# Gaussian Mixture Copula Processes for Irregular Time Series

arXiv: https://arxiv.org/abs/2605.23632 · HTML: https://mlsanity.com/paper/2605.23632/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 70 references that are in the map.
- First posted on arXiv 22 May 2026.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Time2Vec: Learning a Vector Representation of Time](https://mlsanity.com/paper/1907.05321/) (2019)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [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)
- [Transformer Neural Processes: Uncertainty-Aware Meta Learning Via Sequence Modeling](https://mlsanity.com/paper/2207.04179/) (2022)
- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Meta-Learning Stationary Stochastic Process Prediction with Convolutional Neural Processes](https://mlsanity.com/paper/2007.01332/) (2020)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [TACTiS: Transformer-Attentional Copulas for Time Series](https://mlsanity.com/paper/2202.03528/) (2022)
- [Copula Flows for Synthetic Data Generation](https://mlsanity.com/paper/2101.00598/) (2021)
- [The Gaussian Neural Process](https://mlsanity.com/paper/2101.03606/) (2021)
- [Autoregressive Conditional Neural Processes](https://mlsanity.com/paper/2303.14468/) (2023)
- [TACTiS-2: Better, Faster, Simpler Attentional Copulas for Multivariate Time Series](https://mlsanity.com/paper/2310.01327/) (2023)
- [Practical Conditional Neural Processes Via Tractable Dependent Predictions](https://mlsanity.com/paper/2203.08775/) (2022)
- [Deep Archimedean Copulas](https://mlsanity.com/paper/2012.03137/) (2020)
- [Deep Generative Quantile-Copula Models for Probabilistic Forecasting](https://mlsanity.com/paper/1907.10697/) (2019)
- [HyperIMTS: Hypergraph Neural Network for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.17431/) (2025)
- [Effortless, Simulation-Efficient Bayesian Inference using Tabular Foundation Models](https://mlsanity.com/paper/2504.17660/) (2025)
- [Physiome-ODE: A Benchmark for Irregularly Sampled Multivariate Time Series Forecasting Based on Biological ODEs](https://mlsanity.com/paper/2502.07489/) (2025)

## Abstract (arXiv)

We introduce Gaussian Mixture Copula Processes (GMCP), a conditional copula process for irregularly sampled multivariate time series (IMTS) that is expressive and marginalization consistent at the same time. Existing conditional copula processes achieve only one of the two. Gaussian copula processes are consistent by construction but confined to elliptical dependence, whereas attentional copulas such as TACTiS-2 are far more expressive but neither preserve the marginals they are built on nor guarantee that integrating out a target point returns the joint over the remaining ones. GMCP closes this gap by inferring the parameters of a Gaussian Mixture Copula from the context in a way that guarantees consistent marginalization. This yields a non-elliptical copula process with tractable likelihood and consistency guarantees. Our experiments show that copulas equipped with MargFlow, a normalizing-flow model for the univariate marginals fitted in isolation, outperform jointly trained baselines in probabilistic IMTS forecasting. We further provide evidence that GMCP is more expressive than Gaussian copula processes, and on par with the inconsistent TACTiS-2.