K-Nearest Neighbor Estimation of Functional Nonparametric Regression Model under NA Samples

Xueping Hu,Liuliu Wang,Jingya Wang,Keming Yu

doi:10.3390/axioms11030102

Xueping Hu, Liuliu Wang + Show 2 more

Open Access

https://doi.org/10.3390/axioms11030102

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Journal: Axioms	Publication Date: Feb 25, 2022
Citations: 4	License type: CC BY 4.0

Affiliation: Anqing Normal University, Brunel University London

Abstract

Functional data, which provides information about curves, surfaces or anything else varying over a continuum, has become a commonly encountered type of data. The k-nearest neighbor (kNN) method, as a nonparametric method, has become one of the most popular supervised machine learning algorithms used to solve both classification and regression problems. This paper is devoted to the k-nearest neighbor (kNN) estimators of the nonparametric functional regression model when the observed variables take values from negatively associated (NA) sequences. The consistent and complete convergence rate for the proposed kNN estimator is first provided. Then, numerical assessments, including simulation study and real data analysis, are conducted to evaluate the performance of the proposed method and compare it with the standard nonparametric kernel approach.

Highlights

Functional data analysis (FDA) is a branch of statistics that analyzes data providing information about curves, surfaces or anything else varying over a continuum
We focus on the k-nearest neighbors
Whereas k-nearest neighbor (kNN) regression under negatively associated (NA) sequences has not been explored in the literature, in this paper, we extend the kNN estimation of functional data from the case of independent samples to NA sequences

Summary

Introduction

Functional data analysis (FDA) is a branch of statistics that analyzes data providing information about curves, surfaces or anything else varying over a continuum. M(·) is the unknown regression operator from SF to R, and the random error satisfies E( |χ) = 0, a.s. For the estimation of model (1), Ferraty and Vieu [5] investigated the classical functional Nadaraya-Watson (N-W) kernel type estimator of m(·) and obtained the asymptotic properties with rates in the case of α-mixing functional data. The papers [11,12,13,14,15,16,17,18], and others, obtained the asymptotic behavior of nonparametric regression estimators for functional data in independent and dependent cases. Kudraszow and Vieu [19] obtained asymptotic results for a kNN generalized regression estimator when the observed variables take values in an abstract space.

Assumptions and Main Results

A Real Study

Proof of Theorem

Conclusions and Future Research