# pTSE: A Multi-model Ensemble Method for Probabilistic Time Series Forecasting

arXiv: https://arxiv.org/abs/2305.11304 · HTML: https://mlsanity.com/paper/2305.11304/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 55 references that are in the map.
- First posted on arXiv 16 May 2023.

## Builds on (most cited first, 10 of 10)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Temporal Pattern Attention for Multivariate Time Series Forecasting](https://mlsanity.com/paper/1809.04206/) (2018)
- [Diverse Trajectory Forecasting with Determinantal Point Processes](https://mlsanity.com/paper/1907.04967/) (2019)
- [Criteria for Classifying Forecasting Methods](https://mlsanity.com/paper/2212.03523/) (2022)
- [Ensembles of Localised Models for Time Series Forecasting](https://mlsanity.com/paper/2012.15059/) (2020)
- [If You Like It, GAN It. Probabilistic Multivariate Times Series Forecast With GAN](https://mlsanity.com/paper/2005.01181/) (2020)
- [Continual Causal Inference with Incremental Observational Data](https://mlsanity.com/paper/2303.01775/) (2023)

## Built on by (most cited first, 16 of 16)

- [Time-LLM: Time Series Forecasting by Reprogramming Large Language Models](https://mlsanity.com/paper/2310.01728/) (2023)
- [Continual Learning in Predictive Autoscaling](https://mlsanity.com/paper/2307.15941/) (2023)
- [Self-supervised Adaptive Pre-training of Multilingual Speech Models for Language and Dialect Identification](https://mlsanity.com/paper/2312.07338/) (2023)
- [Small but Mighty: Enhancing Time Series Forecasting with Lightweight LLMs](https://mlsanity.com/paper/2503.03594/) (2025)
- [Non-collective Calibrating Strategy for Time Series Forecasting](https://mlsanity.com/paper/2506.03176/) (2025)
- [Enhancing Asynchronous Time Series Forecasting with Contrastive Relational Inference](https://mlsanity.com/paper/2309.02868/) (2023)
- [Enhancing Multilingual Speech Recognition through Language Prompt Tuning and Frame-Level Language Adapter](https://mlsanity.com/paper/2309.09443/) (2023)
- [GMTR: Graph Matching Transformers](https://mlsanity.com/paper/2311.08141/) (2023)
- [DarkShot: Lighting Dark Images with Low-Compute and High-Quality](https://mlsanity.com/paper/2312.16805/) (2023)
- [TRLS: A Time Series Representation Learning Framework via Spectrogram for Medical Signal Processing](https://mlsanity.com/paper/2401.05431/) (2024)
- [TEN-GUARD: Tensor Decomposition for Backdoor Attack Detection in Deep Neural Networks](https://mlsanity.com/paper/2401.05432/) (2024)
- [T-Pixel2Mesh: Combining Global and Local Transformer for 3D Mesh Generation from a Single Image](https://mlsanity.com/paper/2403.13663/) (2024)
- [Strategic Arms with Side Communication Prevail Over Low-Regret MAB Algorithms](https://mlsanity.com/paper/2408.17101/) (2024)
- [Towards General Deepfake Detection with Dynamic Curriculum](https://mlsanity.com/paper/2410.11162/) (2024)
- [Attr-Int: A Simple and Effective Entity Alignment Framework for Heterogeneous Knowledge Graphs](https://mlsanity.com/paper/2410.13409/) (2024)
- [Mask6D: Masked Pose Priors For 6D Object Pose Estimation](https://mlsanity.com/paper/2507.06486/) (2025)

## Abstract (arXiv)

Various probabilistic time series forecasting models have sprung up and shown remarkably good performance. However, the choice of model highly relies on the characteristics of the input time series and the fixed distribution that the model is based on. Due to the fact that the probability distributions cannot be averaged over different models straightforwardly, the current time series model ensemble methods cannot be directly applied to improve the robustness and accuracy of forecasting. To address this issue, we propose pTSE, a multi-model distribution ensemble method for probabilistic forecasting based on Hidden Markov Model (HMM). pTSE only takes off-the-shelf outputs from member models without requiring further information about each model. Besides, we provide a complete theoretical analysis of pTSE to prove that the empirical distribution of time series subject to an HMM will converge to the stationary distribution almost surely. Experiments on benchmarks show the superiority of pTSE overall member models and competitive ensemble methods.