Liang Weijuan, Zhang Qingzhao, Ma Shuangge
School of Statistics, Renmin University of China, Beijing, China.
Department of Statistics and Data Science, School of Economics, The Wang Yanan Institute for Studies in Economics, and Fujian Key Lab of Statistics, Xiamen University, Xiamen, China.
Comput Stat Data Anal. 2023 Oct;186. doi: 10.1016/j.csda.2023.107782. Epub 2023 May 25.
Functional data analysis has been extensively conducted. In this study, we consider a partially functional model, under which some covariates are scalars and have linear effects, while some other variables are functional and have unspecified nonlinear effects. Significantly advancing from the existing literature, we consider a model with interactions between the functional and scalar covariates. To accommodate long-tailed error distributions which are not uncommon in data analysis, we adopt the quantile technique for estimation. To achieve more interpretable estimation, and to accommodate many practical settings, we assume that the functional covariate effects are locally sparse (that is, there exist subregions on which the effects are exactly zero), which naturally leads to a variable/model selection problem. We propose respecting the "main effect, interaction" hierarchy, which postulates that if a subregion has a nonzero effect in an interaction term, then its effect has to be nonzero in the corresponding main functional effect. For estimation, identification of local sparsity, and respect of the hierarchy, we propose a penalization approach. An effective computational algorithm is developed, and the consistency properties are rigorously established under mild regularity conditions. Simulation shows the practical effectiveness of the proposed approach. The analysis of the Tecator data further demonstrates its practical applicability. Overall, this study can deliver a novel and practically useful model and a statistically and numerically satisfactory estimation approach.
功能数据分析已得到广泛开展。在本研究中，我们考虑一个部分功能模型，在此模型下，一些协变量是标量且具有线性效应，而其他一些变量是函数型的且具有未指定的非线性效应。与现有文献相比有显著进展的是，我们考虑一个函数型协变量与标量协变量之间存在交互作用的模型。为了适应数据分析中并不罕见的长尾误差分布，我们采用分位数技术进行估计。为了实现更具可解释性的估计，并适应许多实际情况，我们假设函数型协变量效应在局部是稀疏的（即存在一些子区域，在这些子区域上效应恰好为零），这自然导致了一个变量/模型选择问题。我们提出遵循“主效应、交互作用”层次结构，该结构假定如果一个子区域在交互项中有非零效应，那么它在相应的主函数型效应中也必须是非零的。为了进行估计、识别局部稀疏性并遵循层次结构，我们提出一种惩罚方法。开发了一种有效的计算算法，并在温和的正则性条件下严格建立了一致性性质。模拟结果表明了所提方法的实际有效性。对Tecator数据的分析进一步证明了其实际适用性。总体而言，本研究能够提供一个新颖且实用的模型以及一种在统计和数值上令人满意的估计方法。
