Description

The Kalman Filter (KF) was introduced by Kalman (1960) as a new approach to discrete data linear filtering using state space methods. It involves the estimation of the state of the system on the basis of noisy measurements considered as a linear combination of state variables.

The solution is in the form of a recursive algorithm that, given the state space model of the system, provides an estimate of the state at each sampling instant that is "optimal" in the sense of least square error.

The Kalman Filter estimates a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements (Welch and Bishop, 2004). The Kalman Filter consists of two sets of equations.

1. Time update (or predictor) equations, which project the current state and error covariance estimates forward (in time) to obtain the a priori estimates for the next time step.

2. Measurement update (or analysis or corrector) equations, which perform the feedback step, incorporating new measurements into the a priori estimate to obtain an improved a posteriori estimate.

Software

• A Matlab toolbox

• SSPIR toolbox for the R language.

Advantages

The advantages of KF are: it is powerful, fast, supports estimation of past, present and future states, its disadvantage is that it is applicable only to linear or nearly linear problems. It is interesting to note that the derivation provided by Kalman applies to arbitrary random signals, described by up to a second order average statistical properties and does not assume Gaussian properties of the states.

Disadvantages

The process and/or the measurement relationships must be linear.

Case studies

Data assimilation for real-time runoff forecasting (case study)

See also

• Extended Kalman Filter

• Ensemble Kalman Filter

References and Further reading

There are many books written and applications are numerous in many different subjects, from engineering, computing to biological and environmental sciences.

• A page on the Kalman Filter maintained by Greg Welch and Gary Bishop at the University of North Carolina at Chapel Hill, including

• Welch, G. and Bishop, G., 2004. An introduction to the Kalman Filter. TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill, NC 27599-3175.

• Wikipedia page on the Kalman Filter.

• Brown, R.G. and Hwang, P.C.Y., 1992. Introduction to random signals and applied Kalman filtering. J. Wiley and Sons, New York. ( Amazon.com, Amazon.co.uk)

• Harvey, A.C., 1990. Forecasting, structural time series models and the Kalman filter. Cambridge University Press, Cambridge. ( Amazon.com, Amazon.co.uk)

• Kalman, R., 1960. New approach to linear filtering and prediction problems. ASME Trans., Journal Basic Eng, 82-D(35-45).

• Young, P.C., 1984. Recursive Estimation and Time Series Analysis. Springer-Verlag, Berlin. ( Amazon.com, Amazon.co.uk)

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