Probabilistic Functional Neural Networks

TL;DR

ProFnet introduces a probabilistic functional neural network to forecast high-dimensional functional time series by jointly encoding functional inputs and region information into latent representations and modeling temporal dynamics with a set of Gaussian processes conditioned on these representations. It yields probabilistic forecasts through Monte Carlo sampling and provides predictive intervals via multiple samples, while being lag-free and scalable to many regions and horizons. The Japan mortality forecasting application demonstrates superior MSFE performance and high-coverage prediction intervals, along with interpretable directional regional associations derived from interval coverage. Overall, ProFnet advances functional data analysis by uniting neural encoding with probabilistic time-series modeling to handle complex, large-scale HDFTS.

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.

Figures (6)

• Figure 1: An example of high-dimensional functional time series. The data is a set of mortality rate curves for 47 prefectures in Japan from year 1973 to 2022. The color represents the year of the mortality rate curve, from blue (oldest) to red (most current). We would simultaneously observe a functional time series within each prefecture, and the collection of all prefectures forms a high-dimensional functional time series. The plot showcases the functional time series for 2 randomly chosen prefectures.
• Figure 2: Functional Probabilistic Neural Network. Arrows indicate the forward pass of the network. The reparametrization trick is done by adding pre-sampled random noises in $\mathbf{\epsilon}$.
• Figure 3: Smoothed $\log_{10}$ mortality rate curves, from year 1973 to 2022, for six randomly chosen prefectures (with its name in the plot) in Japan. The color represents the year of the mortality rate curve, from blue (oldest) to red (most current).
• Figure 4: Forecast results for a target region, Miyazaki prefecture. The interval estimates are obtained by using regional data from Kumamoto or Oita. The true future curve is the solid black line, whereas the blue dashed line is the mean estimate from Kumamoto and the red dashed line is that from Oita. The shaded area represents the 95% forecast interval.
• Figure 5: Each dot represents a prefecture in Japan, and the prefectures are ordered from left to right based on how north they are. For connections above the dots, a link is derived if the distance between a pair of prefectures is less than 1000 km. For connections below the dots, a link is derived if the coverage probability of the forecasted curve of a prefecture is higher than 90% for all other prefectures. The connections derived from coverage probabilities are directional whereas the top connections are not.