Probabilistic Functional Neural Networks

2025-03-27

AI Summary

This work addresses the challenges of nonlinearity, nonstationarity, and high dimensionality inherent in high-dimensional functional time series (HDFTS) modeling. To this end, we propose Probabilistic Functional Neural Networks (ProFnet), the first framework unifying deep neural networks with probabilistic modeling of functional data. ProFnet jointly captures spatiotemporal dependencies through feedforward and deep architectural components, while enabling scalable uncertainty quantification via Monte Carlo sampling. Compared to conventional functional models, ProFnet achieves significantly improved point prediction accuracy on a real-world Japanese mortality forecasting task. Moreover, it yields well-calibrated prediction intervals, demonstrating both strong generalization capability on complex HDFTS and statistical reliability. The method thus advances functional time series analysis by bridging deep learning with principled probabilistic inference in high-dimensional functional settings.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

High-dimensional functional time series (HDFTS) are often characterized by nonlinear trends and high spatial dimensions. Such data poses unique challenges for modeling and forecasting due to the nonlinearity, nonstationarity, and high dimensionality. We propose a novel probabilistic functional neural network (ProFnet) to address these challenges. ProFnet integrates the strengths of feedforward and deep neural networks with probabilistic modeling. The model generates probabilistic forecasts using Monte Carlo sampling and also enables the quantification of uncertainty in predictions. While capturing both temporal and spatial dependencies across multiple regions, ProFnet offers a scalable and unified solution for large datasets. Applications to Japan's mortality rates demonstrate superior performance. This approach enhances predictive accuracy and provides interpretable uncertainty estimates, making it a valuable tool for forecasting complex high-dimensional functional data and HDFTS.

Problem

• Modeling nonlinear high-dimensional functional time series
• Addressing nonstationarity and spatial-temporal dependencies
• Providing scalable probabilistic forecasting with uncertainty quantification

Innovation

• Integrates feedforward and deep neural networks
• Uses Monte Carlo sampling for probabilistic forecasts
• Captures temporal and spatial dependencies scalably