Decomposition of Time Series Dynamic Linear Models

Abstract

This paper derives the admissible decompositions for a time series dynamic linear model, assuming only that the model is observable. The decompositions depend on factorizations of the characteristic polynomial of the state evolution matrix G into relatively prime factors. This generalizes the method of West (1997) which considers one decomposition in the particular case where G is diagonalizable. Conditions are derived for a decomposition to be independent. These results show that no autoregressive process of order d has an independent decomposition for any integer d. Two illustrations of this procedure are discussed in detail.where, for a suitable integer d, h t is a d  1 state vector, F is a d  1 vector of known terms and the d  d state evolution matrix G is known; m t and the d  1 state evolution noise vector x t are assumed to be mutually independent and the evolution noise variance matrix E½x t x 0 t ¼ W . West and Harrison (1997) propose that the trend terms are included in the state vector h t so that the model (1.2) is essentially derived from (1.1). However, this proposal is indeterminate in the sense that several nontrivially different trend specifications can result in exactly the same model (1.2). This raises the questions of how many decompositions (1.1) are admissible for a given TSDLM with specified F and G and which of these decompositions can be regarded as useful and meaningful.West (1997) identifies a restricted class of TSDLMs in which each model is expressed as a linear combination of AR or ARMA models of a low order. Huerta andWest (1999a, 1999b) use this result in a Bayesian analysis of AR(d) models, where d is large, and West et al. (1999) describe an interesting application in the field of clinical neurophysiology. This approach realizes only the one decomposition (referred to here as the irreducible decomposition) and the argument depends on the existence of interim AR models which have complex-valued parameters. It is also required that the d eigenvalues of the state evolution matrix are distinct; however, there are economic and several other applications where G has at least one eigenvalue which is repeated several times. For example, if a TSDLM includes a polynomial trend then G has a repeated eigenvalue unity whose multiplicity is one more than the polynomial degree (Godolphin and Harrison, 1975). The TSDLM representations of the forward-shifted models of Box and Jenkins (1976, §5.3) require G to have repeated eigenvalue zero, as shown by Johnson (1999) who generalizes a result of Godolphin and Stone (1980). Models with polynomial trends or forward-shifted components or indeed any components resulting from a repeated eigenvalue structure are not decomposable by West's method. Furthermore, Pierce (1979), Hillmer andTiao (1982) and Pole et al. (1994, §3.6) focus attention on decompositions of time series whose component terms are mutually independent; however, it sometimes happens that the irreducible decomposition does not possess this property even though other independent deco...

Blind Signal Separation of Mixtures of Chaotic Processes: A Comparison Between Independent Component Analysis and State Space Modeling

“…The concept of modeling a univariate time series by a decomposition into several components has a long history in statistics. A good summary of the earlier work is given in the introduction of Godolphin & Johnson [2003]. As an example for an important application, we mention seasonal adjustment in econometric data; in the 1990s state space methods have been developed for this task [Kitagawa & Gersch, 1996;Kato et al, 1996].…”

“…In this paper we will employ state space models with block-diagonal state transition matrix for the purpose of a decomposition into independent components. This approach has also been studied by Godolphin & Johnson [2003]; however, we note that block-diagonal state transition matrices have been used much earlier in the system identification community [Pagan, 1975;Engle & Watson, 1981].…”

“…(1) and (2) can be adapted to the purpose of modeling independent components by choosing a specific structure for the parameter matrices A, C, S η and S . For this purpose we choose a common block-diagonal structure for A and S η [Godolphin & Johnson, 2003]:…”

“…Note that in the univariate case N = 1, as discussed by Godolphin & Johnson [2003], it would suffice to use C j = (1 0 . .…”

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