J. M. Angulo

Possible long-range dependence in fractional random fields ☆

Abstract Many existing stochastic models have been developed for description and analysis of Markov diffusion. This paper outlines the new concept of α -duality which lays the foundation for an extension of Markov diffusion to fractional diffusion. The theory of Riesz and Bessel potentials and the corresponding potential spaces play a key role in this new approach. We establish the existence of an important subclass of fractional random fields, namely that of fractional Riesz–Bessel motions, which extends the class of fractional Brownian motions. As a result, the scope of Markov diffusion is widened to cover random fields with long-range dependence.

Fractional Generalized Random Fields on Bounded Domains

Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.

A state-space model approach to optimum spatial sampling design based on entropy

We consider the spatial sampling design problem for a random field X. This random field is in general assumed not to be directly observable, but sample information from a related variable Y is available. Our purpose in this paper is to present a state-space model approach to network design based on Shannons definition of entropy, and describe its main points with regard to some of the most common practical problems in spatial sampling design. For applications, an adaptation of Ko et al.s (1995) algorithm for maximum entropy sampling in this context is provided. We illustrate the methodology using piezometric data from the Velez aquifer (Malaga, Spain).

Fractional Generalized Random Fields of Variable Order

Abstract We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on ℝ n . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz–Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.

Stochastic fractional-order differential models with fractal boundary conditions

A class of stochastic fractional-order differential models with homogeneous boundary conditions on a fractal set is introduced. The corresponding solution class satisfies a weak-sense Markov condition with respect to domains with fractal boundary. Some examples are given which provide a stochastic version of fractal drums.

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.

Scaling limit solution of a fractional Burgers equation

A fractional version of the heat equation, involving fractional powers of the negative Laplacian operator, with random initial conditions of exponential type, is introduced. Two cases are considered, depending on whether the Hopf-Cole transformation of such random initial conditions coincides, in the mean-square sense, with the gradient of the fractional Riesz-Bessel motion introduced in Anh et al. (J. Statist. Plann. Inference 80 (1999) 95-110), or with a quadratic function of such a random field. The scaling limits of the random fields defined by the Hopf-Cole transformation of the solutions to the fractional heat equation introduced in the two cases considered are then calculated via their spectral representations.

Optimal Spatial Sampling Design in a Multivariate Framework

The problem of spatial sampling design for estimating a multivariate random field from information obtained by sampling related variables is considered. A formulation assigning different degrees of importance to the variables and locations involved is introduced. Adopting an entropy-based approach, an objective function is defined as a linear combination in terms of the amount of information on the variables and/or the locations of interest contained in the data. In the multivariate Gaussian case, the objective function is obtained as a geometric mean of conditional covariance matrices. The effect of varying the degrees of importance for the variables and/or the locations of interest is illustrated in some numerical examples.

A study on sampling design for optimal prediction of space–time stochastic processes

Abstract. Optimal selection of sampling strategies is considered for the prediction of spatio-temporal processes in a state-space-model framework. General conditions are assumed in relation to the basic elements of the problem: modelling space-time interaction, formulating prediction objectives, defining the type and structure of sampling configurations, and formulating optimality criteria. An empirical study, involving a diversity of cases selected within two different examples, is carried out with the aim of illustrating some aspects of interest inherent to the problem considered, with special emphasis on highlighting the important effect of the space-time interaction structure on the ratios of information associated with different possible sampling configurations.

Semi-parametric statistical approaches for space-time process prediction

The problem of estimation and prediction of a spatial-temporal stochastic process, observed at regular times and irregularly in space, is considered. A mixed formulation involving a non- parametric component, accounting for a deterministic trend and the effect of exogenous variables, and a parametric component representing the purely spatio-temporal random variation is proposed. Correspondingly, a two-step procedure, first addressing the estimation of the non- parametric component, and then the estimation of the parametric component is developed from the residual series obtained, with spatial-temporal prediction being performed in terms of suitable spatial interpolation of the temporal variation structure. The proposed model formula-tion, together with the estimation and prediction procedure, are applied using a Gaussian ARMA structure for temporal modelling to space-time forecasting from real data of air pollution concentration levels in the region surrounding a power station in northwest Spain.