Anisotropic Gaussian Random Fields Research Articles

Higher-order derivative of local times for space–time anisotropic Gaussian random fields

Let X={X(t),t∈RN} be a centered space–time anisotropic Gaussian random field values in Rd. Under some general conditions, the existence and smoothness (in the sense of Meyer-Watanabe) of the higher-order derivative of the local times of X(t) are studied. Moreover, we show that the derivatives of the local time of X(t) is jointly continuous on Rd×[0,1]N. The existing results on local times of fractional Brownian motion and other Gaussian random fields are extended to higher-order derivative of local times of more general space–time anisotropic Gaussian random fields.

Gaussian random field-based characterization and reconstruction of cancellous bone microstructure considering the constraint of correlation structure

The macro scale physical properties of cancellous bone materials are governed by the microstructural features, which is of great significance for the multi-scale research of cancellous bone and the inverse design of bone-mimicking materials. Therefore, it is essential to characterize the natural cancellous bone samples, and reconstruct the microstructures with the biomimetic osteointegration and mechanical properties. In this research, a novel approach for the characterization and reconstruction of cancellous bone was proposed, based on the medical image analysis and anisotropic three-dimensional Gaussian random field (GRF). The geometric similarity, i.e. the interface curvature distribution (ISD), was meticulously studied, which is important to the osteointegration ability. And the mechanical properties were validated by the stress-strain curves under the large compressive strain simulated by the smoothed particle hydrodynamic (SPH) method. In addition, the effects of the generation parameters of GRF-based biomimetic microstructures on the apparent properties were analyzed. The ISD results demonstrated that both GRF and micro-CT groups had the similar columnar morphological properties, while the latter had more hyperbolic features. And it was found that the GRF-based biomimetic microstructures and the natural bone samples based on micro-CT (MCT) had the similar failure mode. The concordance correlation coefficient between MCT and GRF pairs was 0.8685, with a Pearson ρ value of 0.8804, and significance level p<0.0001. The Bland-Altman LoA was 0.1647 MPa with 95 % (1.96SD) lower and upper bound value between −0.2892 and 0.6185 MPa. The two groups had almost the same elastic modulus with the mean absolute percentage error (MAPE) of 7.84 %. While the yield stress and total conversion energy of the GRF-based samples were lower than those of the natural bone samples, and the MAPE were 16.99 % and 16.27 %, respectively. Although it meant the lower structural efficiency, the huge design space of this approach and advanced 3D printing technology can provide great potential for the design of orthopedic implants.

Spatially varying anisotropy for Gaussian random fields in three-dimensional space

Isotropic covariance structures can be unreasonable for phenomena in three-dimensional spaces. In the ocean, the variability of a response may vary with depth, and ocean currents may lead to spatially varying anisotropy. We construct a class of non-stationary anisotropic Gaussian random fields (GRFs) in three dimensions through stochastic partial differential equations (SPDEs), where computations are done efficiently using Gaussian Markov random field approximations. A key novelty is the parametrization of the spatially varying anisotropy through vector fields.In a simulation study, we find that simple stationary models obtain reasonable parameter estimates with a moderate number of observations and a single realization, whereas the most complex non-stationary anisotropic model requires dense observations and multiple realizations. Further, we construct a stationary and a non-stationary GRF prior for salinity in an ocean mass outside Trondheim, Norway, based on simulations from the complex numerical ocean model SINMOD. These GRF priors are then evaluated using in-situ measurements collected with an autonomous underwater vehicle. We find that the new model outperforms the stationary anisotropic GRF prior for real-time prediction of unobserved locations both in terms of root mean square error and continuous rank probability score.

Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields

We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.

Efficient generation of anisotropic N-field microstructures from 2-point statistics using multi-output Gaussian random fields

The ability to efficiently generate microstructure instances corresponding to specified 2-point statistics is a crucial capability in rigorously studying random heterogeneous materials within the Integrated Computational Materials Engineering and Materials Informatics frameworks. However, the lack of computationally efficient, statistically expressive models for achieving this transformation is a recurring roadblock in many foundational Materials Informatics challenges. In this article, we present a theoretical and computational framework for generating stationary, periodic microstructural instances corresponding to specified stationary, periodic 2-point statistics by stochastically modeling the microstructure as an N-output Gaussian Random Field. First, we illustrate how 2-point statistics can be used to parameterize anisotropic Gaussian Random Fields. Second, we derive analytic relationships between the 2-point statistics and the spatially resolved sampled microstructures, within the approximation of a N-output Gaussian Random Field. Finally, we propose the algorithms necessary to efficiently sample these fields in O(SlnS) computational complexity and while incurring O(S) memory cost. We also discuss the current limitations of the proposed framework, and its usefulness to future Materials Informatics workflows.

Characterization of anisotropic Gaussian random fields by Minkowski tensors

Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. Anisotropy in such systems can be sensitively and comprehensively characterized by the so-called Minkowski tensors (MTs) from integral geometry. Here, we analytically calculate expected MTs of arbitrary rank for the level sets of Gaussian random fields. The explicit expressions for interfacial MTs are confirmed in detailed simulations. We demonstrate how the MTs detect and characterize the anisotropy of the level sets, and we clarify which shape information is contained in the MTs of different rank. Using an irreducible representation of the MTs in the Euclidean plane, we show that higher-rank tensors indeed contain additional anisotropy information compared to a rank two tensor. Surprisingly, we can nevertheless predict this information from the second-rank tensor if we assume that the random field is Gaussian. This relation between tensors of different rank is independent of the details of the model. It is, therefore, useful for a null hypothesis test that detects non-Gaussianities in anisotropic random fields.

Anisotropic Gaussian random fields: criteria for hitting probabilities and applications

We develop criteria for hitting probabilities of anisotropic Gaussian random fields with associated canonical pseudo-metric given by a class of gauge functions. This yields lower and upper bounds in terms of general notions of capacity and Hausdorff measure, respectively, therefore extending the classical estimates with the Bessel-Riesz capacity and the $\gamma$-dimensional Hausdorff measure. We apply the criteria to a system of linear stochastic partial differential equations driven by space-time noises that are fractional in time and either white or colored in space.

Disks as Inhomogeneous, Anisotropic Gaussian Random Fields

Abstract We model astrophysical disk surface brightness fluctuations as an inhomogeneous, anisotropic, time-dependent Gaussian random field. The field locally obeys the stochastic partial differential equation of a Matérn field, which has a power spectrum that is flat at large scales and falls off as a power law at small scales. We provide a series of pedagogical examples and along the way provide a convenient parameterization for the local covariance. We then consider two applications to disks. In the first we generate an animation of a disk. In the second, by integrating over an animation of a disk, we generate synthetic light curves and show that the high frequency slope of the resulting power spectrum depends on the local covariance model. We finish with a summary and a brief discussion of other possible astrophysical applications.

Efficient Generation of Anisotropic N-Field Microstructures From 2-Point Statistics Using Multi-Output Gaussian Random Fields

The ability to efficiently generate microstructure instances corresponding to specified two-point statistics is a crucial capability in rigorously studying random heterogeneous materials within the Integrated Computational Materials Engineering and Materials Informatics frameworks. However, the lack of computationally efficient, statistically expressive models for achieving this transformation is a recurring roadblock in many foundational Materials Informatics challenges. In this article, we present a theoretical and computational framework for generating stationary, periodic microstructural instances corresponding to specified stationary, periodic two-point statistics by stochastically modeling the microstructure as an N-output Gaussian Random Field. First, we illustrate how two-point statistics can be used to parameterize anisotropic Gaussian Random Fields. Second, we derive analytic relationships between the two-point statistics and the spatially resolved sampled microstructures, within the approximation of a N-output Gaussian Random Field. Finally, we propose the algorithms necessary to efficiently sample these fields in O (S ln S) computational complexity and while incurring O (S) memory cost. We also discuss the current limitations of the proposed framework, and its usefulness to future Materials Informatics workflows.

On intersections of independent space–time anisotropic Gaussian fields

Let XH={XH(s),s∈RN1} and XK={XK(t),t∈RN2} be two independent centered space–time anisotropic Gaussian random fields taking values in Rd. In this paper, we study the existence of intersections of XH and XK. Furthermore, we determine the Hausdorff dimensions of the set of intersection times and the set of intersection points of the random fields, respectively. Our results generalize the corresponding results of Chen and Xiao (2012).

Hausdorff Measure of the Range of Space–Time Anisotropic Gaussian Random Fields

Let $$X=\{X(t)\in {{\mathbb {R}}}^d, t\in {{\mathbb {R}}}^N\}$$ be a centered space–time anisotropic Gaussian random field with stationary increments, whose components are independent but may not be identically distributed. Under certain mild conditions, we determine the exact Hausdorff measure function for the range $$X([0,1]^N)$$ . Our result extends those in Talagrand (Ann Probab 23:767–775, 1995) for fractional Brownian motion and Luan and Xiao (J Fourier Anal Appl 18:118–145, 2012) for time-anisotropic and space-isotropic Gaussian random fields.

On critical points of Gaussian random fields under diffeomorphic transformations

Let {X(t),t∈M} and {Z(t′),t′∈M′} be smooth Gaussian random fields parameterized on Riemannian manifolds M and M′, respectively, such that X(t)=Z(f(t)), where f:M→M′ is a diffeomorphic transformation. We study the expected number and height distribution of the critical points of X in connection with those of Z. As an important case, when X is an anisotropic Gaussian random field, then we show that its expected number of critical points becomes proportional to that of an isotropic field Z, while the height distribution remains the same as that of Z.

Continuity in law of some additive functionals of bifractional Brownian motion

ABSTRACTLet be a bifractional Brownain motion with indices and . We prove the continuity in law, in some anisotropic Besov spaces, with respect to H and K. Our result generalizes those obtained by Jolis and Viles [Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theor. Probab. 20(2) (2007), pp. 133–152] of the fractional Brownian motion local time and gives a new result for the generalized fractional derivatives with kernel depending on slowly varying function of the local time of . Notice that their result was generalized by Wu and Xiao [Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844] for wide class of anisotropic gaussian random fields satisfying some condition (A) which is not satisfied by . To prove our result, we use the decomposition in law of given by Lei and Nualart [A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624]. Our result is also new in the space of continuous functions.

Supervised Classification of the Scalar Gaussian Random Field Observations under a Deterministic Spatial Sampling Design

Given training sample, the problem of classifying a scalar Gaussian random field observation into one of two populations specified by different parametric mean models and common parametric covariance function is considered. Such problems are usually called as supervised classification or contextual classification problems. This paper concerns with classification procedures associated with Bayes Discriminant Function (BDF) under deterministic spatial sampling design. In the case of parametric uncertainty, the ML estimators of unknown parameters are plugged in the BDF. The actual risk and the approximation of the expected risk (AER) associated with aforementioned plug-in BDF are derived. This is the extension of the previous one to the case of complete parametric uncertainty, i.e., when all mean functions and covariance function parameters are unknown. Stationary geometrically anisotropic Gaussian random field with exponential covariance function sampled on regular 2-dimensional lattice is used for illustrative examples.

Polar Functions for Anisotropic Gaussian Random Fields

LetXbe an (N,d)-anisotropic Gaussian random field. Under some general conditions onX, we establish a relationship between a class of continuous functions satisfying the Lipschitz condition and a class of polar functions ofX. We prove upper and lower bounds for the intersection probability for a nonpolar function andXin terms of Hausdorff measure and capacity, respectively. We also determine the Hausdorff and packing dimensions of the times set for a nonpolar function intersectingX. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise, and the operator-scaling Gaussian random field with stationary increments.

On the use of the Radon transform to estimate longshore currents from video imagery

Larnier, S., Almar, R., Cienfuegos, R., Lejay, A., 2014. On the use of the Radon transform to estimate longshore currents from video imagery. In: Green, A.N. and Cooper, J.A.G. (eds.), Proceedings 13th International Coastal Symposium (Durban, South Africa), Journal of Coastal Research, Special Issue No. 70, pp. 023–028, ISSN 0749-0208.In nearshore applications, the estimation of longshore currents is of primary importance since it controls the alongshore sediment transport and coastal evolution. Direct estimation of longshore currents using in-situ instruments is difficult and costly, especially under highly energetic wave climates. Low cost remote sensing systems based on video observations constitute a promising alternative when the drifting sea foam left after the passage of breaking waves is visible. In this paper we describe a method based on longshore timestacks that necessitates less data than approaches using the full video frames. Our approach uses the Radon transform applied on the time series derived from timestacks to produce an estimate of the longshore component of nearshore surface currents detected from the foam signature in video images. The Radon transform can be used to separate the wave crests from the drifting foam part. The identification of the longshore drift is enhanced with both a temporal and a spatial filters. The corresponding sinogram from the Radon transform is computed in order to find the angle of the alongshore drifting that is further converted into the longshore component of the surface currents. For the estimation of longshore currents, our approach is first tested using synthetic timestack examples created using anisotropic Gaussian random current fields. Comparisons between estimates derived by our algorithm and manual operator detection from videos are performed showing good agreement. These videos come from a field campaign conducted in the Mataquito River mouth area in the Maule region (Chile). Field test comparisons were also made against in situ current meter from the 2008 Truc Vert experiment in Aquitaine (France). It was taking place during an energetic event.

Objective Bayesian Analysis of Geometrically Anisotropic Spatial Data

Anisotropic models are often used in spatial statistics to analyze spatially referenced data. Within a Bayesian framework we develop default priors for the anisotropic Gaussian random field model with and without including a nugget parameter accounting for the effects of microscale variations and measurement errors. We present Jeffreys priors and a reference prior and study their posterior propriety. Moreover, we obtain that the predictive distributions at ungauged locations have finite variance. We also show that the seemingly uninformative uniform prior for the anisotropy parameters, ratio and angle, yields an improper posterior. Finally, we find that the proposed priors have good frequentist properties and we illustrate our approach by analyzing two data sets for which we discuss model choice as well as predictions and uncertainty estimates.

Spatiotemporal hierarchical modelling of extreme precipitation in Western Australia using anisotropic Gaussian random fields

We discuss an approach for the statistical modelling of extreme precipitation events in South-West Australia over space and time, using a latent spatiotemporal process where precipitation maxima follow a generalised extreme value distribution. Temporal features are captured by modelling trends on the location and scale parameters. Spatial features are captured using anisotropic Gaussian random fields. Site specific explanatory variables are also incorporated. We fit several models using Bayesian inferential methods to precipitation extremes recorded at 36 weather stations around the Western Australian state capital city of Perth over the period 1907–2009. Model choice is performed using the DIC criterion. The best fitting model shows significant non-stationarity over time, with extreme precipitation events becoming less frequent. Extreme precipitation events are stronger at coastal locations, with the intensity decreasing as we head to the higher and drier areas to the North-East.

On intersections of independent anisotropic Gaussian random fields

Let \(X^H = \{ X^H (s),s \in \mathbb{R}^{N_1 } \} \) and \(X^K = \{ X^K (t),t \in \mathbb{R}^{N_2 } \} \) be two independent anisotropic Gaussian random fields with values in ℝd with indices \(H = (H_1 ,...,H_{N_1 } ) \in (0,1)^{N_1 } ,K = (K_1 ,...,K_{N_2 } ) \in (0,1)^{N_2 } \), respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let \(E_1 \subseteq \mathbb{R}^{N_1 } ,E_2 \subseteq \mathbb{R}^{N_2 } \) and F ⊂ ℝd be Borel sets. A necessary condition and a sufficient condition for \(\mathbb{P}\{ (X^H (E_1 ) \cap X^K (E_2 )) \cap F \ne \not 0\} > 0\) in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space \((\mathbb{R}^{N_1 + N_2 + d} ,\tilde \rho )\) are proved, where \(\tilde \rho \) is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.

Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields.

This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.