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Structural adaptation via directional regularity: rate accelerated estimation in multivariate functional data

Omar Kassi, Sunny G. W. Wang

TL;DR

This work introduces directional regularity as a direction-aware notion of smoothness for multivariate functional data and shows that adapting to the maximizing regularity via a data-driven change-of-basis can accelerate convergence in smoothing and related tasks. It develops estimators for the maximizing angle and a robust identification procedure, accompanied by non-asymptotic performance guarantees. Theoretical results are complemented by a novel anisotropic simulator and empirical demonstrations, illustrating improved smoothing rates and reliable anisotropy detection. The framework offers a practical pre-processing step to exploit directional anisotropy in FDA, with broad potential in smoothing, anisotropic detection, and related multivariate tasks.

Abstract

We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence can be obtained through a change-of-basis by adapting to the directional regularity of a multivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Possible applications of the directional regularity approach are discussed, and we advocate its consideration as a standard pre-processing step in multivariate functional data analysis.

Structural adaptation via directional regularity: rate accelerated estimation in multivariate functional data

TL;DR

This work introduces directional regularity as a direction-aware notion of smoothness for multivariate functional data and shows that adapting to the maximizing regularity via a data-driven change-of-basis can accelerate convergence in smoothing and related tasks. It develops estimators for the maximizing angle and a robust identification procedure, accompanied by non-asymptotic performance guarantees. Theoretical results are complemented by a novel anisotropic simulator and empirical demonstrations, illustrating improved smoothing rates and reliable anisotropy detection. The framework offers a practical pre-processing step to exploit directional anisotropy in FDA, with broad potential in smoothing, anisotropic detection, and related multivariate tasks.

Abstract

We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence can be obtained through a change-of-basis by adapting to the directional regularity of a multivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Possible applications of the directional regularity approach are discussed, and we advocate its consideration as a standard pre-processing step in multivariate functional data analysis.
Paper Structure (20 sections, 10 theorems, 71 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 10 theorems, 71 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Directional Regularity
  3. Data setting and definition
  4. Examples
  5. Methodology
  6. Estimating equations
  7. Identification issues
  8. Correction for the remainder term
  9. Theoretical Properties
  10. Numerical Properties
  11. Anisotropic simulator
  12. Parameter settings and error measures
  13. Empirical Results
  14. Discussion
  15. Computational Considerations
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\mathbf t \in \mathcal{T}$ and assume that there exists basis vectors $(\mathbf{u_1}(\mathbf t), \mathbf{u_2}(\mathbf t)) \in \mathbb{S}$ such that $H_{\mathbf{u_1}}(\mathbf{t}) < H_{\mathbf{u_2}}(\mathbf{t})$. Moreover, suppose that the functions $L_{\mathbf{u_1}}$ and $L_{\mathbf{u_2}}$ are c

Figures (5)

  • Figure 1: In general, the worst regularity $H_1$ will be obtained on the canonical basis, leading to isotropy (here $H_1 < H_2)$. This can lead to slower rates of convergence. A change-of-basis from $(\mathbf{e_1}, \mathbf{e_2})$ to $(\mathbf{u_1}, \mathbf{u_2})$ enables one to obtain an anisotropic process. Locating the basis ($\mathbf{u_1}, \mathbf{u_2}$) is equivalent to locating the angle between $\mathbf{u_1}$ and $\mathbf{e_1}$.
  • Figure 2: Plot showing the remainder term for different angles $\alpha$. $\mathcal{F}(\alpha) \rightarrow \infty$ as $\alpha \rightarrow \pi/2$, for the sum of fBms as presented in Example \ref{['ex:fbm']} and $\Delta = M_0^{-1/4}$. Going closer to the boundary $0$ and $\pi/2$ can dramatically affect the estimation of the true angles.
  • Figure 3: Results for risk of estimated angles $\alpha$ (with correction). $N$ is the number of curves, $M_0$ the number of observed points along each curve, and $\sigma$ is the noise level.
  • Figure 4: Results for risk of estimated angles $\alpha$ (with correction). $N$ is the number of curves, $M_0$ the number of observed points along each curve, and $\sigma$ is the noise level.
  • Figure 5: Illustration of the region in which the thresholding parameter $\tau$ should fall in the case of anisotropy ($\overline H \neq \underline H$).

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Example 1: Sums and products of two fractional Brownian motions
  • Example 2: Sums and products of two independent Ornstein-Uhlenbeck processes
  • Example 3: Multi-fractional Brownian sheet
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • ...and 5 more