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A WoldLike Decomposition of 2D Discrete Homogeneous Random Fields
"... this paper we consider the structure of twodimensional discrete homogeneous random fields. We extend the results of Helson and Lowdenslager (1962), Korezlioglu and Loubaton (1986), Kallianpur et al. (1990), and Chiang (1991), to show that the two, three, and fourfold Woldtype decompositions are ..."
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this paper we consider the structure of twodimensional discrete homogeneous random fields. We extend the results of Helson and Lowdenslager (1962), Korezlioglu and Loubaton (1986), Kallianpur et al. (1990), and Chiang (1991), to show that the two, three, and fourfold Woldtype decompositions are special cases of the countablyinfinitefold decomposition presented in this paper. The countablyinfinitefold decomposition arises from a set of new totalorder and nonsymmetricalhalf plane (NSHP) definitions imposed on the random field. These order definitions are obtained by rotating the NSHP support by angles of rational tangent, rather than considering only the vertical and horizontal orientations. A family of real, zeromean, random variables fy(n; m); (n; m) 2 Z
On the Linear Prediction of some L p Random Fields
 J. Austral. Math. Soc
, 1999
"... This work is concerned with the prediction problem for a class of L p random fields. For this class of fields, we derive prediction error formulas, spectral factorizations, and orthogonal decompositions. AMS 1991 Subject Classification: 60G25, 60G60, 60G10. Key Words: Prediction, random field ..."
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Cited by 2 (0 self)
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This work is concerned with the prediction problem for a class of L p random fields. For this class of fields, we derive prediction error formulas, spectral factorizations, and orthogonal decompositions. AMS 1991 Subject Classification: 60G25, 60G60, 60G10. Key Words: Prediction, random fields. y Research supported in part by an NSF Postdoctoral Fellowships and by a NSFNATO Postdoctoral fellowship, while at CEREMADE, Universit'e ParisDauphine, 75775 Paris Cedex, France and at CERMA, ENPC, La Courtine, 93167 Noisy le Grand Cedex, France. 1 Introduction We study in these notes the prediction problem for a class of random fields which do not necessarily have finite variance. More precisely, we study zero mean random fields fXmn g for which there exist a finite nonnegative Borel measure ¯ on the torus and p 2 (1; 1) with the property E fi fi fi fi fi fi M X m=\GammaM N X n=\GammaN amnXmn fi fi fi fi fi fi p i Z fi fi fi fi fi fi M X m=\GammaM N X n=\GammaN a...
Orthogonal Decompositions of 2D Nonhomogeneous Discrete Random Fields
"... Imposing a totalorder on a 2D discrete random field induces an orthogonal decomposition of the random field into two components: A purelyindeterministic field and a deterministic one. The purelyindeterministic component is shown to have a 2D whiteinnovations driven movingaverage representatio ..."
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Imposing a totalorder on a 2D discrete random field induces an orthogonal decomposition of the random field into two components: A purelyindeterministic field and a deterministic one. The purelyindeterministic component is shown to have a 2D whiteinnovations driven movingaverage representation. The 2D deterministic random field can be perfectly predicted from the field's "past" with respect to the imposed total order definition. The deterministic field is further orthogonally decomposed into an evanescent field, and a remote past field. The evanescent field is generated by the columntocolumn innovations of the deterministic field with respect to the imposed nonsymmetricalhalfplane totalordering definition. The presented decomposition can be obtained with respect to any nonsymmetricalhalfplane totalordering definition, for which the nonsymmetricalhalfplane boundary line has rational slope. Corresponding author:Telephone: +9727461842, email: francos@newton.bgu.ac....
Modified Whittle Estimation of Multilateral Spatial Models
, 2003
"... We consider the estimation of parametric models for stationary spatial or spatiotemporal data on a ddimensional 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 & ..."
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We consider the estimation of parametric models for stationary spatial or spatiotemporal data on a ddimensional 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 continuousfrequency 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 discretefrequency 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 minimumdistance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed nonnegative, discretefrequency, Whittle function, without edgeeffect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almostunbiased 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 reordering as a triangular array. When the data are nonGaussian, the asymptotic variances of all parameter estimates are likely to be affected, and we provide a consistent, nonnegative definite, estimate of the asymptotic variance matrix.
Asymptotic Normality of the Sample Mean and Covariances of Evanescent Fields in Noise
, 2007
"... We consider the asymptotic properties of the sample mean and the sample covariance sequence of a field composed of the sum of a purelyindeterministic and evanescent components. The asymptotic normality of the sample mean and sample covariances is established. A Bartletttype formula for the asympto ..."
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We consider the asymptotic properties of the sample mean and the sample covariance sequence of a field composed of the sum of a purelyindeterministic and evanescent components. The asymptotic normality of the sample mean and sample covariances is established. A Bartletttype formula for the asymptotic covariance matrix of the sample covariances of this field, is derived.
1 Nouveaux modèles paramétriques 3D.
"... Résumé – Le travail présenté dans cet article se situe dans le cadre de la modélisation paramétrique des champs stochastiques tridimensionnels (3D). Nous introduisons des modèles paramétriques issus de la décomposition de Wold 3D et mettrons en évidence la structure spatiale et spectrale de la par ..."
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Résumé – Le travail présenté dans cet article se situe dans le cadre de la modélisation paramétrique des champs stochastiques tridimensionnels (3D). Nous introduisons des modèles paramétriques issus de la décomposition de Wold 3D et mettrons en évidence la structure spatiale et spectrale de la partie évanescente d’une texture 3D. Abstract – This paper deal with the multidimensional (3D) parametric model of stochastic processes. We introduce a new models and new results based on 3D Wold decomposition. By this way we can provide both spatial and spectral properties of the evanescent component of 3D texture. 1.