Statistical Glossary
Kalman Filter (Equations):
The basic mathematics behind the idea of Kalman filter may be described as follows - Consider, for example, a Markov chain - i.e. a random series with Markov property - described by the following equation:
(1) |
where
- is the value of the vector-values Markov chain
is a known matrix which describes regular causal link between the current state
and the next state
The Kalman filter for time series specified by equation (1) may be described by the following recursive expression:
(2) |
where
is the "Kalman gain".
The expression (2) is only one of several mathematically- equivalent forms. It shows that the Kalman filter output
of the Markov chain (1) ; the second term is a linear function of the prediction error (
The theory of Kalman filtering specifies expressions for calculation of the optimal matrix in equation (2) from the known covariance matrix
Besides the simplest case described by (1-2) , the classical theory of Kalman filters covers more complex settings, e.g. when the state vector is not observable - i.e. only a linear function
is known
where
There are also generalizations of the Kalman filter theory for continuous time , expressed in terms of differential equations (not difference equations, like above); and there are extensions to non-linear filtering.