Publication

The Auxiliary Variable Trick for deriving Kalman Smoothers

2004
Reports

Abstract

We present a Forward-Backward Kalman smoother derivation that functions for small observation noise. Whilst this smoother can be found by judicious transformation of the standard Forward-Backward equations, we introduce an auxiliary variable trick which greatly simplifies the derivation, based on the probabilistic interpretation of the Kalman Filter, allowing one to work directly with moments of the distribution. The trick is of potential interest for the simple derivation of smoothing type inference in other related systems.

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2004
Reports

Abstract

Official source

Related concepts (25)

Kalman filter

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

Particle filter

Particle filters, or sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to find approximate solutions for filtering problems for nonlinear state-space systems, such as signal processing and Bayesian statistical inference. The filtering problem consists of estimating the internal states in dynamical systems when partial observations are made and random perturbations are present in the sensors as well as in the dynamical system.

Ensemble Kalman filter

The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting.

Related publications (1)

The Auxiliary Variable Trick for deriving Kalman Smoothers

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MATHICSE Technical Report : Efficient state/Parameter estimation in nonlinear unsteady PDEs by reduced basis ensemble Kalman filter