backward recursion that refines filter estimates using future data (RTS algorithm).

When to use which

Use the filter for online problems where future observations are not yet available: navigation, real-time control, online forecasting. The output at time $t$ uses only data through $t$ and arrives one step at a time as the data does. Computational cost is $O(p^3)$ per step in the state dimension $p$.

Use the smoother for offline problems where the entire data record is available and the goal is the best possible estimate at every time point: scientific reanalysis, retrospective study of recorded experiments, interpolating missing observations. The smoother runs the filter forward and then a separate backward pass that refines each $x_t$ estimate using all observations after time $t$. The cost is roughly twice the filter’s.

Where the algorithm shows up

• State-space time series. The filter computes the likelihood of an ARMA model written in state-space form, which is how statsmodels.tsa.statespace.SARIMAX evaluates the likelihood for fitting. See 2.6.2 Dynamic Linear Model for the time-series framing.
• Robotics and control. State estimation for vehicles and robots, fusing IMU, GPS, and other sensors, is a canonical Kalman application. The Apollo and Artemis missions use extended versions of the algorithm.
• Bayesian regression with drifting coefficients. Putting a state-space prior on regression coefficients ${\beta }_t$ that evolve as a random walk turns ordinary regression into a Kalman filter on the coefficient vector: the basis for recursive least squares and time-varying regression in econometrics.
• Gaussian process regression in time. When a GP uses a kernel that admits a state-space representation (Matérn family with half-integer smoothness), GP regression on a 1-D time index can be done in $O(T)$ via the Kalman filter rather than $O(T^3)$ via direct conditioning.

See Also

• 2.6.2 Dynamic Linear Model: the model the filter operates on, in time-series language
• State Space Models in time series: the applied time-series treatment
• 3.10 Gaussian Processes: Kalman is the temporal-state-space form of GP regression
• 3.11.5 EM Algorithm: used to learn the unknown $\Phi ,Q,R$ in a state-space model when they are not given