# Smooth Sailing: Lipschitz-Driven Uncertainty Quantification for Spatial Association

arXiv: https://arxiv.org/abs/2502.06067 · HTML: https://mlsanity.com/paper/2502.06067/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 79 references that are in the map.
- First posted on arXiv 9 Feb 2025.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models](https://mlsanity.com/paper/1612.08468/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [SatCLIP: Global, General-Purpose Location Embeddings with Satellite Imagery](https://mlsanity.com/paper/2311.17179/) (2023)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Mission Critical -- Satellite Data is a Distinct Modality in Machine Learning](https://mlsanity.com/paper/2402.01444/) (2024)
- [Recent Advances in Domain Adaptation for the Classification of Remote Sensing Data](https://mlsanity.com/paper/2104.07778/) (2021)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [Clarabel: An interior-point solver for conic programs with quadratic objectives](https://mlsanity.com/paper/2405.12762/) (2024)
- [Inverse Density as an Inverse Problem: The Fredholm Equation Approach](https://mlsanity.com/paper/1304.5575/) (2013)
- [Nearest Neighbor Sampling for Covariate Shift Adaptation](https://mlsanity.com/paper/2312.09969/) (2023)
- [Regression coefficient estimation from remote sensing maps](https://mlsanity.com/paper/2407.13659/) (2024)
- [Consistency of common spatial estimators under spatial confounding](https://mlsanity.com/paper/2308.12181/) (2023)
- [Methodological monotheism across fields of science in contemporary quantitative research](https://mlsanity.com/paper/2208.05373/) (2022)
- [Consistent Validation for Predictive Methods in Spatial Settings](https://mlsanity.com/paper/2402.03527/) (2024)

## Built on by (most cited first, 1 of 1)

- [Wrong Model, Right Uncertainty: Spatial Associations for Discrete Data with Misspecification](https://mlsanity.com/paper/2509.01776/) (2025)

## Abstract (arXiv)

Estimating associations between spatial covariates and responses - rather than merely predicting responses - is central to environmental science, epidemiology, and economics. For instance, public health officials might be interested in whether air pollution has a strictly positive association with a health outcome, and the magnitude of any effect. Standard machine learning methods often provide accurate predictions but offer limited insight into covariate-response relationships. And we show that existing methods for constructing confidence (or credible) intervals for associations can fail to provide nominal coverage in the face of model misspecification and nonrandom locations - despite both being essentially always present in spatial problems. We introduce a method that constructs valid frequentist confidence intervals for associations in spatial settings. Our method requires minimal assumptions beyond a form of spatial smoothness and a homoskedastic Gaussian error assumption. In particular, we do not require model correctness or covariate overlap between training and target locations. Our approach is the first to guarantee nominal coverage in this setting and outperforms existing techniques in both real and simulated experiments. Our confidence intervals are valid in finite samples when the noise of the Gaussian error is known, and we provide an asymptotically consistent estimation procedure for this noise variance when it is unknown.