Linear-Cost Covariance Functions for Gaussian Random Fields

A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The approach is applicable to a wide range of data from physical measurements and numerical simulations. It is based on the well-known invariance of the Gaussian under linear operators, in particular differentiation. Instead of using a generic covariance function to represent data from an unknown field, the space of possible covariance functions is restricted to allow only Gaussian random fields that fulfill the homogeneous differential equation. The resulting tailored kernel functions lead to more reliable regression compared to using a generic kernel and makes some hyperparameters directly interpretable. For differential equations representing laws of physics such a choice limits realizations of random fields to physically possible solutions. Source terms are added by superposition and their strength estimated in a probabilistic fashion, together with possibly unknown hyperparameters with physical meaning in the differential operator.

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MODELS OF COVARIANCE FUNCTIONS OF GAUSSIAN RANDOM FIELDS ESCAPING FROM ISOTROPY, STATIONARITY AND NON NEGATIVITY

Image Analysis & Stereology ◽

10.5566/ias.v33.p75-81 ◽

2014 ◽

Vol 33 (1) ◽

pp. 75

Author(s):

Pablo Gregori ◽

Emilio Porcu ◽

Jorge Mateu

Keyword(s):

Image Analysis ◽

Random Fields ◽

Fourier Transforms ◽

Gaussian Random Fields ◽

Space Time ◽

Linear Combinations ◽

Covariance Functions ◽

Recent Advances ◽

Special Cases ◽

Covariance Models

This paper represents a survey of recent advances in modeling of space or space-time Gaussian Random Fields (GRF), tools of Geostatistics at hand for the understanding of special cases of noise in image analysis. They can be used when stationarity or isotropy are unrealistic assumptions, or even when negative covariance between some couples of locations are evident. We show some strategies in order to escape from these restrictions, on the basis of rich classes of well known stationary or isotropic non negative covariance models, and through suitable operations, like linear combinations, generalized means, or with particular Fourier transforms.

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Discrete sample estimation for gaussian random fields generated by stochastic partial differential equations

Communication in Statistics- Theory and Methods ◽

10.1080/03610929808828954 ◽

1998 ◽

Vol 14 (4) ◽

pp. 883-903

Author(s):

Jaroslav Mohapl

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Random Fields ◽

Stochastic Partial Differential Equations ◽

Gaussian Random Fields ◽

Partial Differential ◽

Discrete Sample

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Comparison of Nonlinear Spatial Correlation Models by the Influence of the Data Augmentation to the Classification Risk

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2002.7.1.15200 ◽

2002 ◽

Vol 7 (1) ◽

pp. 31-42

Author(s):

J. Šaltytė ◽

K. Dučinskas

Keyword(s):

Spatial Correlation ◽

Random Fields ◽

Data Augmentation ◽

Gaussian Random Fields ◽

Classification Rule ◽

Numerical Comparison ◽

First Order ◽

Bayesian Risk ◽

Correlation Models

The Bayesian classification rule used for the classification of the observations of the (second-order) stationary Gaussian random fields with different means and common factorised covariance matrices is investigated. The influence of the observed data augmentation to the Bayesian risk is examined for three different nonlinear widely applicable spatial correlation models. The explicit expression of the Bayesian risk for the classification of augmented data is derived. Numerical comparison of these models by the variability of Bayesian risk in case of the first-order neighbourhood scheme is performed.

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