论文标题
最小二乘雅各比多项式近似的多元非参数回归
Multivariate nonparametric regression by least squares Jacobi polynomials approximations
论文作者
论文摘要
在这项工作中，我们研究了基于多元非参数回归（MNPR）问题的稳定解的随机正交投影量估计器。更确切地说，给定与MNPR问题维度相对应的整数$ d \ geq 1 $，正整数$ n \ geq 1 $和一个真实的参数$α\ geq- \ geq - \ frac {1} {2} {2} {2} $ $，我们表明，相当大的$ d- $变量回归功能是相当大的$ d- $ varsission函数的，它的设置既定范围均可及其稳定的投影。 $ d-$ variate jacobi多项式带有参数$（α，α）。$相关的Uni-Variate jacobi多项式最多具有$ n $的学位，并且其张量产品超过$ \ MATHCAL U = [0,1]^d，$相对于相关的多个jacobi jacobi jacobi jacobi jacobi jacobi jacobi jacobi rate。 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 $(N+1)^d\times (N+1)^d$ random projection matrix is well conditioned.此外，我们提供了该估算器的平方集成以及$ l^2- $风险错误。 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 $s> \frac{d}{2}.$ 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观察到$ n $ i.i.d.的数据采样点$ \ mathbf x_i $遵循$ d- $ variate beta分布。最后，我们通过一些具有合成数据和实际数据的数值模拟来说明我们提出的多元非参数估计器的性能。
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 $d\geq 1$ corresponding to the dimension of the MNPR problem, a positive integer $N\geq 1$ and a real parameter $α\geq -\frac{1}{2},$ 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 $(N+1)^d\times (N+1)^d$ 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 $s> \frac{d}{2}.$ 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.
