# Probabilistic Time Series Forecasts with Autoregressive Transformation Models

arXiv: https://arxiv.org/abs/2110.08248 · HTML: https://mlsanity.com/paper/2110.08248/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 69 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Accurate Uncertainty Estimation and Decomposition in Ensemble Learning](https://mlsanity.com/paper/1911.04061/) (2019)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [Gaussian Process Conditional Density Estimation](https://mlsanity.com/paper/1810.12750/) (2018)
- [Deep Conditional Transformation Models](https://mlsanity.com/paper/2010.07860/) (2020)
- [Most Likely Transformations](https://mlsanity.com/paper/1508.06749/) (2015)
- [Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain](https://mlsanity.com/paper/1804.02921/) (2018)
- [Deep transformation models: Tackling complex regression problems with neural network based transformation models](https://mlsanity.com/paper/2004.00464/) (2020)
- [deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression](https://mlsanity.com/paper/2104.02705/) (2021)
- [Multivariate Conditional Transformation Models](https://mlsanity.com/paper/1906.03151/) (2019)
- [Adaptive Explainable Neural Networks (AxNNs)](https://mlsanity.com/paper/2004.02353/) (2020)
- [Stationary vine copula models for multivariate time series](https://mlsanity.com/paper/2008.05990/) (2020)
- [Neural Mixture Distributional Regression](https://mlsanity.com/paper/2010.06889/) (2020)
- [Implicit Copulas: An Overview](https://mlsanity.com/paper/2109.04718/) (2021)
- [Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models](https://mlsanity.com/paper/1602.08154/) (2016)

## Built on by (most cited first, 7 of 7)

- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Distributional Gradient Boosting Machines](https://mlsanity.com/paper/2204.00778/) (2022)
- [Deep interpretable ensembles](https://mlsanity.com/paper/2205.12729/) (2022)
- [Bernstein Flows for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2202.05650/) (2022)
- [How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression](https://mlsanity.com/paper/2405.05429/) (2024)
- [Why Did This Model Forecast This Future? Closed-Form Temporal Saliency Towards Causal Explanations of Probabilistic Forecasts](https://mlsanity.com/paper/2206.00679/) (2022)
- [Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals](https://mlsanity.com/paper/2505.14164/) (2025)

## Abstract (arXiv)

Probabilistic forecasting of time series is an important matter in many applications and research fields. In order to draw conclusions from a probabilistic forecast, we must ensure that the model class used to approximate the true forecasting distribution is expressive enough. Yet, characteristics of the model itself, such as its uncertainty or its feature-outcome relationship are not of lesser importance. This paper proposes Autoregressive Transformation Models (ATMs), a model class inspired by various research directions to unite expressive distributional forecasts using a semi-parametric distribution assumption with an interpretable model specification. We demonstrate the properties of ATMs both theoretically and through empirical evaluation on several simulated and real-world forecasting datasets.