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Time Series Forecasting via Direct Per-Step Probability Distribution Modeling

Linghao Kong, Xiaopeng Hong

TL;DR

Time Series Forecasting via Direct Per-Step Probability Distribution Modeling introduces interPDN, a probabilistic TSF framework that forecasts per-step discrete distributions rather than scalars. It employs interleaved dual branches with non-uniform supports and a coarse-scale branch to enforce long-term consistency via self-supervised losses. The method achieves state-of-the-art results across nine real-world datasets and is supported by comprehensive ablations demonstrating the value of per-step distribution forecasting, interleaved supports, and cross-scale constraints. By avoiding fixed distributional assumptions and leveraging uncertainty-aware predictions, interPDN offers robust, scalable TSF suitable for real-world deployment.

Abstract

Deep neural network-based time series prediction models have recently demonstrated superior capabilities in capturing complex temporal dependencies. However, it is challenging for these models to account for uncertainty associated with their predictions, because they directly output scalar values at each time step. To address such a challenge, we propose a novel model named interleaved dual-branch Probability Distribution Network (interPDN), which directly constructs discrete probability distributions per step instead of a scalar. The regression output at each time step is derived by computing the expectation of the predictive distribution on a predefined support set. To mitigate prediction anomalies, a dual-branch architecture is introduced with interleaved support sets, augmented by coarse temporal-scale branches for long-term trend forecasting. Outputs from another branch are treated as auxiliary signals to impose self-supervised consistency constraints on the current branch's prediction. Extensive experiments on multiple real-world datasets demonstrate the superior performance of interPDN.

Time Series Forecasting via Direct Per-Step Probability Distribution Modeling

TL;DR

Time Series Forecasting via Direct Per-Step Probability Distribution Modeling introduces interPDN, a probabilistic TSF framework that forecasts per-step discrete distributions rather than scalars. It employs interleaved dual branches with non-uniform supports and a coarse-scale branch to enforce long-term consistency via self-supervised losses. The method achieves state-of-the-art results across nine real-world datasets and is supported by comprehensive ablations demonstrating the value of per-step distribution forecasting, interleaved supports, and cross-scale constraints. By avoiding fixed distributional assumptions and leveraging uncertainty-aware predictions, interPDN offers robust, scalable TSF suitable for real-world deployment.

Abstract

Deep neural network-based time series prediction models have recently demonstrated superior capabilities in capturing complex temporal dependencies. However, it is challenging for these models to account for uncertainty associated with their predictions, because they directly output scalar values at each time step. To address such a challenge, we propose a novel model named interleaved dual-branch Probability Distribution Network (interPDN), which directly constructs discrete probability distributions per step instead of a scalar. The regression output at each time step is derived by computing the expectation of the predictive distribution on a predefined support set. To mitigate prediction anomalies, a dual-branch architecture is introduced with interleaved support sets, augmented by coarse temporal-scale branches for long-term trend forecasting. Outputs from another branch are treated as auxiliary signals to impose self-supervised consistency constraints on the current branch's prediction. Extensive experiments on multiple real-world datasets demonstrate the superior performance of interPDN.

Paper Structure

This paper contains 25 sections, 20 equations, 8 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Backbone of Each Branch
  5. Probabilistic Generation Module
  6. Interleaved Dual Branches
  7. Bi-scale Temporal Branches
  8. Bi-scale Temporal Branches
  9. Loss Function
  10. Experiments
  11. Datasets and Baselines
  12. Datasets and Baselines
  13. Main Results
  14. Ablation Study
  15. Conclusion
  16. ...and 10 more sections

Figures (8)

  • Figure 1: Different ways of per-step time series modeling: (a) Scalar estimation; (b) Probability distribution parameter prediction; (c) Expectation forecasting via discrete probability distribution
  • Figure 2: (a) Overall model architecture for direct per-step probability distribution forecasting ; (b) Backbone structure of each branch; (c) Dual-branch design corresponding to interleaved support sets.
  • Figure 3: MSE versus $\alpha$ and $\beta$ at four forecast horizons on ETTh1 dataset
  • Figure 4: MSE versus $\gamma$ at four forecast horizons on ETTh1
  • Figure 5: Probability distributions predicted by Branch 1 and Branch 2 on the interleaved support sets at a specific time step. An orange dashed box magnifies the interval [0.710, 0.745], showing the expectations predicted by dual branches and the error of the combined prediction relative to ground truth.
  • ...and 3 more figures