# Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flows

arXiv: https://arxiv.org/abs/2002.06103 · HTML: https://mlsanity.com/paper/2002.06103/

## Key facts

- Cited by 144 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2020.

## Built on by (most cited first, 30 of 144)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion](https://mlsanity.com/paper/2407.01392/) (2024)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [Lag-Llama: Towards Foundation Models for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2310.08278/) (2023)
- [Predict, Refine, Synthesize: Self-Guiding Diffusion Models for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2307.11494/) (2023)
- [Diffusion Models for Time Series Applications: A Survey](https://mlsanity.com/paper/2305.00624/) (2023)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [ScoreGrad: Multivariate Probabilistic Time Series Forecasting with Continuous Energy-based Generative Models](https://mlsanity.com/paper/2106.10121/) (2021)
- [Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2410.03024/) (2024)
- [Graph-Augmented Normalizing Flows for Anomaly Detection of Multiple Time Series](https://mlsanity.com/paper/2202.07857/) (2022)
- [Modeling Temporal Data as Continuous Functions with Stochastic Process Diffusion](https://mlsanity.com/paper/2211.02590/) (2022)
- [TACTiS: Transformer-Attentional Copulas for Time Series](https://mlsanity.com/paper/2202.03528/) (2022)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [ProbTS: Benchmarking Point and Distributional Forecasting across Diverse Prediction Horizons](https://mlsanity.com/paper/2310.07446/) (2023)
- [Temporal Latent Auto-Encoder: A Method for Probabilistic Multivariate Time Series Forecasting](https://mlsanity.com/paper/2101.10460/) (2021)
- [MG-TSD: Multi-Granularity Time Series Diffusion Models with Guided Learning Process](https://mlsanity.com/paper/2403.05751/) (2024)
- [Attacks and Defenses for Generative Diffusion Models: A Comprehensive Survey](https://mlsanity.com/paper/2408.03400/) (2024)
- [Aurora: Towards Universal Generative Multimodal Time Series Forecasting](https://mlsanity.com/paper/2509.22295/) (2025)
- [TACTiS-2: Better, Faster, Simpler Attentional Copulas for Multivariate Time Series](https://mlsanity.com/paper/2310.01327/) (2023)
- [Pushing the Limits of Pre-training for Time Series Forecasting in the CloudOps Domain](https://mlsanity.com/paper/2310.05063/) (2023)
- [DiffMotion: Speech-Driven Gesture Synthesis Using Denoising Diffusion Model](https://mlsanity.com/paper/2301.10047/) (2023)
- [Detecting Multivariate Time Series Anomalies with Zero Known Label](https://mlsanity.com/paper/2208.02108/) (2022)
- [Multivariate Time Series Forecasting with Latent Graph Inference](https://mlsanity.com/paper/2203.03423/) (2022)
- [Graph Mixture of Experts and Memory-augmented Routers for Multivariate Time Series Anomaly Detection](https://mlsanity.com/paper/2412.19108/) (2024)
- [Multivariate Quantile Function Forecaster](https://mlsanity.com/paper/2202.11316/) (2022)
- [Multi-scale Attention Flow for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2205.07493/) (2022)
- [Adjusting for Autocorrelated Errors in Neural Networks for Time Series](https://mlsanity.com/paper/2101.12578/) (2021)
- [SSD-TS: Exploring the Potential of Linear State Space Models for Diffusion Models in Time Series Imputation](https://mlsanity.com/paper/2410.13338/) (2024)

## Abstract (arXiv)

Time series forecasting is often fundamental to scientific and engineering problems and enables decision making. With ever increasing data set sizes, a trivial solution to scale up predictions is to assume independence between interacting time series. However, modeling statistical dependencies can improve accuracy and enable analysis of interaction effects. Deep learning methods are well suited for this problem, but multivariate models often assume a simple parametric distribution and do not scale to high dimensions. In this work we model the multivariate temporal dynamics of time series via an autoregressive deep learning model, where the data distribution is represented by a conditioned normalizing flow. This combination retains the power of autoregressive models, such as good performance in extrapolation into the future, with the flexibility of flows as a general purpose high-dimensional distribution model, while remaining computationally tractable. We show that it improves over the state-of-the-art for standard metrics on many real-world data sets with several thousand interacting time-series.