RF in Spatial Statistics

• A random field (RF): \(\{ Z(\mathbf{s}) \; : \; \mathbf{s} \in D \}\), where \(D\) is an index set.

• RF are used extensively in spatial statistics (Cressie 1993), where sample units are conceptualized as elements of an index-set \(D\).

• Inference based on RF relies on further assumptions. The usual assumptions depend heavily on the spatial structure/geometry of the observed spatial data.

Spatial sample units in practice

Multiple Spatial Resolutions

• Change of Support: Predicting a process on one spatial resolution (or scale) using data collected from a different resolution (Gelfand et al. 2001);

  • Downscaling: A specific type of change of support where coarse aggregate data is used to infer values at a finer resolution (Zheng et al. 2025).

• Spatial Data Fusion: Analyzing the same phenomenon when observations are simultaneously available at multiple resolutions (Moraga et al. 2017).

• Spatial Misalignment: Handling response and explanatory variables that are observed on different spatial resolutions (Godoy et al. 2026a).

Set-indexed Random Fields

• A set-indexed random field \(Z(\mathbf{s})\) should be well defined when \(\mathbf{s} \in D\) is not a singleton.
  • \(\{Z(\mathbf{s}) \, : \, \mathbf{s}\) \(\subset\) \(D\}\)
• Set-indexed by sets are not a completely new idea:
  • Itô calculus, Brownian sheet (Adler and Taylor 2007, Ch 1.4)
  • Stochastic integrals: \(Z(B) = v(B)^{-1} \int_B Z(\mathbf{x}) \mathrm{d} x\)

Typical Assumptions¹

• The assumptions regarding the index set \(D\) are inherited from Geostatistics.

• Realizations observed over areal units (or blocks) are an aggregation of the point-level process \(Z(\mathbf{s})\): \[ Z(B) = {\lvert B \rvert}^{-1} \int_{B} Z(\mathbf{x}) \mathrm{d}\mathbf{x}, \]

• Covariances involving aggregations are as follows: \[ \mathrm{Cov}[Z(B), Z(\mathbf{s})] = {\lvert B \rvert}^{-1} \int_{B} \mathrm{Cov}[Z(\mathbf{x}), Z(\mathbf{s})] \mathrm{d}\mathbf{x} \]

Limitations

• Since no analytical solutions are available, Monte Carlo techniques are used to approximate the covariances (Gelfand et al. 2001): \[ \begin{align} \mathrm{Cov}[Z(B), Z(\mathbf{s})] & = {\lvert B \rvert}^{-1} \int_{B} \mathrm{Cov}[Z(\mathbf{x}), Z(\mathbf{s})] \mathrm{d}\mathbf{x} \\ & \approx L^{-1} \sum_{k} \mathrm{Cov}[Z(\mathbf{s}_k), Z(\mathbf{s})] \end{align} \]

• There is no consensus in the literature about how to choose \(L\) and these approximations may introduce unquantifiable biases (Gonçalves and Gamerman 2018).

An Alternative

• A more flexible index set: The class of non-empty, closed and bounded sets in \(D\) (denoted \(\mathcal{C}_D\)).

  • \(\{Z(\mathbf{s}) \, : \, \mathbf{s} \in \mathcal{C}_D\}\) (Godoy et al. 2026b).
  • Isotropic Gaussian Process (GP) are defined with their covariance function depending on a distance between sets.

• Successful in practice: Competitive with areal models, often better than models for data fusion.

• Lacking theoretical foundation: no formal proof of validity of covariance functions, does not allow for smooth covariance functions.

Research objectives

• Derive a theoretically sound RF for spatial data, which allows for modeling areal, point-referenced, and mixed spatial data seamlessly.

• To achieve our goal, we will:

  • Define an new “appropriate” distance for sets (i.e., elements of \(\mathcal{C}_D\)).
  • Review properties of covariance functions.
  • Prove that covariance functions equipped with proposed distance yield valid RFs.

The Hausdorff distance (HD)

• Definition \[ h(A_1, A_2) = \inf \{ r \geq 0 \, : \, A_1 \subseteq {\rm B}_r(A_2), A_2 \subseteq {\rm B}_r(A_1) \}, \] where \(A_1 \subset D\) and \(A_2 \subset D\) are two non-empty sets.

• Intuition: given a reference metric space \((D, d)\), the Hausdorff distance quantifies the greatest distance one would have to travel from a point in one set to reach the other set.

• Limitations: Computationally expensive to compute (Knauer et al. 2011), no results establishing positive-definite functions of this distance.

HD, Balls & Length spaces

• Definition: A metric space \((D, d)\) is a length space if \(d(x, y)\) equals the infimum of the lengths of paths connecting \(x \in D\) and \(y \in D\) (Burago et al. 2001).

• Property: Length spaces possess approximate midpoints.

• Lemma: Let \((D, d)\) be a length space. Then balls expand linearly: \[ \rm B_{r}(\rm B_{k}(x)) = \rm B_{r + k}(x). \]

• Consequence: \(h(\rm B_r(x), \rm B_k(y)) = d(x, y) + \lvert r - k \rvert\).

Minimum Enclosing Balls

• We denote the smallest ball containing a set \(A \subset \mathcal{C}_D\) by \(\mathcal{B}(A)\).

  • Radius: \({\rm R}(A) = \inf_{x \in D} \inf \{ r \geq 0 : A \subset \rm B_r(x) \}\)

  • Set of centers: \(\mathcal{E}(A) = \{ x \in D : {\rm R}(x, A) = {\rm R}(A) \}.\)

  • Chebyshev center: \(c(A) \in \mathcal{E}(A)\)

• \(\mathcal{B}(A)\) always exists for closed and bounded sets on normed metric spaces (Garkavi 1970) and on complete manifolds (such as sphere and torus) (Burago et al. 2001).

The ball-Hausdorff distance

• Definition: Let \((D, d)\) be a length-space. Define \(\mathcal{C}_D\) as the class of non-empty, closed and bounded sets in \(D\). The ball-Hausdorff distance is defined as: \[ bh(A_1, A_2) = d(c(A_1), c(A_2)) + \lvert R(A_1) - R(A_2) \rvert, \] where \(A_1, A_2 \in \mathcal{C}_D\).

• Applied context: The class \(\mathcal{C}_D\) encompasses most types of data we encounter in Spatial Statistics:

  • Areal and point-referenced data points are closed and bounded sets (both on \(\mathbb{R}^n\) and \(\mathbb{S}^{n - 1}\))

Additional results

• Remark: On the real line, the ball-Hausdorff distance is equivalent to the Hausdorff distance.

• Remark: If we use the \(\lVert \cdot \rVert_1\) distance for sets in \(\mathbb{R}^p\), the ball-Hausdorff distance can be isometrically embedded into \((\mathbb{R}^{p + 1}, \lVert \cdot \rVert_1)\).

• Theorem: Let \((D, d)\) be a pseudometric space. An upper-bound for the ball-Hausdorff distance is given by: \[ bh(A_1, A_2) \leq d(c(A_1), c(A_2)) + \max \{R(A_1), R(A_2)\}, \] where \(A_1, A_2 \in \mathcal{C}_D\).

Comparison with Hausdorff distance

Time Comparisons

• Orders of magnitude faster than the Hausdorff distance in the scenarios examined here (5x-186x times faster).

Covariance Functions

• Covariance function (CF): \(K : D \times D \to \mathbb{R}_{+}\)

• Isotropic CFs: \(K \{ d(\mathbf{s}_1, \mathbf{s}_2) \}\).

• Positive definiteness (PD): The CF of a RF must satisfy: \(\sum_{i, j = 1}^{n} c_i c_j K\{d(\mathbf{s}_i, \mathbf{s}_j)\} \geq 0\).

• Isotropic CF of the Euclidean distance: \(\Phi_p\) is the class of valid isotropic CF on \(\mathbb{R}^p\): \(\Phi_1 \supset \Phi_2 \supset \cdots \supset \Phi_{\infty} = \bigcap_{k = 1}^{\infty} \Phi_{k}\)

• Notable members of \(\Phi_{\infty}\): Matérn and Powered Exponential.

CND functions & Embeddings

• Let \((D, d)\) and \((D^\ast, d^\ast)\) denote metric spaces.

• Isometric embedding: \(\phi \,: \, D \to D^\ast\) such that \(d^\ast(\phi(s), \phi(t)) = d(s, t)\), for any \(s, t \in D\) (Wells and Williams 1975).

• Conditionally Negative Definite Function: \(g \, : \, D \times D \to \mathbb{R}_{+}\) satisfying: \[\sum_{i = 1}^{m} \sum_{j = 1}^{m} b_i b_j g(s_i, s_j) \leq 0, \, \sum b_i = 0.\] for \(s_1, \ldots, s_m \in D\).

Schoenberg’s Theorem

• Theorem: A pseudometric space \((D, d)\) can be isometrically embedded in a Hilbert space if and only if \(d^2\) is CND.

• Consequence: Let \((D, d)\) be a metric space such that \(d\) is a CND pseudometric. Then, any function belonging to the class \(\Phi_{\infty}\) is PD on \((D, d^{1/2})\).

Theorem: ball-Hausdorff distance properties

Let \((D, d)\) be a length space where the function \(d\) is CND. Define \(\mathcal{C}_D\) as the class of non-empty, bounded sets in \(D\). Then,

1. \(bh(\cdot, \cdot)\) is CND,
2. \((\mathcal{C}_D, \sqrt{bh})\) can be embedded in a Hilbert space.

Intuition

1. Follows from the fact that CND functions form a convex cone.

2. Follows from the Theorem on CND functions and embeddings.

Corollary: PEXP covariance function

Let \((D, d)\) be a length space where the function \(d\) is CND. Then, the Powered Exponential (PEXP) covariance function \[ K(h; \, \theta) = \sigma^2 \exp \left\{ - \left( \frac{h}{\phi} \right)^{\nu} \right\} \] is a valid family on \((\mathcal{C}_D, bh)\) for \(\nu \in (0, 1]\).

Corollary: Matérn covariance function

Let \((D, d)\) be a length space where the function \(d\) is CND. Then, the Matérn covariance function \[ K(h; \, \theta) = \sigma^2 \frac{1}{2^{\nu - 1}\Gamma(\nu)} {\left(\frac{h}{\phi}\right)}^{\nu} K_{\nu} \left( \frac{h}{\phi} \right) \] is a valid family on \((\mathcal{C}_D, \sqrt bh)\).

Recap

• The ball-Hausdorff distance is based on the Hausdorff distance between minimum enclosing balls.

• Its existence is guaranteed for most scenarios that are relevant in spatial statistics applications.

• Conditions for the CND property of the distance and an algorithm for its computation have been proposed.

• Rich families of covariance functions are readily available for the (element-wise) square-root of the ball-Hausdorff distance.

Do the same properties hold for the Hausdorff distance?

• One way to assess whether a covariance function is not PD is as follows:
  1. Compute the distance matrix for your dataset (based on the distance function you are interested at)
  2. Compute the covariance matrices based on the covariance functions for a real datataset
  3. Compute the smallest eigenvalue (denoted \(\lambda_1\)) associated with those covariance matrices
  4. If \(\lambda_1 < 0\), we have strong evidence that the covariance function is not valid.

Assessment: Matérn (\(\nu = 2.5\))

Assessment: PEXP (\(\nu = 1\))

The data

Marginal ECDFs

Model

• Model: \((Y(\mathbf{s}_i) \mid X(\mathbf{s}_i), z(\mathbf{s}_i)) \sim \mathcal{N}(\alpha + \beta^\top X(\mathbf{s}_i) + z(\mathbf{s}_i), \tau^2)\).

• \(\mathbf{z} \sim \mathrm{NNGP}(\mathbf{0}, K(\cdot, \cdot))\), where

  • \(K(\mathbf{s}_i, \mathbf{s}_j; \theta) = v(\mathbf{s}_i) v(\mathbf{s}_j) \exp \{ - bh(\mathbf{s}_i, \mathbf{s}_j) / \phi \}\)
  • \(v(\mathbf{s}) = \mathbb{1}(\lvert \mathbf{s} \rvert > 0) \sigma_a + \mathbb{1}(\lvert \mathbf{s} \rvert = 0) \sigma_p\)
  • \(1 / \phi \sim Exp(\lambda_\phi)\), \(\log (\sigma_k) \sim N(0, 1)\), \(\alpha \sim N(0, 1)\) , \(\beta \sim N(0, 1)\)

• Inference using Stan

Physical vs Statistical Model

Change of Support

Summary

• Contribution:
  • A new distance for (bounded) sets;
  • Computationally efficient (and feasible) algorithm for computing the proposed distance;
  • Proved that standard isotropic covariance families (e.g., Matérn¹, powered exponential) remain valid in this generalized setting;
  • Valid set-indexed isotropic RF have the potential to simplify many statistical problems in spatial statistics.

Future Work & Limitations

• The proposed distance is a pseudometric (\(bh(A, B) = 0\) does not imply \(A \equiv B\)). However, introducing a nugget effect alleviates that problem.

• Defining cross-covariance functions in this context would be a huge deal for spatial misalignment!

• Other topics in this context:

  • How to define stationarity (when not assuming isotropy)?
  • What about anisotropy?
  • Mean-square differentiability may require more general concepts of differentiation.