Department of Population Health Sciences, Georgia State University, Atlanta, Georgia, USA.
Biometrics. 2023 Dec;79(4):3319-3331. doi: 10.1111/biom.13838. Epub 2023 Feb 27.
We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and real datasets.
我们考虑一般的非线性函数-标量（FOS）回归模型，其中函数响应以一般未知的非线性形式依赖于多个标量预测因子。现有的方法要么假设特定的模型形式（例如，加性模型），要么直接在与标量预测因子数量相等的维度的空间中估计非线性函数，这只能应用于具有少数几个标量预测因子的模型。为了克服这些缺点，受神经网络中经典通用逼近定理的启发，我们开发了一种功能通用逼近定理，可用于逼近一般的非线性 FOS 映射，并可轻松应用于功能数据分析框架。利用这个定理和利用光滑性规律，我们开发了一种新的方法来拟合一般的非线性 FOS 回归模型并进行预测。我们的新方法对模型形式没有任何特定的假设，并且避免了在与标量预测因子数量相等的维度的空间中直接估计非线性函数。通过估计一系列双变量函数，我们的方法可以应用于具有相对较多标量预测因子的模型。通过对各种模拟和真实数据集的实证研究，证明了所提出方法的良好性能。
