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A Note on Local Linear Regression for Time Series in Banach Spaces

Florian Heinrichs

TL;DR

This work extends local linear regression to time series valued in Banach spaces $V$ to estimate a smoothly varying mean operator $\mu:[0,1]\to V$ and its derivatives under non-stationarity. It develops the $V$-valued local linear estimator $(\hat{\mu}_{h_n}(t),\widehat{D\mu}_{h_n}(t))$ and bias-reduced Jackknife variants $\tilde{\mu}_n,\widetilde{D\mu}_n$, establishing uniform convergence rates under mild moment conditions and smoothness of $\mu$ (Fréchet differentiability). The paper compares these estimators to the Nadaraya-Watson approach, shows favorable performance for estimating $\mu$ in simulations, and demonstrates practical applications to EEG-based eye-movement reconstruction and pedestrian/bag detection in videos. The theoretical results are complemented by Monte Carlo experiments and real-data case studies, with code available for replication. Overall, the methodology enables robust kernel-based smoothing and derivative estimation for functional time series in Banach spaces, with direct applicability to neuroscience and computer vision data.

Abstract

This work extends local linear regression to Banach space-valued time series for estimating smoothly varying means and their derivatives in non-stationary data. The asymptotic properties of both the standard and bias-reduced Jackknife estimators are analyzed under mild moment conditions, establishing their convergence rates. Simulation studies assess the finite sample performance of these estimators and compare them with the Nadaraya-Watson estimator. Additionally, the proposed methods are applied to smooth EEG recordings for reconstructing eye movements and to video analysis for detecting pedestrians and abandoned objects.

A Note on Local Linear Regression for Time Series in Banach Spaces

TL;DR

This work extends local linear regression to time series valued in Banach spaces to estimate a smoothly varying mean operator and its derivatives under non-stationarity. It develops the -valued local linear estimator and bias-reduced Jackknife variants , establishing uniform convergence rates under mild moment conditions and smoothness of (Fréchet differentiability). The paper compares these estimators to the Nadaraya-Watson approach, shows favorable performance for estimating in simulations, and demonstrates practical applications to EEG-based eye-movement reconstruction and pedestrian/bag detection in videos. The theoretical results are complemented by Monte Carlo experiments and real-data case studies, with code available for replication. Overall, the methodology enables robust kernel-based smoothing and derivative estimation for functional time series in Banach spaces, with direct applicability to neuroscience and computer vision data.

Abstract

This work extends local linear regression to Banach space-valued time series for estimating smoothly varying means and their derivatives in non-stationary data. The asymptotic properties of both the standard and bias-reduced Jackknife estimators are analyzed under mild moment conditions, establishing their convergence rates. Simulation studies assess the finite sample performance of these estimators and compare them with the Nadaraya-Watson estimator. Additionally, the proposed methods are applied to smooth EEG recordings for reconstructing eye movements and to video analysis for detecting pedestrians and abandoned objects.

Paper Structure

This paper contains 9 sections, 3 theorems, 34 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Empirical Results
  4. Monte Carlo Simulation Study
  5. Case Study
  6. EEG-Data Analysis
  7. Video Analysis
  8. Proof of Theorem \ref{['thm:lle']}
  9. Additional Empirical Results

Key Result

Theorem 3

Let $h_n\searrow 0$ as $n\to\infty$ and define $K^*(x) = 2\sqrt{2}K(\sqrt{2}x)-K(x)$ and $\kappa_2 = \int_{-1}^1 x^2 K(x){\,\mathrm{d}} x$. Under Assumption assump, it holds and

Figures (5)

  • Figure 1: Different choices of the mean operators $\mu$. Left: $\mu_1$. Right: $\mu_2$.
  • Figure 2: Different choices of the error process ${\varepsilon}$. Left: (FAR-BM). Center: (tvBM) Right: (tvFAR1).
  • Figure 3: Reconstruction of target position based on EEG data, where the data was smoothed with different methods.
  • Figure 4: Sup norm of the residuals for various estimators.
  • Figure 5: CUSUM process of the residuals' sup norm for various estimators.

Theorems & Definitions (4)

  • Remark 2
  • Theorem 3
  • Proposition 4
  • Proposition 5