Reconstructing the Kalman Filter for Stationary and Non Stationary Time Series

A Kalman filter, suitable for application to a stationary or a non-stationary time series, is proposed. It works on time series with missing values. It can be used on seasonal time series where the associated state space model may not satisfy the traditional observability condition. A new concept called an 'extended normal random vector' is introduced and used throughout the paper to simplify the specification of the Kalman filter. It is an aggregate of means, variances, covariances and other information needed to define the state of a system at a given point in time. By working with this aggregate, the algorithm is specified without direct recourse to those relatively complex formulae for calculating associated means and variances, normally found in traditional expositions of the Kalman filter. A computer implementation of the algorithm is also described where the extended normal random vector is treated as an object; the operations of addition, subtraction and multiplication are overloaded to work on instances of this object; and a form of statistical conditioning is implemented as an operator.