Multivariate nonparametric regression by least squares Jacobi polynomials approximations

2022/02/02 by Asma BenSaber, Sophie Dabo-Niang, BenSaber, Asma +4
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Mathematical functions and polynomials #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.2202.01283

25 pages, 5 figures

arxiv created 2022/02/02 · openalex publication_date 2022/02/02 · arxiv updated 2022/02/04 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

In this work, we study a random orthogonal projection based least squares estimator for the stable solution of a multivariate nonparametric regression (MNPR) problem. More precisely, given an integer d≥ 1 corresponding to the dimension of the MNPR problem, a positive integer N≥ 1 and a real parameter α≥ -(1)/(2), we show that a fairly large class of d-variate regression functions are well and stably approximated by its random projection over the orthonormal set of tensor product d-variate Jacobi polynomials with parameters (α,α). The associated uni-variate Jacobi polynomials have degree at most N and their tensor products are orthonormal over \mathcal U=[0,1]^d, with respect to the associated multivariate Jacobi weights. In particular, if we consider n random sampling points \mathbf X_i following the d-variate Beta distribution, with parameters (α+1,α+1), then we give a relation involving n, N, α to ensure that the resulting (N+1)^d× (N+1)^d random projection matrix is well conditioned. Moreover, we provide squared integrated as well as L²-risk errors of this estimator. Precise estimates of these errors are given in the case where the regression function belongs to an isotropic Sobolev space H^s(I^d), with s> (d)/(2). Also, to handle the general and practical case of an unknown distribution of the \mathbf X_i, we use Shepard's scattered interpolation scheme in order to generate fairly precise approximations of the observed data at n i.i.d. sampling points \mathbf X_i following a d-variate Beta distribution. Finally, we illustrate the performance of our proposed multivariate nonparametric estimator by some numerical simulations with synthetic as well as real data.

Related

• A Stable Jacobi polynomials based least squares regression estimator associated with an ANOVA decomposition model
• Random matrices based schemes for stable and robust nonparametric and functional regression estimators
• On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression
• A Computational Framework for Multivariate Convex Regression and its Variants
• Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function