部分函数型数据的回归模型与应用

Regression Models and Applications for Partially Functional Data

摘要: 随着科技的不断发展，各种类型的数据不断涌现，与传统的离散数据相比，函数型数据可以更好地反映出数据的连续性和变化趋势。在实际应用中，我们通常需要根据函数型数据的特征来建立模型。针对响应变量为标量型，预测变量包含函数型和标量型的情况，传统的线性模型已经不能满足需求，为此，本文提出了一种广义部分函数型线性模型，它不仅可以处理标量型和函数型的复合变量，还能够通过预测变量和响应变量之间的关系来估计连接函数，更加符合实际情况。具体来说，本文采用函数型主成分分析方法对函数型变量进行降维，然后利用最大似然估计法对回归系数进行估计，通过局部线性回归的方法对连接函数进行估计，这样我们就可以得到最终的估计值，从而实现对复合变量的建模和预测。为了验证该模型的有效性，本文将其应用于人口密度的研究中。结果表明，该模型可以有效地反应预测变量和响应变量之间的关系，得到了符合预期的结论。这项研究也使得广义部分函数型线性模型有了进一步的应用推广。

Abstract: With the development of technology, various types of data are emerging. Compared with traditional discrete data, functional data can better reflect the continuity and trend of data. In practical appli-cations, we usually need to establish models based on the features of functional data. For the case where the response variable is scalar and the predictor variable contains both functional and scalar variables, traditional linear models cannot meet the requirements. To address this issue, this paper proposes a generalized partially functional linear model, which can not only handle scalar and func-tional composite variables but also estimate the link function through the relationship between the predictor and response variables, making it more practical. Specifically, this paper uses the func-tional principal component analysis method to reduce the dimension of functional variables, and then estimates the regression coefficients using the likelihood estimation method and the link function using the local linear regression method. In this way, we can obtain the final estimate and achieve modeling and prediction of composite variables. To verify the effectiveness of the model, this article applies it to the study of population density. The results show that the model can effec-tively reflect the relationship between the predictor and response variables and obtain expected conclusions. This study also promotes the further application and promotion of the generalized par-tially linear functional linear model.