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Dynamical local Fréchet curve regression in manifolds

M. D. Ruiz-Medina, A. Torres-Signes

TL;DR

This work tackles dynamic, nonparametric prediction of Fréchet means for time-correlated curves on manifolds. It develops two parallel pipelines: an extrinsic approach that embeds log-mapped data into a time-varying tangent Hilbert space and performs projection-based local linear regression before mapping back via the exponential, and an intrinsic approach that employs nonlinear weighted Fréchet means with proven asymptotic optimality. The authors establish the requisite geometric and probabilistic assumptions, provide explicit estimators (including spectral expansions and weighted moments), and compare methods through extensive simulations on the sphere and a real Earth’s magnetic field dataset from NASA’s MAGSAT, with NW-type Fréchet regression as a benchmark. The results demonstrate finite-sample performance gains for the intrinsic method in precision, alongside practical applicability to geophysical curve prediction, with bandwidth and manifold geometry playing key roles. The framework advances Riemannian functional data analysis by detailing both extrinsic and intrinsic local Fréchet regression for dynamical curve data on manifolds.

Abstract

The present paper solves the problem of local linear approximation of the Fréchet conditional mean in an extrinsic and intrinsic way from time correlated bivariate curve data evaluated in a manifold (see Torres et al, 2025, on global Fréchet functional regression in manifolds). The extrinsic local linear Fréchet functional regression predictor is obtained in the time-varying tangent space by projection into an orthornormal eigenfunction basis in the ambient Hilbert space. The conditions assumed ensure the existence and uniqueness of this predictor, and its computation via exponential and logarithmic maps. A weighted Fréchet mean approach is adopted in the computation of an intrinsic local linear Fréchet functional regression predictor. The asymptotic optimality of this intrinsic local approximation is also proved. The finite sample size performance of the empirical version of both, extrinsic and intrinsic local functional predictors, and of a Nadaraya-Watson type Fréchet curve predictor is illustrated in the simulation study undertaken. As motivating real data application, we consider the prediction problem of the Earth's magnetic field from the time-varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft.

Dynamical local Fréchet curve regression in manifolds

TL;DR

This work tackles dynamic, nonparametric prediction of Fréchet means for time-correlated curves on manifolds. It develops two parallel pipelines: an extrinsic approach that embeds log-mapped data into a time-varying tangent Hilbert space and performs projection-based local linear regression before mapping back via the exponential, and an intrinsic approach that employs nonlinear weighted Fréchet means with proven asymptotic optimality. The authors establish the requisite geometric and probabilistic assumptions, provide explicit estimators (including spectral expansions and weighted moments), and compare methods through extensive simulations on the sphere and a real Earth’s magnetic field dataset from NASA’s MAGSAT, with NW-type Fréchet regression as a benchmark. The results demonstrate finite-sample performance gains for the intrinsic method in precision, alongside practical applicability to geophysical curve prediction, with bandwidth and manifold geometry playing key roles. The framework advances Riemannian functional data analysis by detailing both extrinsic and intrinsic local Fréchet regression for dynamical curve data on manifolds.

Abstract

The present paper solves the problem of local linear approximation of the Fréchet conditional mean in an extrinsic and intrinsic way from time correlated bivariate curve data evaluated in a manifold (see Torres et al, 2025, on global Fréchet functional regression in manifolds). The extrinsic local linear Fréchet functional regression predictor is obtained in the time-varying tangent space by projection into an orthornormal eigenfunction basis in the ambient Hilbert space. The conditions assumed ensure the existence and uniqueness of this predictor, and its computation via exponential and logarithmic maps. A weighted Fréchet mean approach is adopted in the computation of an intrinsic local linear Fréchet functional regression predictor. The asymptotic optimality of this intrinsic local approximation is also proved. The finite sample size performance of the empirical version of both, extrinsic and intrinsic local functional predictors, and of a Nadaraya-Watson type Fréchet curve predictor is illustrated in the simulation study undertaken. As motivating real data application, we consider the prediction problem of the Earth's magnetic field from the time-varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft.
Paper Structure (14 sections, 2 theorems, 57 equations, 20 figures, 1 table)

This paper contains 14 sections, 2 theorems, 57 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Local linear Fréchet regression for Hilbert--valued response and regressors
  3. Background and assumptions
  4. Assumptions
  5. Extrinsic local linear Fréchet curve regression in manifolds
  6. Intrinsic local Fréchet curve regression
  7. Local Fréchet curve prediction based on Nadaraya-Watson (NW) type estimation in $\mathcal{M}$
  8. Intrinsic local linear Fréchet curve prediction in $\mathcal{M}$
  9. Simulation study
  10. Extrinsic local linear Fréchet curve prediction evaluated in $\mathcal{M}$
  11. Intrinsic local linear Fréchet curve predictor evaluated in $\mathcal{M}$
  12. Real data application
  13. Extrinsic local linear Fréchet prediction
  14. Supplementary material of real-data application

Key Result

Lemma 1

Under assumptions (i)-(vii), the following identities hold for the local moments introduced in equation (glm): For every $t\in \mathcal{T},$ and for $j=0,1,2,$ as $h\to \infty,$ where, for any positive natural $l,$$K_{h}^{(l)}$ has been introduced in equation (eqvi), and for certain $v\in \mathcal{T}_{x(s)}\mathcal{M}.$ Here, $\mathcal{D}(K_{h})=\max_{x,y\in \hbox{Supp}(K_{h})} d_{\mathcal{M}}(x

Figures (20)

  • Figure 1: Spherical curve regressor observations. Left--hand side plot, times $s=$10 (red), 20 (green), and 30 (blue). Center plot, time $s=40$ (red), $50$ (green), $60$ (blue), and right--hand side plot, time $s= 70$ (red), $80$ (green), $90$ (blue), $100$ (cyan)
  • Figure 2: Localized uniform grid of $20000\times 20000$ nodes at the left--hand side, and empirical Fréchet curve mean at the right--hand side
  • Figure 3: Spherical curve response observations generated at times $s=10$ (black), $20$ (pink), $30$ (cyan) for the left-hand-side plot, $s=40$ (black), $50$ (pink), 60 (cyan) for the center plot, $70$ (red), $80$ (green), $90$ (blue), $100$ (cyan) for the right-hand-side plot
  • Figure 4: Local Fréchet curve predictor (time-varying NW-type local approximation) in red color, and the corresponding response curve value in black color at times $s=10, 20, 30, 40, 50, 60, 70, 80, 90$
  • Figure 5: NW-type local Fréchet curve predictor. Histogram of the one-dimensional time projections values of the sample mean of the computed quadratic geodesic functional errors at the left-hand side, and the temporal empirical mean of the pointwise values of the quadratic geodesic functional errors at the right-hand side at each sampled time ($40$ sampled times are displayed)
  • ...and 15 more figures

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • proof
  • Proposition 1
  • proof