Spatial Functional k-Nearest-Neighbour Regression under Polynomial Dependence
Abstract
This paper investigates non-parametric regression estimation when the explanatory variable takes values in a separable Hilbert space and observations are sampled over an increasing regular spatial lattice. Under a rigorous field-to-field independence setup between covariates and errors, we explore the structural and asymptotic concentration properties of the functional k-nearest-neighbour (k-NN) estimator. By establishing a sharp pathwise deterministic sandwiching framework for the data-driven random bandwidth, we successfully decouple the local infinite-dimensional small-ball profile from the polynomial covariance decay of the neighborhood indicators. Pointwise convergence rates are derived across short-range, critical, and long-range spatial regimes, revealing a combined penalty term that reflects both covariate spatial interaction and response error memory. Furthermore, uniform consistency over compact subsets is established for unbounded sub-Gaussian error processes through metric entropy and indicator boundary shell chaining. Structured polynomial simulations confirm our theoretical rates and exemplify the precise mechanics of the spatial long-range bottleneck.
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BibTeX
@misc{bouka2026spatial,
title = {{Spatial Functional $k$-Nearest-Neighbour Regression under Polynomial Dependence}},
author = {Stéphane Bouka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02921},
eprint = {2610.02921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02921},
}
RIS
TY - UNPB
TI - Spatial Functional k-Nearest-Neighbour Regression under Polynomial Dependence
AU - Stéphane Bouka
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02921
UR - https://arxiv.org/abs/2610.02921
AB - This paper investigates non-parametric regression estimation when the explanatory variable takes values in a separable Hilbert space and observations are sampled over an increasing regular spatial lattice. Under a rigorous field-to-field independence setup between covariates and errors, we explore the structural and asymptotic concentration properties of the functional k-nearest-neighbour (k-NN) estimator. By establishing a sharp pathwise deterministic sandwiching framework for the data-driven random bandwidth, we successfully decouple the local infinite-dimensional small-ball profile from the polynomial covariance decay of the neighborhood indicators. Pointwise convergence rates are derived across short-range, critical, and long-range spatial regimes, revealing a combined penalty term that reflects both covariate spatial interaction and response error memory. Furthermore, uniform consistency over compact subsets is established for unbounded sub-Gaussian error processes through metric entropy and indicator boundary shell chaining. Structured polynomial simulations confirm our theoretical rates and exemplify the precise mechanics of the spatial long-range bottleneck.
ER -