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Uncertainty Quantification of Spline Predictors on Compact Riemannian Manifolds

Charlie Sire, Mike Pereira

Abstract

To predict smooth physical phenomena from observations, spline interpolation provides an interpretable framework by minimizing an energy functional associated with the Laplacian operator. This work proposes a methodology to construct a spline predictor on a compact Riemannian manifold, while quantifying the uncertainty inherent in the classical deterministic solution. Our approach leverages the equivalence between spline interpolation and universal kriging with a specific covariance kernel. By adopting a Gaussian random field framework, we generate stochastic simulations that reflect prediction uncertainty. However, on compact manifolds, the covariance kernel depends on the generally unknown spectrum of the Laplace-Beltrami operator. To address this, we introduce a finite element approximation based on a triangulation of the manifold. This leads to the use of intrinsic Gaussian Markov Random Fields (GMRF) and allows for the incorporation of anisotropies through local modifications of the Riemannian metric. The method is validated using a temperature study on a sphere, where the operator's spectrum is known, and is further extended to a test case on a cylindrical surface.

Uncertainty Quantification of Spline Predictors on Compact Riemannian Manifolds

Abstract

To predict smooth physical phenomena from observations, spline interpolation provides an interpretable framework by minimizing an energy functional associated with the Laplacian operator. This work proposes a methodology to construct a spline predictor on a compact Riemannian manifold, while quantifying the uncertainty inherent in the classical deterministic solution. Our approach leverages the equivalence between spline interpolation and universal kriging with a specific covariance kernel. By adopting a Gaussian random field framework, we generate stochastic simulations that reflect prediction uncertainty. However, on compact manifolds, the covariance kernel depends on the generally unknown spectrum of the Laplace-Beltrami operator. To address this, we introduce a finite element approximation based on a triangulation of the manifold. This leads to the use of intrinsic Gaussian Markov Random Fields (GMRF) and allows for the incorporation of anisotropies through local modifications of the Riemannian metric. The method is validated using a temperature study on a sphere, where the operator's spectrum is known, and is further extended to a test case on a cylindrical surface.
Paper Structure (34 sections, 10 theorems, 78 equations, 8 figures, 3 algorithms)

This paper contains 34 sections, 10 theorems, 78 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Spline interpolation solution on compact manifolds
  3. Formulation of the spline prediction problems
  4. Analytical solution to the spline prediction problems
  5. Equivalence with kriging
  6. Finite element approximation of Gaussian Random Fields
  7. Finite element statistical model
  8. Predictive distribution of the finite element model
  9. Predictive posterior simulations
  10. Overview of the approach
  11. Simulating an intrinsic GMRF
  12. Computing the posterior expectation
  13. Generating the conditional simulations
  14. Overall simulation algorithm
  15. Anisotropy and parameter estimation
  16. ...and 19 more sections

Key Result

Proposition 1

The solution to the problem $\mathcal{P}_\tau$, which corresponds to the interpolating spline when $\tau = 0$ and to the smoothing spline when $\tau > 0$, is given by with

Figures (8)

  • Figure 1: Predictions of the SST ($^\circ C$) on Earth, illustrated in 2D using the spherical coordinates $(\theta,\phi).$$n=20$ observation points are shown as black dots. Left: true values. Middle: isotropic splines. Right : Anisotropic splines.
  • Figure 2: Comparison of the prediction errors and scores for the isotropic and anisotropic SST predictions on the Earth.
  • Figure 3: Comparison between the uncertainty quantification in the isotropic and the anisotropic SST prediction on the Earth, along the longitude closest to $170^\circ$.
  • Figure 4: Predictions on the cylinder. Left: true function. Middle: isotropic spline. Right: anisotropic spline. $n=20$ observation points are shown as black dots.
  • Figure 5: Comparison of the prediction errors and scores for the isotropic and anisotropic predictions on the cylinder.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 1: Spline interpolation problem on $\mathcal{M}$
  • Definition 2: Smoothing spline problem on $\mathcal{M}$
  • Proposition 1: Spline prediction wahbabook
  • Proposition 2: Convergence of the conditional GRF.
  • Proposition 3: Distribution of the random vector $\left(\mathbf{Z}\mid A=a\right)$.
  • Corollary 1: Distribution of the finite-element approximation of the GRF.
  • Proposition 4: Convergence of the finite element conditional GRF
  • Proposition 5: Simulation of intrinsic GMRF
  • proof
  • Proposition 6: Computation of the posterior expectation.
  • ...and 5 more