Spatial Functional Deep Neural Network Model: A New Prediction Algorithm

Abstract

Accurate prediction of spatially dependent functional data is critical for various engineering and scientific applications. In this study, a spatial functional deep neural network model was developed with a novel non-linear modeling framework that seamlessly integrates spatial dependencies and functional predictors using deep learning techniques. The proposed model extends classical scalar-on-function regression by incorporating a spatial autoregressive component while leveraging functional deep neural networks to capture complex non-linear relationships. To ensure a robust estimation, the methodology employs an adaptive estimation approach, where the spatial dependence parameter was first inferred via maximum likelihood estimation, followed by non-linear functional regression using deep learning. The effectiveness of the proposed model was evaluated through extensive Monte Carlo simulations and an application to Brazilian COVID-19 data, where the goal was to predict the average daily number of deaths. Comparative analysis with maximum likelihood-based spatial functional linear regression and functional deep neural network models demonstrates that the proposed algorithm significantly improves predictive performance. The results for the Brazilian COVID-19 data showed that while all models achieved similar mean squared error values over the training modeling phase, the proposed model achieved the lowest mean squared prediction error in the testing phase, indicating superior generalization ability.