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Locally Adaptive Conformal Inference for Operator Models

Trevor Harris, Yan Liu

TL;DR

Operator models map functions to functions and demand uncertainty quantification for function-valued outputs. The authors propose Local Sliced Conformal Inference (LSCI), which builds test-specific, locally adaptive prediction sets by depth-based conformity scores under local exchangeability, rather than relying on global exchangeability. They develop Φ-depths, local scoring, slice normalization, and a theory bounding the coverage gap, and demonstrate tighter, more adaptive sets on synthetic GP tasks and real spatiotemporal datasets with robustness to bias and certain covariate shifts. The work delivers distribution-free, finite-sample guarantees for operator-model uncertainty and offers practical benefits for spatiotemporal forecasting and physics emulation.

Abstract

Operator models are regression algorithms between Banach spaces of functions. They have become an increasingly critical tool for spatiotemporal forecasting and physics emulation, especially in high-stakes scenarios where robust, calibrated uncertainty quantification is required. We introduce Local Sliced Conformal Inference (LSCI), a distribution-free framework for generating function-valued, locally adaptive prediction sets for operator models. We prove finite-sample validity and derive a data-dependent upper bound on the coverage gap under local exchangeability. On synthetic Gaussian-process tasks and real applications (air quality monitoring, energy demand forecasting, and weather prediction), LSCI yields tighter sets with stronger adaptivity compared to conformal baselines. We also empirically demonstrate robustness against biased predictions and certain out-of-distribution noise regimes.

Locally Adaptive Conformal Inference for Operator Models

TL;DR

Operator models map functions to functions and demand uncertainty quantification for function-valued outputs. The authors propose Local Sliced Conformal Inference (LSCI), which builds test-specific, locally adaptive prediction sets by depth-based conformity scores under local exchangeability, rather than relying on global exchangeability. They develop Φ-depths, local scoring, slice normalization, and a theory bounding the coverage gap, and demonstrate tighter, more adaptive sets on synthetic GP tasks and real spatiotemporal datasets with robustness to bias and certain covariate shifts. The work delivers distribution-free, finite-sample guarantees for operator-model uncertainty and offers practical benefits for spatiotemporal forecasting and physics emulation.

Abstract

Operator models are regression algorithms between Banach spaces of functions. They have become an increasingly critical tool for spatiotemporal forecasting and physics emulation, especially in high-stakes scenarios where robust, calibrated uncertainty quantification is required. We introduce Local Sliced Conformal Inference (LSCI), a distribution-free framework for generating function-valued, locally adaptive prediction sets for operator models. We prove finite-sample validity and derive a data-dependent upper bound on the coverage gap under local exchangeability. On synthetic Gaussian-process tasks and real applications (air quality monitoring, energy demand forecasting, and weather prediction), LSCI yields tighter sets with stronger adaptivity compared to conformal baselines. We also empirically demonstrate robustness against biased predictions and certain out-of-distribution noise regimes.

Paper Structure

This paper contains 45 sections, 1 theorem, 48 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Local exchangeability
  4. Conformal inference
  5. Adaptive conformal inference
  6. local sliced Conformal Inference
  7. $\Phi$-depth
  8. Data depth.
  9. $\Phi$-depths.
  10. Central regions.
  11. Projection class.
  12. Method
  13. Local $\Phi$-scoring.
  14. Slice variance normalization.
  15. Local Conformal inference sets
  16. ...and 30 more sections

Key Result

Proposition 3.1

Let $d:\mathcal{F}\times\mathcal{F}\to [0, 1]$ be a bounded pre-metric and suppose the residual process is locally exchangeable (Section subsec:local-exch-background). Then

Figures (4)

  • Figure 1: Residual functions from a neural operator model applied to energy demand (Section \ref{['sec:real_experiments']}). (Left) Residuals vary smoothly across inputs, showing nonstationary amplitude and shape. (Middle / Right) Local residual covariance structures at two distant inputs. Their anisotropy and orientation differ substantially, illustrating the failure of global exchangeability and the need for geometrically adaptive local conformal methods.
  • Figure 2: LSCI empirical coverage ($\alpha=0.1$) on homoskedastic regression across many $H$–$\varphi$ and $\lambda$-$M$ localization settings. Coverage in either case not strongly impacted by localization.
  • Figure 3: a. Constant bias $2\sin(4\pi t)$, re-normed to the given bias level, added to each prediction. b. Conditional bias $2c\Vert f\Vert_2 \sin(4\pi t + \Vert f\Vert_2)$. c. Local covariate shift via a moving $\sigma$ "bump" (Section \ref{['sec:simulation_details']}). d. Spectral covariate shift via a rotating $\sigma$ "spike" through the harmonics of $f$ (Section \ref{['sec:simulation_details']})
  • Figure 4: Spatial uncertainty as a function of seasonality. LSCI adapts over time to the seasonal patterns.

Theorems & Definitions (1)

  • Proposition 3.1