Anisotropic Gaussian Random Field

• Anisotropic Gaussian Random Fields are stochastic processes defined by direction-dependent scaling exponents and geometric metrics, enabling detailed modeling of self-similar and operator-scaling behaviors.
• They employ anisotropic function spaces such as Hölder and Besov spaces to characterize sample path regularity, ensuring accurate depiction of long-range dependence and spatial inhomogeneity.
• Applications include geostatistics, texture analysis, and environmental modeling, with methods like SPDEs and penalized complexity priors enhancing covariance parameter estimation.

An anisotropic Gaussian random field (AGRF) is a family of finite-dimensional Gaussian processes indexed over a multi-dimensional parameter space in which the covariance structure and associated sample path properties exhibit direction-dependent (anisotropic) behavior. Anisotropy is typically quantified by specifying a set of scaling exponents, a deformation matrix, or a spatially varying metric, allowing a wide range of phenomena—including self-similar, operator-scaling, spatially inhomogeneous, and locally oriented random fields—to be rigorously modeled and analyzed.

1. Fundamental Definitions and Canonical Metrics

Let $I \subset \mathbb{R}^N$ be a bounded or open index set. An $(N,d)$-Gaussian random field is a centered Gaussian process

$X = \{ X(t) \in \mathbb{R}^d : t \in I \}.$

The canonical metric associated to $X$ is

$$d(s, t) := \sqrt{\mathbb{E}[ \|X(s) - X(t)\|^2 ] }, \quad s, t \in I,$$

with $\| \cdot \|$ the usual Euclidean norm in $\mathbb{R}^d$.

Anisotropy is introduced via a geometric metric

$$\rho(s, t) = \sum_{j=1}^N |s_j - t_j|^{H_j}, \quad H_j \in (0,1],$$

which encodes the (possibly distinct) Hölder exponents in each coordinate direction. The basic anisotropy condition is the domination

$$d(s, t) \leq c \, \rho(s, t) \qquad \forall s,t \in I,$$

for some $c > 0$, which ensures that increments of $X$ are controlled in the geometry defined by the exponents $H$ (Söhl, 2012).

2. Operator Scaling and Self-Similarity: Structure and Spectra

A broad class of AGRFs is provided by operator-scaling Gaussian random fields (OSGRFs). For $d \ge 1$, let $E \in \mathbb{R}^{d \times d}$ have eigenvalues with positive real parts ($E \in \mathcal{E}^+$), and fix $H > 0$ (“Hurst index”). An OSGRF satisfies the scaling

$$\{ X(a^E x), x \in \mathbb{R}^d \} \overset{d}{=} \{ a^H X(x), x \in \mathbb{R}^d \} \;\; \forall a>0,$$

where $a^E = \exp(E \log a)$ (Clausel et al., 2013, Clausel--Lesourd et al., 2011). These fields admit harmonizable spectral representations: $$X(x) = \int_{\mathbb{R}^d} ( e^{i \langle x, \xi \rangle } - 1 )\, \rho(\xi)^{-H - \frac12 \mathrm{tr}(E)}\, \widehat{W}(d\xi ),$$ with $\rho$ a continuous, $E^T$-homogeneous pseudo-norm: $\rho(a^{E^T} \xi) = a \rho(\xi)$.

In spaces of dimension 2 or higher, AGRFs widely generalize fractional Brownian motion to fields where scaling behavior, regularity, and long-range dependence are directionally dependent, accommodating phenomena such as stretched correlation contours and non-isotropic regularity (Makogin et al., 2014, Clausel et al., 2013, Clausel et al., 2013).

3. Sample Path Regularity and Anisotropic Function Spaces

Sample path regularity of an AGRF is most precisely characterized using anisotropic Hölder or Besov spaces $B^s_{p,q}(\mathbb{R}^d, D)$, where $D$ specifies the dilation geometry: $$\|f\|_{\dot{B}^{s}_{p,q}(\mathbb{R}^2, D)} = \sum_{\ell=1}^{2} \left( \int_0^1 \| \Delta^{M_\ell}_{t e_\ell} f \|_{L^p}^q t^{-sq/\lambda_\ell - 1} dt \right)^{1/q},$$ with $(\lambda_\ell, e_\ell)$ the eigenpairs of $D$ (Clausel et al., 2013, Clausel et al., 2013). For an OSGRF $X$ with parameter $E_0$ and index $H_0$, the almost sure local Besov regularity exponent is

$$\alpha_{X, {\rm loc}}(D, p, q) = H_0 \Longleftrightarrow D=E_0,$$

i.e., the maximal (“sharpest”) smoothness is captured only when the geometry of the analysis matches that of the field (Clausel et al., 2013).

Chung-type laws of the iterated logarithm (LIL) and exact local/uniform moduli of continuity for AGRFs, including those lacking stationary increments, have been established. Set $A(x,y) = \sum_{j=1}^{k} |x_j-y_j|^{\alpha_j}$. Then, for such $X$ (with strong local nondeterminism),

$$\liminf_{r \to 0} \frac{\sup_{A(x,x_0)\le r} |X(x) - X(x_0)| }{ r (\log\log 1/r)^{-1/Q} } = K \quad \text{a.s.,} \quad Q := \sum_{j=1}^k \alpha_j^{-1}$$

(Lee et al., 2021). The correct geometric scaling is essential for sharp modulus and LIL results.

4. Covariance Structure, Parameterization, and Inference

For spatially stationary and geometrically anisotropic AGRFs on $\mathbb{R}^2$, the covariance function takes the form

$$C(h) = \sigma^2 \varphi( \sqrt{ h^T \Omega h }; \theta ),$$

where $\Omega$ is a positive-definite deformation matrix parameterized by anisotropy direction and axial ratio, and $\varphi$ is e.g. a Matérn kernel (Petrakis et al., 2012, Villazón et al., 20 Aug 2024). This structure leads to elliptical isolevel sets, with principal axes given by eigenvectors of $\Omega$.

SPDE-based modeling incorporates anisotropy via a spatially varying positive-definite matrix field $H(x)$: $$\left( \kappa^2(x) - \nabla \cdot [ H(x) \nabla ] \right)^{\alpha/2} u(x) = W(x),$$ which generalizes the Whittle-Matérn model to include both geometric and spatially inhomogeneous anisotropy (Lee et al., 2020, Berild et al., 2023, Fuglstad et al., 2013). Model parameterization can employ eigen-decomposition, “half-angle” (via vector parameterization for SPD matrices with det 1), or basis expansion for the diffusion and range parameters (Llamazares-Elias et al., 3 Sep 2024).

Recent advances exploit penalized complexity (PC) priors to regularize both the range and anisotropy, shrinking toward infinite correlation length and isotropy, ensuring practical identifiability in high-dimensional settings (Llamazares-Elias et al., 3 Sep 2024). Deep neural networks can provide statistically efficient and computationally rapid alternatives to maximum likelihood for covariance parameter estimation in large spatial datasets (Villazón et al., 20 Aug 2024).

5. Geometric and Probabilistic Level-Set Theory

The characterization of level sets and excursion geometry of AGRFs employs Minkowski tensors and integral geometry. For a smooth, stationary Gaussian field $G$, the 2nd-rank Minkowski tensor of the excursion set

$$w_1^{0,2}(\rho) = \varphi(\rho) \sqrt{\frac{2}{\pi}} (\lambda_1 \lambda_2)^{1/4} \cdots,$$

with $(\lambda_1, \lambda_2)$ eigenvalues of the gradient covariance matrix, fully encodes the principal axes, alignment, and anisotropy of the level set geometry (Klatt et al., 2021, Chingangbam et al., 2021). For Gaussian fields, higher-order Minkowski tensors can be expressed solely in terms of $w_1^{0,2}$; deviations serve as non-Gaussianity diagnostics.

Shape parameters such as the axis-ratio $\alpha, \beta$ (from the eigenvalues of averaged contour Minkowski tensors) provide coordinate-free scalar measures of anisotropy, with $\alpha=1$ indicating isotropy and $\alpha<1$ quantifying the degree of directional alignment in contour ensembles (Chingangbam et al., 2021).

6. Hitting Probabilities, Polar Sets, and Capacity Theory

For AGRFs on parameter space $I \subset \mathbb{R}^N$ with anisotropy exponents $H = (H_1,\ldots,H_N)$, introduce $Q = \sum_{j=1}^{N} 1/H_j$. A fundamental result is the upper bound for hitting probabilities: for Borel $F \subset \mathbb{R}^d$,

$$\mathbb{P}\{\exists\, t \in I: X(t) + Y(t) \in F\} \le C_2\,\mathcal{H}_{d-Q}(F),$$

where $\mathcal{H}_\alpha$ denotes the $\alpha$-dimensional Hausdorff measure and $Y$ is an independent Lipschitz perturbation (Söhl, 2012). Consequently, all sets $F$ with $\dim_{\rm H}(F) < d - Q$ are polar (almost surely not hit by $X$). This polar set threshold is sharp and extends classic isotropic results to the anisotropic regime, with analogous lower bounds expressed in terms of capacity (Hinojosa-Calleja et al., 2020).

The framework accommodates additive perturbations by independent fields with bounded Hölder regularity. The composition of the metric structure (domination by the anisotropic metric $\rho$), a uniform lower bound on eigenvalues of the local covariance, and Gaussian regularity allows sharp control over small-ball probabilities and the geometry of the field's support (Söhl, 2012, Lee et al., 2023).

7. Applications and Extensions

Anisotropic GRF theory underpins the statistical modeling of oriented textures, spatial and spatiotemporal environmental fields, astrophysical imaging, ocean mass distribution, and geostatistical inference with direction-dependent correlation. Models with spatially varying anisotropy, such as those defined by SPDEs with locally varying diffusion tensors, enable representations that adapt to environmental inhomogeneity, flow, and structural orientation (Lee et al., 2020, Berild et al., 2023, Fuglstad et al., 2013).

In texture analysis and synthesis, fields such as the Locally Anisotropic Fractional Brownian Field enable assignment of prescribed orientation at every point, employing tangent field formulations and spectral or turning-band simulation schemes (Polisano et al., 2014). In geostatistics, neural networks offer fast and flexible parameter estimation for covariance models with complex (e.g. Matérn) geometric anisotropy (Villazón et al., 20 Aug 2024).

The combination of spectral, functional analytic, and geometric approaches provides a rich toolkit to describe, identify, and simulate classically and non-classically anisotropic random functions across scientific domains.

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References

• J. Söhl, "Polar sets of anisotropic Gaussian random fields" (Söhl, 2012)
• M. Clausel and B. Vedel, "A strong optimality result for anisotropic self--similar textures" (Clausel et al., 2013); "An optimality result about sample path properties of Operator Scaling Gaussian Random Fields" (Clausel et al., 2013); "Explicit constructions of operator scaling Gaussian fields" (Clausel--Lesourd et al., 2011)
• L. Lee, Y. Xiao, "Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields" (Lee et al., 2021)
• Fuglstad et al., "Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy" (Fuglstad et al., 2013)
• Petrakis & Hristopulos, "Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields" (Petrakis et al., 2012)
• Berild & Fuglstad, "Spatially Varying Anisotropy for Gaussian Random Fields in Three-Dimensional Space" (Berild et al., 2023)
• J. Cheng, A. Schwartzman, "On critical points of Gaussian random fields under diffeomorphic transformations" (Cheng et al., 2019)
• Villazón, Alegría & Emery, "Neural Networks for Parameter Estimation in Geometrically Anisotropic Geostatistical Models" (Villazón et al., 20 Aug 2024)
• C.B. Abdalla et al., "The geometrical meaning of statistical isotropy of smooth random fields in two dimensions" (Chingangbam et al., 2021)
• G. Matheron et al., "Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors" (Klatt et al., 2021)
• D. Makogin, Y. Mishura, "Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet" (Makogin et al., 2014)
• Polisano et al., "Texture Modeling by Gaussian fields with prescribed local orientation" (Polisano et al., 2014)
• Y. Xiao et al., "Local times of anisotropic Gaussian random fields and stochastic heat equation" (Lee et al., 2023)
• J. Li, D. Wu, "Anisotropic Gaussian random fields: Criteria for hitting probabilities and applications" (Hinojosa-Calleja et al., 2020)
• V. Simpson, D. Bolin, E. Fuglstad, "A parameterization of anisotropic Gaussian fields with penalized complexity priors" (Llamazares-Elias et al., 3 Sep 2024)
• M. Clausel and B. Vedel, "A strong optimality result for anisotropic self--similar textures" (Clausel et al., 2013)