A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method.

提出了部分最小二乘回归来估计功能对功能的回归模型，其中功能响应和多个功能预测变量由具有二次和交互作用的随机曲线组成。函数对函数回归模型的直接估计通常是一个不适定的问题。为了克服该困难，实际上，通常将属于无限维空间的功能数据投影到基函数的有限维空间中。函数对函数回归模型转换为基础展开系数的多元回归模型。在所提出方法的估计阶段，通过有限维基函数展开法对功能变量进行近似。我们表明，通过函数响应，多个函数预测变量和函数预测变量的二次/交互项构造的偏最小二乘回归等效于使用函数变量的基数展开构造的偏最小二乘回归。从功能变量基础展开的偏最小二乘回归，我们为函数对函数回归模型的系数函数的偏最小二乘估计提供了一个明确的公式。因为通常没有指定模型的真实形式，所以我们提出了模型选择的正向程序。通过几次蒙特卡洛实验和两次经验数据分析，检验了该方法的有限样本性能，发现结果与现有方法相比具有优势。