Brief Intro

• Taking into account spatial dependence possibly present in data is a foremost aspect of spatial statistics.

  • “Everything is related to everything else, but near things are more related than distant things.”

• Important for proper uncertainty quantification in GLMMs (and friends)

• Useful for spatial interpolation (more on this later)

Types of Data

• There are two broad categories of spatial data:

  • Areal
  • Point-referenced

• Models are tailored for each of these two data types.

• They differ in assumption and how they quantify the “proximity” between sample units.

Areal data

Areal data

Point-referenced data

Spatial data in practice

Challenges

• 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. 2026).

Objectives

• Develop a spatial model which:

  1. Provides an unified framework for point-referenced, areal, and even mixed spatial data
  2. Allows for modeling data at disparate spatial resolutions

• How? Adapting what is done for point-referenced data!

• Challenge: A meaningful and computationally feasible distance function between spatial units regardless of their type (e.g., point or area).

What is a Set? and Why sets?

• In mathematics, a set is simply a collection of distinct objects (elements).

• In spatial statistics, we usually work with coordinates or areas (or regions) in an index set \(D\).

• Instead of distinguishing between points and areas, we may regard them both as sets.

• How do we quantify distance between sets?

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), hard to make GPs work under this distance.

What is a Ball?

• In a metric space \((D, d)\), an open ball of radius \(r\) centered at \(x\) is: \[ \mathrm{B}_r(x) = \{ y \in D : d(x, y) < r \} \]

Balls?

Minimum Enclosing Balls (MEB)

The ball-Hausdorff Distance

• We define the distance between two sets \(A_1\) and \(A_2\) as: \[ bh(A_1, A_2) = d(c(A_1), c(A_2)) + \lvert R(A_1) - R(A_2) \rvert \]

• Intuition: Distance = (How far are the centers?) + (How different are the sizes?).

• Efficiency: Computationally much cheaper than the standard Hausdorff distance (5x to 186x faster).

Comparison with Hausdorff distance

Further theoretical justification

• For a spatial model to be valid, its covariance matrix must be positive definite.

  • This property guarantees that any linear-combination of our spatial variables has a non-negative variance.

• I proved theorems establishing conditions for the positive-definiteness of covariance functions under the ball-Hausdorff distance.

• Impact: We can define GPs directly on sets and achieve the initial research objectives

Data Fusion in California

• Fusing in situ (points) and Satellite (areal) data.

• Comparing results to Physical model (gold standard)

Statistical vs. Physical Model (1/2)

Statistical vs. Physical Model (2/2)

Potential Applications in Ecology?

• Species Distribution Models (SDMs): Fusing opportunistic sightings (points) with range maps or habitat surveys (polygons).

• Spatial misalignment: Species’ abundance (and occurrence) are really observed at the same spatial scale as environmental variables.

• Remote Sensing Integration: Moving beyond “pixel as point” assumptions to treat sensor footprints as geometric sets.

A Brief Summary

• Unified Framework: Points and areas are treated consistently as sets.

• The “Balls” Shortcut: Ball-Hausdorff distance is much faster than the classical distance between sets, while keeping the relationship between sets.

• Theoretically Sound: Proven validity of GPs for sets.

Thank you!!