Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-adaptive functional Gaussian Process regression

MD Ruiz-Medina, AE Madrid, A Torres-Signes, JM Angulo

Abstract

This paper proposes a new formulation of functional Gaussian Process regression in manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification of these measures with infinite-product Gaussian measures is then obtained via the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectra constitute the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application undertaken illustrate the finite sample and asymptotic properties of the proposed functional regression predictor.

Time-adaptive functional Gaussian Process regression

Abstract

This paper proposes a new formulation of functional Gaussian Process regression in manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification of these measures with infinite-product Gaussian measures is then obtained via the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectra constitute the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application undertaken illustrate the finite sample and asymptotic properties of the proposed functional regression predictor.
Paper Structure (14 sections, 2 theorems, 34 equations, 13 figures)

This paper contains 14 sections, 2 theorems, 34 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian measures in separable Hilbert spaces
  4. Time-varying angular spectral analysis in manifolds of FGPs
  5. Notation and background for time-adaptive EBFGP regression
  6. The FGP regression methodology
  7. Sequential Empirical Bayes approach for adaptive FGP regression
  8. Conditional conjugate infinite-dimensional Gaussian families
  9. Functional bias and variance analysis
  10. Simulation study
  11. Subfamily 1
  12. Subfamily 2
  13. Synthetic data analysis
  14. Final comments

Key Result

Theorem 1

Let $a\in \mathbb{H}$, and $\mathcal{R}\in L_{1}^{+}(\mathbb{H})$. There exists a unique Gaussian probability measure $\mu_{a,\mathcal{R}}$ on $(\mathbb{H},\mathcal{B}(\mathbb{H}))$, with $\mathcal{B}(\mathbb{H})$ being the Borel $\sigma$-algebra on the separable Hilbert space $\mathbb{H}$, such tha This measure $\mu_{a,\mathcal{R}}$ is the restriction to $\mathbb{H}$ (isometrically identified in

Figures (13)

  • Figure 1: Informative priors for hyperparameters characterizing spatiotemporal covariance function subfamilies 1 and 2. Regularity hyperparameter priors for $\gamma$ (top-left), $\nu$ (top-center) and $\varpi$ (top-right). Memory and noise hyperparameter priors for $\beta$ (bottom-left), $\alpha$ (bottom-center), and $\sigma$ (bottom-right)
  • Figure 2: Subfamily 1. Realization of conditional FGP spherical functional time series model (left-hand side), spherical functional posterior mean (center), and observation spherical functional time series model (right-hand side), $T_{\mathbb{T}}=\left\{1, 11, 21, 31, 41, 51, 61, 71, 81, 91\right\}\subset \mathbb{T}$
  • Figure 3: Subfamily 1. Time-varying EMQEs. $T=50, N=500, TR=\log(T)\simeq 4, M=100, R=200$ (top-left); $T=300, N=150, TR=\log(T)\simeq 6, M=50, R=400$ (top-right); $T=300, N=200, TR= \log(T)\simeq 6, M=50, R=400$ (center-left); $T=500, N=50, TR= \log(T)\simeq 6, M=100, R=400$, (center-right); $T=300, N=200, TR=[T^{\varrho}]_{-}, \varrho=1/2.45, M=50, R=400$, (bottom-left); $T=300, N=250, TR=[T^{\varrho}]_{-}, \varrho=1/2.45, M=50, R=400$ (bottom-right)
  • Figure 4: Subfamily 1. Theoretical and estimated (posterior) time linear correlation of the projected FGP model, at the eigenspaces specified by the logarithmic and power-function truncation schemes analyzed. Blue dotted line for the theoretical model, and red dotted line for the posterior model. $T=50, N=500, TR=\log(T)\simeq 4, M=100, R=200$ (top-left); $T=50, N=50, TR=\log(T)\simeq 4, M=50, R=400$ (top-right); $T=50, N=50, TR=\log(T)\simeq 4, M=50, R=500$ (center-left); $T=50, N=500, TR=6, M=100, R=200$ (center-right); $T=50, N=50, TR=6, M=50, R=400$ (bottom-left); $T=500, N=50, TR=6, M=100, R=400$ (bottom-right)
  • Figure 5: Subfamily 1. Empirical mean of the bias over the $R$ replicates. $T=500, N=50, TR=\log(T)\simeq 6, M=100, R=400$ (top-left); $T=500, N=150, TR=\log(T)\simeq 6, M=50, R=200$ (top-right); $T=300, N=150, TR=\log(T)\simeq 6, M=50, R=400$ (bottom-left); $T=300, N=200, TR=[T^{\varrho}]_{-}= 10, \varrho=1/2.45, M=50, R=400$ (bottom-center); $T=300, N=250, TR=[T^{\varrho}]_{-}= 10, \varrho=1/2.45, M=50, R=400$ (bottom-right)
  • ...and 8 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Definition 1
  • Lemma 1