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Deep Functional Factor Models: Forecasting High-Dimensional Functional Time Series via Bayesian Nonparametric Factorization

ICML 2024 · Time series › Causal and generative time series · approach: Probabilistic and conformal forecasting

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Proposes DF²M, a Bayesian nonparametric factor model (IBP loadings) with a multi-task GP whose temporal kernel is built from deep sequence networks (LIN/LSTM/GRU/attention), trained by sparse variational inference, for forecasting high-dimensional functional time series.

Problem

Forecasting high-dimensional functional time series with nonlinear, non-Markovian temporal dependence while keeping the model explainable, where standard deep learning is a black box and linear VAR-type functional models are Markovian and linear.

Approach

Combine an IBP-sparse functional factor model with a multi-task GP whose temporal kernel is a deep kernel over the full history of latent factors (via LSTM/GRU/attention), and infer it with sparse variational inference using inducing points shared across factors.

Key results

• DF2M-LSTM reduces MSPE vs LSTM on Japanese Mortality at h=1 (3.753 vs 4.989) and on Global Mortality (7.672 vs 13.28).
• DF2M-ATTN achieves MAPE 1.119 vs ATTN 3.166 on Japanese Mortality at h=1, showing large gains over the attention baseline.
• On Stock Intraday, DF2M-ATTN and ATTN perform similarly, and DF2M-LIN beats DF2M-LSTM and DF2M-GRU, the only exceptions to DF2M's consistent improvements.

Measured effects reported in the paper

Surprising findings

• On Stock Intraday, the Markovian linear variant DF2M-LIN outperforms the LSTM and GRU variants, which the authors attribute to weak long-term dependence in financial data.
• DF2M-ATTN and ATTN perform similarly on Stock Intraday, the only dataset where DF2M does not consistently win.

Limitations

• Relies on simple spatial kernels, neglecting intricate relationships in the observation space (stated by authors).
• Evaluated only on short series (n under 50 time steps), limiting evidence for longer horizons.
• Baselines are standard deep models with linear output heads; comparison to classical functional time series methods is only described in related work, not experiments.

Experimental setup

Datasets: Japanese Mortality Energy Consumption (London smart meter) Global Mortality (Human Mortality Database) Stock Intraday (WRDS S&P 100)

Benchmarks: Japanese Mortality Energy Consumption Global Mortality Stock Intraday

Metrics: MSPE MAPE

Baselines: LIN LSTM GRU ATTN

Builds on: Indian buffet process multi-task Gaussian process Gaussian process dynamical model deep kernel learning automatic differentiation variational inference

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