Spectral factorization of wide sense stationary processes on Z2

Korezlioglu, H.

doi:10.1016/0047-259x(86)90092-8

Cited by 36 publications

(24 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This extends the Wold representation theorem, and there is a corresponding unilateral infinite AR representation if also f (λ) is everywhere positive; see Whittle (1954), Helson and Lowdenslager (1958), Guyon (1982), Korezlioglu and Loubaton (1986).…”

Section: Introductionmentioning

confidence: 83%

Modified whittle estimation of multilateral spatial models

Robinson¹,

Sanz²

2003

Working Paper Series

View full text Add to dashboard Cite

We consider the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d ≥ 2. The achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the "edge effect", which worsens with increasing d. The other is the difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, especially in case of multilateral models, due mainly to the Jacobian term. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the latter problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by * Research supported by ESRC Grant R000239936. Thanks are due to Fabrizio Iacone for carrying out the numerical work reported in Section 3. 1 first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. A Monte Carlo study of finite sample behaviour is included. The asymptotic regime allows increase in both directions, unlike the usual random fields formulation, with the central limit theorem established after re-ordering as a triangular array. When the data are non-Gaussian, the asymptotic variances of all parameter estimates are likely to be affected, and we provide a consistent, non-negative definite, estimate of the asymptotic variance matrix.

show abstract

Section: Introductionmentioning

confidence: 83%

Modified whittle estimation of multilateral spatial models

Robinson¹,

Sanz²

2003

Working Paper Series

View full text Add to dashboard Cite

show abstract

“…Remark. The above model of an evanescent field is not the most general one, [16], [18], [8]; for an example see [4].…”

Section: The Evanescent Componentmentioning

confidence: 99%

“…Evanescent processes were first introduced in [16] (on R). In Korezlioglu and Loubaton [18], "horizontal" and "vertical" total-orders and the corresponding horizontally and vertically evanescent components of a homogeneous random field on Z 2 are defined. In Kallianpur [17], as well as in Chiang [3], similar techniques are employed to obtain four-fold orthogonal decompositions of regular (non-deterministic) homogeneous random fields.…”

Section: Introductionmentioning

confidence: 99%

Spectral representation and asymptotic properties of certain deterministic fields with innovation components

Cohen

Francos

2003

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.

show abstract

“…Several authors have studied random fields on the lattice points of the plane. Some of the important results in this field are included in the work of Helson and Lowdenslager [2], where a generalization of Szeg6's theorem to half-planes is proved; Chiang [1], where the regularity problem for the half-planes is discussed; Kallianpur and Mandrekar [7], where a Wold-Halmos decomposition Theorem is proved; Korezlioglu and Loubaton [8], where spectral factorizations are considered, Soltani [15], dealing with regularity and quarter-plane moving average representation; and Miamee [10], where an extension of Szego's theorem for third quadrant is given. Another problem which has been proved to be useful in the prediction theory of stationary stochastic processes is the idea of the angle between past and future.…”

Section: Astract (Con Inue On Ritv If Necomary and Idendfy By Blocmentioning

confidence: 99%

On the Angle for Stationary Random Fields.

Miamee¹,

Niemi²

1985

View full text Add to dashboard Cite

Spectral factorization of wide sense stationary processes on Z2

Cited by 36 publications

References 10 publications

Modified whittle estimation of multilateral spatial models

Modified whittle estimation of multilateral spatial models

Spectral representation and asymptotic properties of certain deterministic fields with innovation components

On the Angle for Stationary Random Fields.

Contact Info

Product

Resources

About