A Factorization problem and the problem of predicting non-stationary vector-valued stochastic processes

Rissanen, Jorma; Barbosa, Luiz Sérgio de Oliveira

doi:10.1007/bf00534844

Cited by 10 publications

(1 citation statement)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another notable solution to the prediction problem for {X(t)} , an ARMA process with time-dependent coefficients observed on (-oo, T], is given by Rissanen and Barbosa [14,15] (see also [13] and [3], pp. 147-155).…”

Section: Introductionmentioning

confidence: 99%

Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

Spencer

Anh

1991

J. Aust. Math. Soc. Series B, Appl. Math

View full text Add to dashboard Cite

This paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.

show abstract

Section: Introductionmentioning

confidence: 99%

Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

Spencer

Anh

1991

J. Aust. Math. Soc. Series B, Appl. Math

View full text Add to dashboard Cite

show abstract

S

Hazewinkel¹

1993

Encyclopaedia of Mathematics

View full text Add to dashboard Cite

Introduction. In [6 Grothendieck et al.], chapter 16, there is a comprehensive discussion of (higher order left) differential operators on schemes and hence on commutative algebras. Even more details can be found in [5 Gabriel], especially in connection with group schemes and their infinitesimal structure.These discussions include a recursive description of these algebras of differential operators in terms of commutators with (left) multiplication operators; see propositions 16.8.8 and 16.8.9 on pages 42 and 43 of [6 Grothendieck et al.]. This description will be recalled later (sections 6, 7; for the affine case only).More or less recently there has arisen substantial interest in whether something more or less similar can be done for non-commutative algebras. Straightforward generalizations definitely do not quite work.The contexts in which this interest arises are, among more, Batalin-Vilkovisky algebras, Gerstenhaber algebras, homotopy algebras, higher derived algebras. See [1 Akman; 2 Akman; 3 Akman et al.] and, especially, the references in these papers. Also the deformation theory of algebras and diagrams à la Gerstenhaber-Shack, [4 Doubek]. The context 'homotopy algebras' is currently probably the most important. For some of this look at the last sections of [9 Markl].There is in addition interest in such algebras in the realm of formal groups; for that consult the chapter on 'divided power algebras and sequences and algebras of differential operators' in [8 Hazewinkel].The setting for this whole note is that of associative algebras over a base ring k that is commutative and unital. Often it is good to think of k as the ring of integers Z. Example. Algebras of differential operators on k[X,Y ] . Certainly a very natural algebra of differential operators on k[X,Y] is formed by the collection of operators that are finite sums of the formLet me point out a few features of this algebra, which I will denote Diff naive (k[X,Y ]) 1 . First of all it is indeed a unital associative (and non-commutative) algebra over k. Second, there is a notion of degree. An expression like (2.1) is of degree ! n if f i, j = 0 for all i + j > n . And this notion satisfies that if D and ! D are of degree ! n,!!! " n , respectively, then D ! D is of degree ! n + n' . Third, it contains all derivations and obvious higher derivations on k[X,Y ] . Fourth, if D is an element of Diff naive (k[X,Y ]) of degree ! n and t is an element of k[X,Y ] , i.e. a polynomial, then the commutator 1.!The reason for the subscript 'naive' will become clear later.

show abstract

On factoring positive operators

Rissanen

1970

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

A Factorization problem and the problem of predicting non-stationary vector-valued stochastic processes

Cited by 10 publications

References 10 publications

Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

S

On factoring positive operators

Contact Info

Product

Resources

About