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Global Fréchet regression from time correlated bivariate curve data in manifolds

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

Abstract

Global Fréchet regression is addressed from the observation of a strictly stationary bivariate curve process, evaluated in a finite--dimensional compact differentiable Riemannian manifold, with bounded positive smooth sectional curvature. The involved univariate curve processes respectively define the functional response and regressor, having the same Fréchet functional mean. The supports of the marginal probability measures of the regressor and response processes are assumed to be contained in a ball, whose radius ensures the injectivity of the exponential map. This map has time--varying origin at the common marginal Fréchet functional mean. A weighted Fréchet mean approach is adopted in the definition of the theoretical loss function. The regularized Fréchet weights are computed, in the time--varying tangent space from the log--mapped regressors. Under these assumptions, and some Lipschitz regularity sample path conditions, when a unique minimizer exists, the uniform weak--consistency of the empirical Fréchet curve predictor is obtained, under mean--square ergodicity of the log--mapped regressor process in the first two moments. A simulated example in the sphere illustrates the finite sample size performance of the proposed Fréchet predictor. Predictions in time of the spherical coordinates of the magnetic field vector are obtained from the time--varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft in the real--data example analyzed.

Global Fréchet regression from time correlated bivariate curve data in manifolds

Abstract

Global Fréchet regression is addressed from the observation of a strictly stationary bivariate curve process, evaluated in a finite--dimensional compact differentiable Riemannian manifold, with bounded positive smooth sectional curvature. The involved univariate curve processes respectively define the functional response and regressor, having the same Fréchet functional mean. The supports of the marginal probability measures of the regressor and response processes are assumed to be contained in a ball, whose radius ensures the injectivity of the exponential map. This map has time--varying origin at the common marginal Fréchet functional mean. A weighted Fréchet mean approach is adopted in the definition of the theoretical loss function. The regularized Fréchet weights are computed, in the time--varying tangent space from the log--mapped regressors. Under these assumptions, and some Lipschitz regularity sample path conditions, when a unique minimizer exists, the uniform weak--consistency of the empirical Fréchet curve predictor is obtained, under mean--square ergodicity of the log--mapped regressor process in the first two moments. A simulated example in the sphere illustrates the finite sample size performance of the proposed Fréchet predictor. Predictions in time of the spherical coordinates of the magnetic field vector are obtained from the time--varying geocentric latitude and longitude of the satellite NASA's MAGSAT spacecraft in the real--data example analyzed.
Paper Structure (13 sections, 2 theorems, 53 equations, 22 figures, 1 table)

This paper contains 13 sections, 2 theorems, 53 equations, 22 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fréchet regression in metric spaces with finite-dimensional Euclidean regressors
  4. Fréchet regression under dependent curve data in Riemannian manifolds
  5. Model assumptions
  6. Regularization of infinite--dimensional Fréchet weights
  7. Formulation of the theoretical and empirical loss functions
  8. Weak-consistency of the predictor
  9. Numerical examples
  10. Real--data example
  11. Final Comments
  12. Proof of Theorem \ref{['th1']}
  13. Analysis of the remaining months December, 1979--May, 1980

Key Result

Theorem 1

Under conditions (i)--( v) in Section s31, and conditions (a)--(b) in Section s32, if assumptions A.1 and C.1 hold, then, for each $x(\cdot )\in \mathcal{X}_{\mathcal{C}_{\mathcal{M}}(\mathcal{T})}.$ Additionally, under assumption B.1 ,

Figures (22)

  • Figure 1: Spherical curve values of the generated regressor process at times $s_{i}=10, 20, 30, 40, 50, 60,70 ,80, 90,100.$ Three sampling times are represented on the left and center plots, and four on the right plot.
  • Figure 2: Uniform spherical grid with $400$ nodes (left--hand side), the empirical curve Fréchet mean $\mu_{\widehat{X}_{0},\mathcal{M}}(\cdot)$ (center), and the response in the time--varying tangent space at times $s_{i}=10,20,30,40,50,60,70,80,90,100$ (right--hand side).
  • Figure 3: Spherical curve values of the generated spherical curve response process at times $s_{i}=10, 20, 30, 40, 50, 60,70 ,80, 90,100.$
  • Figure 4: Joint representation of spherical curve values of the generated spherical curve response and regressor processes at times $s_{i}=10, 20, 30, 40, 50, 60,70 ,80, 90,100.$ Here, red, green, blue, and white colors are used for regressor spherical curve values, while black, magenta, cyan, and yellow colors are used for response spherical curve values.
  • Figure 5: The empirical Fréchet weights are plotted. The dashed lines displayed correspond to their evaluation from generated log--mapped sample curve regressor values at $s_{i},$$i=1,\dots,100,$ and for $40$ log--mapped $\mathcal{M}$--curve arguments of the predictor.
  • ...and 17 more figures

Theorems & Definitions (9)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 5
  • Theorem 1
  • Corollary 1
  • Remark 6
  • proof