Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adaptive estimation of irregular mean and covariance functions

Steven Golovkine, Nicolas Klutchnikoff, Valentin Patilea

TL;DR

This paper addresses nonparametric estimation of the mean and covariance functions for functional data observed with measurement error at discrete times under unknown local regularity. It introduces a smoothing-first, then-estimate framework that uses a data-driven local regularity estimator $\widehat{H}_t$ to adapt bandwidths for trajectory smoothing, enabling adaptive mean $\widehat{\mu}_N^*$ and covariance $\widehat{\Gamma}_N^*$ estimators. The key contributions include a concentration bound for $\widehat{H}_t$, penalized risk-based bandwidth selection that accounts for curve dropouts, and a unified approach applicable to independent or common designs in both sparse and dense regimes; empirical results on MfBm-inspired simulations show competitive or superior performance relative to established methods. The methodology provides robust, easy-to-implement tools for FDA that leverage replication to automatically adapt to unknown trajectory regularity and heteroscedastic measurement errors. The work lays groundwork for extensions to smoother paths and potential CLTs, with practical impact for real-world irregular functional data analysis.

Abstract

Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.

Adaptive estimation of irregular mean and covariance functions

TL;DR

This paper addresses nonparametric estimation of the mean and covariance functions for functional data observed with measurement error at discrete times under unknown local regularity. It introduces a smoothing-first, then-estimate framework that uses a data-driven local regularity estimator to adapt bandwidths for trajectory smoothing, enabling adaptive mean and covariance estimators. The key contributions include a concentration bound for , penalized risk-based bandwidth selection that accounts for curve dropouts, and a unified approach applicable to independent or common designs in both sparse and dense regimes; empirical results on MfBm-inspired simulations show competitive or superior performance relative to established methods. The methodology provides robust, easy-to-implement tools for FDA that leverage replication to automatically adapt to unknown trajectory regularity and heteroscedastic measurement errors. The work lays groundwork for extensions to smoother paths and potential CLTs, with practical impact for real-world irregular functional data analysis.

Abstract

Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.

Paper Structure

This paper contains 18 sections, 4 theorems, 93 equations, 4 figures.

Table of Contents

  1. Introduction
  2. From unfeasible to feasible optimal estimators
  3. Local regularity estimator
  4. Local regularity in quadratic mean
  5. The local regularity estimation method
  6. Concentration properties of the local regularity estimator
  7. From the regularity of the process to the regularity of the trajectories
  8. Adaptive mean and covariance function estimators
  9. Adaptive optimal mean estimation
  10. Adaptive covariance function estimates
  11. The estimator on the diagonal of the covariance function
  12. Implementation aspects
  13. Empirical study
  14. Simulation design
  15. Mean estimation
  16. ...and 3 more sections

Key Result

Lemma 1

Assume $t\mapsto H_t \in (0,1)$ is twice continuously differentiable, and $t\mapsto L_t >0$ is continuous, $t\geq 0$. Then, $X$ defined in eq:X_md satisfies condition H:equivalent with local regularity $H_t$ and Hölder constant $L_t$, provided that, for some $A(0)\geq 0$, the time deformation is

Figures (4)

  • Figure 1: Estimation of the different quantities for the data generating process.
  • Figure 2: Description of the simulated data set.
  • Figure 3: Results for the estimation of $\mu$ in Experiment 1. The ratios are computed using $\text{ISE}_{0}$.
  • Figure 4: Estimation of $\Gamma$ in Experiment 1. The ratios are computed using $\text{2-ISE}_{0}$.

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Details on (\ref{['eq:opt_h_mean0c']})
  • proof : Proof of Theorem \ref{['thm:optim_mu']}
  • proof : Proof of Theorem \ref{['thm:optim_gamma']}
  • proof : Details on \ref{['Dtilde_d']}
  • proof : Proof of Lemma \ref{['lemma_mfbm']}