The Kalman Filter

Time Series Analysis

kalman

filtering

state-space

Recursive optimal estimation for linear-Gaussian state-space models

Published

April 17, 2026

Introduction

The Kalman filter is the workhorse algorithm for state-space models with linear-Gaussian observation and transition equations. It recursively combines the previous state estimate with each new observation to produce an updated state estimate together with its uncertainty, in a closed-form Gaussian update that is provably optimal under the linear-Gaussian assumptions. Originally developed for aerospace tracking, it now underpins virtually every classical time-series filtering application — economic indicators, signal processing, target tracking, financial volatility estimation.

Prerequisites

A working understanding of state-space models, linear algebra, and the multivariate Normal distribution.

Theory

For a linear-Gaussian state-space model with observation \(y_t = F x_t + v_t\) and transition \(x_t = G x_{t-1} + w_t\), the Kalman filter alternates two steps at each time:

Predict: \[x_{t \mid t-1} = G x_{t-1 \mid t-1}, \qquad P_{t \mid t-1} = G P_{t-1 \mid t-1} G^\top + W.\]

Update: \[x_{t \mid t} = x_{t \mid t-1} + K_t (y_t - F x_{t \mid t-1}), \qquad P_{t \mid t} = (I - K_t F) P_{t \mid t-1},\]

with Kalman gain \(K_t = P_{t \mid t-1} F^\top (F P_{t \mid t-1} F^\top + V)^{-1}\).

The smoother (Rauch-Tung-Striebel) is a backward recursion that combines forward filter estimates with future observations to produce smoothed estimates \(x_{t \mid T}\). Non-linear extensions include the extended Kalman filter (linearisation around the current estimate) and the unscented Kalman filter (sigma-point sampling); particle filters handle fully non-Gaussian and non-linear models via Monte Carlo.

Assumptions

Linear Gaussian observation and transition equations with known parameters \((F, G, V, W)\). Parameters can be estimated by MLE on the innovations.

R Implementation

library(dlm)

# Local-level model on Nile
mod <- dlmModPoly(order = 1, dV = 15100, dW = 1470)
filt <- dlmFilter(Nile, mod)

plot(Nile)
lines(dropFirst(filt$m), col = "red", lwd = 2)

# Smoother
smo <- dlmSmooth(filt)
lines(dropFirst(smo$s), col = "blue", lwd = 2)

Output & Results

dlmFilter() returns the filtered state estimates (\(x_{t \mid t}\)) and their covariances. dlmSmooth() produces the retrospective smoothed estimates (\(x_{t \mid T}\)). Plotting both on the original series shows the filter’s adaptation in real time and the smoother’s retrospective best estimate.

Interpretation

A reporting sentence: “The Kalman filter applied to the Nile river-flow series produced one-step-ahead forecasts (filtered states) that adapt to each new observation; the smoothed estimate, using the full series in retrospect, identified a sharp level drop near 1899 that the forward filter takes several years to track.” Distinguish filtered (real-time) from smoothed (retrospective) estimates in any report.

Practical Tips

For non-linear models, use the extended Kalman filter (KFAS::EKF or hand-coded) for mild non-linearity, the unscented Kalman filter for stronger non-linearity, or particle filters (SMCSamplers, SMC2) for arbitrary non-linear / non-Gaussian models.
Uncertainty propagates naturally through both predict and update steps; report state estimates with their estimated standard errors.
The log-likelihood of the model can be computed from the innovations (dlmLL in dlm) and used for MLE-based parameter estimation; the Kalman filter is the engine, MLE is the parameter-fitting wrapper.
The choice of initial state \(x_{0 \mid 0}\) and initial variance \(P_{0 \mid 0}\) matters for the first few time steps; diffuse initialisation (\(P_{0 \mid 0}\) very large) is the standard choice when no prior information is available.
For extremely high-dimensional state spaces (large meteorological models), the ensemble Kalman filter approximates the covariance with a low-rank ensemble; standard Kalman fails in that setting.
Pair filtering with diagnostic checks on the innovations (one-step-ahead residuals); they should be approximately white noise under correct model specification.

R Packages Used

dlm for dlmFilter(), dlmSmooth(), and dlmLL(); KFAS for fast filtering with non-Gaussian observations; FKF for the fast Kalman filter implementation; bsts for Bayesian structural time-series alternatives; nimble for hand-built non-linear filters via probabilistic programming.