In functional data analysis, functional linear regression has attracted significant attention recently. Herein, we consider the case where both the response and covariates are functions. There are two available approaches for addressing such a situation: concurrent and nonconcurrent functional models. In the former, the value of the functional response at a given domain point depends only on the value of the functional regressors evaluated at the same domain point, whereas, in the latter, the functional covariates evaluated at each point of their domain have a non-null effect on the response at any point of its domain. To balance these two extremes, we propose a locally sparse functional regression model in which the functional regression coefficient is allowed (but not forced) to be exactly zero for a subset of its domain. This is achieved using a suitable basis representation of the functional regression coefficient and exploiting an overlapping group-Lasso penalty for its estimation. We introduce efficient computational strategies based on majorization-minimization algorithms and discuss appealing theoretical properties regarding the model support and consistency of the proposed estimator. We further illustrate the empirical performance of the method through simulations and two applications related to human mortality and bidding the energy market.
翻译:在功能性数据分析中,功能性线性回归在最近引起了很大的注意。在这里,我们考虑的是反应和共变都是功能的情况。我们提出了两种可用的办法来应对这种情况:同时的和非连续的功能模型。在前者,功能性反应在某一域点的价值仅取决于在同一域点评价的功能性回归者的价值,而在后者,在其域点评价的功能性共变体对在其域点任何点的反应都产生无结果的影响。为了平衡这两个极端情况,我们提议了一种当地稀疏的功能性回归模型,允许功能性回归系数(但并非被迫)在其域的某一子区完全为零。这是利用功能性回归系数的适当基础并利用重叠的集体-Lasso惩罚来估算的。我们采用了基于主要化-最小化算法的有效计算战略,并讨论了关于拟议估算器模型支持和一致性的理论属性。我们通过模拟和与人类死亡和能源市场招标有关的两种应用进一步说明该方法的经验性表现。