发表机构
Université Paris Cité; CNRS（巴黎西岱大学; 法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带噪声离散观测的函数型线性回归，提出基于傅里叶重建和惩罚最小二乘的两步估计法，在正则性条件下达到极小极大最优速率，并用模拟和气象数据验证。
AI 中文摘要
本文考虑在一种现实采样方案下的标量对函数线性回归模型，其中函数型协变量在规则网格上观测并受到加性噪声污染。我们提出一个两步估计程序：首先，利用基于傅里叶的投影方法从离散噪声观测中重建潜在曲线；其次，在有限维三角空间上通过惩罚最小二乘准则估计斜率函数，并采用数据驱动的方式选择模型维度。我们建立了预测误差的oracle型不等式，分别针对重建曲线和真实潜在曲线。在斜率函数的正则性假设及协变量特征值多项式衰减条件下，我们推导了预测误差的收敛速率，并表明当网格点数量足够大时，我们的估计量达到极小极大最优速率。最后，所提方法在模拟数据和真实气象数据集上进行了说明。
英文摘要
In this paper, we consider the scalar-on-function linear regression model under a realistic sampling scheme in which the functional covariates are observed on a regular grid and contaminated by additive noise. We propose a two-step estimation procedure: first, the underlying curves are reconstructed from the discrete noisy observations using a Fourier-based projection method; second, the slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces, with data-driven selection of the model dimension. We establish oracle-type inequalities for the prediction error, both with respect to the reconstructed curves and to the true latent curves. Under regularity assumptions on the slope function and polynomial decay of the eigenvalues of the covariate, we derive convergence rates for the prediction error and show that our estimator attains the minimax rate when the number of grid points is sufficiently large. Finally, the proposed method is illustrated on simulated data and on a real meteorological dataset.
Comments63 pages, 25 figures
