Multivariate nonparametric regression by least squares Jacobi polynomials approximations
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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 𝒰=[0,1]^d, with respect to the associated multivariate Jacobi weights. In particular, if we consider n random sampling points 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 L^2-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 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 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.
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