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Abstract
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.
Highlights
Being the cornerstone of modern programming practice, objects and classes provide a useful basis for structuring computer code
A specification has been proposed for the state space model that consolidates the usual measurement and transition equations into a single equation (2.1)
All information pertaining to a multivariate normal distribution is consolidated into an entity called a normal random vector object
Summary
Being the cornerstone of modern programming practice, objects and classes provide a useful basis for structuring computer code. The purpose of this paper is to demonstrate that the approach can promote useful ways of thinking in statistical theory We illustrate this point by treating normal random vectors as objects and applying them to the problem of filtering time series. The Kalman filter, for stationary time series, is defined in terms of the resulting object and its operations This normal random vector object is extended to include a special matrix required in the case of non-stationary time series to carry additional information forward through time. The Kalman filter (Kalman, 1960, Kalman and Bucy, 1961) is essentially an algorithm for revising the moments of stochastic components of a linear time series model to reflect information about them contained in time series data It is often used as a stepping-stone to deriving the moments of components at future points of time for forecasting purposes.
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