# TACTiS-2: Better, Faster, Simpler Attentional Copulas for Multivariate Time Series

arXiv: https://arxiv.org/abs/2310.01327 · HTML: https://mlsanity.com/paper/2310.01327/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 73 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 30 of 39)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Monash Time Series Forecasting Archive](https://mlsanity.com/paper/2105.06643/) (2021)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [ETSformer: Exponential Smoothing Transformers for Time-series Forecasting](https://mlsanity.com/paper/2202.01381/) (2022)
- [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)
- [Wild-Time: A Benchmark of in-the-Wild Distribution Shift over Time](https://mlsanity.com/paper/2211.14238/) (2022)
- [Discrete Flows: Invertible Generative Models of Discrete Data](https://mlsanity.com/paper/1905.10347/) (2019)
- [Latent Normalizing Flows for Discrete Sequences](https://mlsanity.com/paper/1901.10548/) (2019)
- [Probabilistic Forecasting with Temporal Convolutional Neural Network](https://mlsanity.com/paper/1906.04397/) (2019)
- [A Survey on Principles, Models and Methods for Learning from Irregularly Sampled Time Series](https://mlsanity.com/paper/2012.00168/) (2020)
- [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)
- [Time Series Data Imputation: A Survey on Deep Learning Approaches](https://mlsanity.com/paper/2011.11347/) (2020)
- [WOODS: Benchmarks for Out-of-Distribution Generalization in Time Series](https://mlsanity.com/paper/2203.09978/) (2022)
- [Copulas as High-Dimensional Generative Models: Vine Copula Autoencoders](https://mlsanity.com/paper/1906.05423/) (2019)

## Built on by (most cited first, 26 of 26)

- [UniTS: A Unified Multi-Task Time Series Model](https://mlsanity.com/paper/2403.00131/) (2024)
- [SciTS: Scientific Time Series Understanding and Generation with LLMs](https://mlsanity.com/paper/2510.03255/) (2025)
- [From Samples to Scenarios: A New Paradigm for Probabilistic Forecasting](https://mlsanity.com/paper/2509.19975/) (2025)
- [Your copula is a classifier in disguise: classification-based copula density estimation](https://mlsanity.com/paper/2411.03014/) (2024)
- [Multivariate Probabilistic Time Series Forecasting with Correlated Errors](https://mlsanity.com/paper/2402.01000/) (2024)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [MVG-CRPS: A Robust Loss Function for Multivariate Probabilistic Forecasting](https://mlsanity.com/paper/2410.09133/) (2024)
- [Winner-takes-all for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2506.05515/) (2025)
- [RDIT: Residual-based Diffusion Implicit Models for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2509.02341/) (2025)
- [Efficiently Generating Correlated Sample Paths from Multi-step Time Series Foundation Models](https://mlsanity.com/paper/2510.02224/) (2025)
- [Neural Mutual Information Estimation with Vector Copulas](https://mlsanity.com/paper/2510.20968/) (2025)
- [Strada-LLM: Graph LLM for traffic prediction](https://mlsanity.com/paper/2410.20856/) (2024)
- [TimePre: Bridging Accuracy, Efficiency, and Stability in Probabilistic Time-Series Forecasting](https://mlsanity.com/paper/2511.18539/) (2025)
- [CAWI: Copula-Aligned Weight Initialization for Randomized Neural Networks](https://mlsanity.com/paper/2605.12580/) (2026)
- [Masked Language Modeling Becomes Conditional Density Estimation for Tabular Data Synthesis](https://mlsanity.com/paper/2405.20602/) (2024)
- [A Scalable Inter-edge Correlation Modeling in CopulaGNN for Link Sign Prediction](https://mlsanity.com/paper/2601.19175/) (2026)
- [Probabilistic Circuits for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2604.27814/) (2026)
- [A Copula-Guided Temporal Dependency Method for Multitemporal Hyperspectral Images Unmixing](https://mlsanity.com/paper/2509.11096/) (2025)
- [How Patterns Dictate Learnability in Sequential Data](https://mlsanity.com/paper/2510.10744/) (2025)
- [Foundation Model Forecasts: Form and Function](https://mlsanity.com/paper/2510.19345/) (2025)
- [Probabilistic Multivariate Time Series Forecasting with Diffusion Copulas](https://mlsanity.com/paper/2605.19685/) (2026)
- [Gaussian Mixture Copula Processes for Irregular Time Series](https://mlsanity.com/paper/2605.23632/) (2026)
- [Navigating the Safety-Fidelity Trade-off: Massive-Variate Time Series Forecasting for Power Systems via Probabilistic Scenarios](https://mlsanity.com/paper/2606.13338/) (2026)
- [TSCoNet: A Two-Stage Copula CNN-LSTM for Uncertainty-Aware Spatio-Temporal Forecasting](https://mlsanity.com/paper/2607.10410/) (2026)
- [SPACE: Sample-cloud Predictive Adaptive Conformal Ellipsoids for Multivariate Time-Series Forecasting](https://mlsanity.com/paper/2608.17333/) (2026)
- [STRIDE: State-Transition Representation via Increment Dynamics and Evolution](https://mlsanity.com/paper/2609.32367/) (2026)

## Abstract (arXiv)

We introduce a new model for multivariate probabilistic time series prediction, designed to flexibly address a range of tasks including forecasting, interpolation, and their combinations. Building on copula theory, we propose a simplified objective for the recently-introduced transformer-based attentional copulas (TACTiS), wherein the number of distributional parameters now scales linearly with the number of variables instead of factorially. The new objective requires the introduction of a training curriculum, which goes hand-in-hand with necessary changes to the original architecture. We show that the resulting model has significantly better training dynamics and achieves state-of-the-art performance across diverse real-world forecasting tasks, while maintaining the flexibility of prior work, such as seamless handling of unaligned and unevenly-sampled time series. Code is made available at https://github.com/ServiceNow/TACTiS.