Partial Linear Additive Integrated Regression Models Based on Functional Gaussian Processes

Abstract

In this paper, proposing a partially linear additive Gaussian process-based integration model for regression problems containing functional-type and vector-valued predictor variables. The model downscales high-dimensional functional data by functional principal component analysis, extracts low-dimensional principal component scores, and constructs additive Gaussian process sub-models, i.e., the non-linear relationship between functional predictor variables and response variables is fitted by independent Gaussian process components, and the vector-valued predictor variables are selected by the linear portion of Lasso regularisation for variable selection. To enhance the model generalisation ability, the predictions of each sub-model are adaptively fused using the Stacking integration strategy. Simulation experiments show that the proposed method can effectively identify irrelevant vector-type variables and significantly outperforms the traditional functional linear model in terms of prediction error; real data analysis further validates its competitiveness in prediction tasks. In addition, the framework proposed in this paper is highly scalable: the feature representation of functional variables can be optimized by introducing multi-kernel learning, and the computational complexity can be reduced by combining with sparse Gaussian process, which can be extended to dynamic data flow modelling or combined with deep integration methods in the future, to provide a flexible and interpretable modelling paradigm for complex and heterogeneous data.