# Robust Probabilistic Time Series Forecasting

arXiv: https://arxiv.org/abs/2202.11910 · HTML: https://mlsanity.com/paper/2202.11910/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 77 references that are in the map.
- First posted on arXiv 24 Feb 2022.

## Builds on (most cited first, 30 of 42)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Practical Black-Box Attacks against Machine Learning](https://mlsanity.com/paper/1602.02697/) (2016)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecasting](https://mlsanity.com/paper/1907.00235/) (2019)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Certified Robustness to Adversarial Examples with Differential Privacy](https://mlsanity.com/paper/1802.03471/) (2018)
- [Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers](https://mlsanity.com/paper/1906.04584/) (2019)
- [Certified Defenses against Adversarial Examples](https://mlsanity.com/paper/1801.09344/) (2018)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [Certified Adversarial Robustness with Additive Noise](https://mlsanity.com/paper/1809.03113/) (2018)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Recurrent Neural Networks for Time Series Forecasting: Current Status and Future Directions](https://mlsanity.com/paper/1909.00590/) (2019)
- [Think Globally, Act Locally: A Deep Neural Network Approach to High-Dimensional Time Series Forecasting](https://mlsanity.com/paper/1905.03806/) (2019)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process](https://mlsanity.com/paper/1904.09080/) (2019)
- [Generalization in Generation: A closer look at Exposure Bias](https://mlsanity.com/paper/1910.00292/) (2019)
- [Label Noise SGD Provably Prefers Flat Global Minimizers](https://mlsanity.com/paper/2106.06530/) (2021)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [Reward Augmented Maximum Likelihood for Neural Structured Prediction](https://mlsanity.com/paper/1609.00150/) (2016)

## Built on by (most cited first, 15 of 15)

- [RobustTSF: Towards Theory and Design of Robust Time Series Forecasting with Anomalies](https://mlsanity.com/paper/2402.02032/) (2024)
- [Robust Multivariate Time-Series Forecasting: Adversarial Attacks and Defense Mechanisms](https://mlsanity.com/paper/2207.09572/) (2022)
- [A Co-training Approach for Noisy Time Series Learning](https://mlsanity.com/paper/2308.12551/) (2023)
- [FTS: A Framework to Find a Faithful TimeSieve](https://mlsanity.com/paper/2405.19647/) (2024)
- [Denoising-Aware Contrastive Learning for Noisy Time Series](https://mlsanity.com/paper/2406.04627/) (2024)
- [Adaptive Sampling for Probabilistic Forecasting under Distribution Shift](https://mlsanity.com/paper/2302.11870/) (2023)
- [Non-parametric Probabilistic Time Series Forecasting via Innovations Representation](https://mlsanity.com/paper/2306.03782/) (2023)
- [Quantifying Robustness: A Benchmarking Framework for Deep Learning Forecasting in Cyber-Physical Systems](https://mlsanity.com/paper/2504.03494/) (2025)
- [PA-RNet: Perturbation-Aware Residual Network for Robust Multimodal Time Series Forecasting](https://mlsanity.com/paper/2508.04750/) (2025)
- [Benchmarking M-LTSF: Frequency and Noise-Based Evaluation of Multivariate Long Time Series Forecasting Models](https://mlsanity.com/paper/2510.04900/) (2025)
- [SEER: Transformer-based Robust Time Series Forecasting via Automated Patch Enhancement and Replacement](https://mlsanity.com/paper/2602.00589/) (2026)
- [Adversarial Robustness of Deep State Space Models for Forecasting](https://mlsanity.com/paper/2604.03427/) (2026)
- [Exposure Bias as Epistemic Underidentification in Recursive Forecasting](https://mlsanity.com/paper/2606.12990/) (2026)
- [Contrastive Time Series Forecasting with Anomalies](https://mlsanity.com/paper/2512.11526/) (2025)
- [Towards Reliable Zero-Shot Crowd Forecasting: Evaluating Time Series Foundation Models for Special Event Pedestrian Forecasting](https://mlsanity.com/paper/2607.17758/) (2026)

## Abstract (arXiv)

Probabilistic time series forecasting has played critical role in decision-making processes due to its capability to quantify uncertainties. Deep forecasting models, however, could be prone to input perturbations, and the notion of such perturbations, together with that of robustness, has not even been completely established in the regime of probabilistic forecasting. In this work, we propose a framework for robust probabilistic time series forecasting. First, we generalize the concept of adversarial input perturbations, based on which we formulate the concept of robustness in terms of bounded Wasserstein deviation. Then we extend the randomized smoothing technique to attain robust probabilistic forecasters with theoretical robustness certificates against certain classes of adversarial perturbations. Lastly, extensive experiments demonstrate that our methods are empirically effective in enhancing the forecast quality under additive adversarial attacks and forecast consistency under supplement of noisy observations.