# Learning Mixture Structure on Multi-Source Time Series for Probabilistic Forecasting

arXiv: https://arxiv.org/abs/2302.11078 · HTML: https://mlsanity.com/paper/2302.11078/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 53 references that are in the map.
- First posted on arXiv 22 Feb 2023.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecasting](https://mlsanity.com/paper/1907.00235/) (2019)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Learning Factored Representations in a Deep Mixture of Experts](https://mlsanity.com/paper/1312.4314/) (2013)
- [Unsupervised Scalable Representation Learning for Multivariate Time Series](https://mlsanity.com/paper/1901.10738/) (2019)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Variational Mixture-of-Experts Autoencoders for Multi-Modal Deep Generative Models](https://mlsanity.com/paper/1911.03393/) (2019)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows](https://mlsanity.com/paper/2002.06103/) (2020)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Disentangled Sequential Autoencoder](https://mlsanity.com/paper/1803.02991/) (2018)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Sequential Neural Models with Stochastic Layers](https://mlsanity.com/paper/1605.07571/) (2016)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [Mixture Models for Diverse Machine Translation: Tricks of the Trade](https://mlsanity.com/paper/1902.07816/) (2019)
- [Probabilistic Recurrent State-Space Models](https://mlsanity.com/paper/1801.10395/) (2018)
- [Learning Stochastic Recurrent Networks](https://mlsanity.com/paper/1411.7610/) (2014)
- [Deep Learning for Time-Series Analysis](https://mlsanity.com/paper/1701.01887/) (2017)
- [Z-Forcing: Training Stochastic Recurrent Networks](https://mlsanity.com/paper/1711.05411/) (2017)
- [Exploring Interpretable LSTM Neural Networks over Multi-Variable Data](https://mlsanity.com/paper/1905.12034/) (2019)
- [Mixture Dense Regression for Object Detection and Human Pose Estimation](https://mlsanity.com/paper/1912.00821/) (2019)

## Built on by (most cited first, 1 of 1)

- [Exploring the Synergy of Quantitative Factors and Newsflow Representations from Large Language Models for Stock Return Prediction](https://mlsanity.com/paper/2510.15691/) (2025)

## Abstract (arXiv)

In many data-driven applications, collecting data from different sources is increasingly desirable for enhancing performance. In this paper, we are interested in the problem of probabilistic forecasting with multi-source time series. We propose a neural mixture structure-based probability model for learning different predictive relations and their adaptive combinations from multi-source time series. We present the prediction and uncertainty quantification methods that apply to different distributions of target variables. Additionally, given the imbalanced and unstable behaviors observed during the direct training of the proposed mixture model, we develop a phased learning method and provide a theoretical analysis. In experimental evaluations, the mixture model trained by the phased learning exhibits competitive performance on both point and probabilistic prediction metrics. Meanwhile, the proposed uncertainty conditioned error suggests the potential of the mixture model's uncertainty score as a reliability indicator of predictions.