The usual function-on-function linear regression model depicts the association between functional variables in the whole rectangular region and the value of response curve at any point is influenced by the entire trajectory of the predictor curve. But in addition to this, there are cases where the value of the response curve at a point is only influenced by the value of the predictor curve in a subregion, such as the historical relationship and the short-term association. We will consider the restricted function-on-function regression model, where the value of response curve at any point is influenced by a subtrajectory of the predictor. We have two major purposes. First, we propose a novel estimation procedure that is more accurate and computational efficient for the restricted function-on-function model with a given subregion. Second, as the subregion is seldom specified in practice, we propose a subregion selection procedure that can lead to models with better interpretation and predictive performance. Algorithms are developed for both model estimation and subregion selection.

中文翻译：

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受限函数对函数线性回归模型

通常的函数对函数线性回归模型描述了整个矩形区域中函数变量之间的关联，响应曲线在任一点的值都受到预测曲线整个轨迹的影响。但除此之外，还有一些情况下，某个点的响应曲线值仅受某个子区域预测曲线值的影响，例如历史关系和短期关联。我们将考虑受限函数对函数回归模型，其中任何点的响应曲线值都受预测变量的子轨迹影响。我们有两个主要目的。首先，我们提出了一种新的估计程序，该程序对于给定子区域的受限函数对函数模型更准确且计算效率更高。第二，由于在实践中很少指定子区域，我们提出了一个子区域选择程序，可以使模型具有更好的解释和预测性能。为模型估计和子区域选择开发了算法。