Reconstructing the Kalman filter for stationary and non stationary time series

A Kalman filter for application to stationary or non-stationary time series is proposed. A major feature is a new initialisation method to accommodate non-stationary time series. The filter works on time series with missing values at any point of time including the initialisation phase. It can also be used where a state space model does not satisfy the traditional observability condition, a situation that can arise with seasonal time series. Another feature of the paper is that the Kalman filter is described in terms of the augmented moments of the state vectors, these being an aggregate of means, variances, covariances and other pertinent information. By doing this, the Kalman filter is specified without direct recourse to those relatively complex formulae for calculating associated means and variances found in traditional expositions. A computer implementation of the Kalman filter is also described where the augmented moments are treated as an object; the operations of addition and multiplication are overloaded to work on instances of this object; and a form of statistical conditioning is implemented as an operator.